From 3fd6a57c6e1cee7a91028584f39f25cc4e12a6d5 Mon Sep 17 00:00:00 2001
From: mhjensen <morten.hjorth-jensen@fys.uio.no>
Date: Fri, 30 Aug 2019 05:47:54 +0200
Subject: [PATCH] Fixed type and shifted Ising model to end for Regression
 slides

---
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 doc/pub/Regression/html/Regression.html       | 786 ++++++++-------
 doc/pub/Regression/ipynb/Regression.ipynb     | 906 +++++++++---------
 .../ipynb/ipynb-Regression-src.tar.gz         | Bin 211 -> 211 bytes
 doc/pub/Regression/pdf/Regression-minted.pdf  | Bin 451746 -> 451753 bytes
 doc/src/Regression/Regression.do.txt          | 432 ++++-----
 112 files changed, 18180 insertions(+), 18192 deletions(-)

diff --git a/doc/pub/Regression/html/._Regression-bs000.html b/doc/pub/Regression/html/._Regression-bs000.html
index 786ff5b64..b5293106b 100644
--- a/doc/pub/Regression/html/._Regression-bs000.html
+++ b/doc/pub/Regression/html/._Regression-bs000.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -402,7 +402,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>
-<center><h4>Aug 29, 2019</h4></center> <!-- date -->
+<center><h4>Aug 30, 2019</h4></center> <!-- date -->
 <br>
 <p>
 
diff --git a/doc/pub/Regression/html/._Regression-bs001.html b/doc/pub/Regression/html/._Regression-bs001.html
index d905d84aa..9708d32ef 100644
--- a/doc/pub/Regression/html/._Regression-bs001.html
+++ b/doc/pub/Regression/html/._Regression-bs001.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs002.html b/doc/pub/Regression/html/._Regression-bs002.html
index 720625ee3..8f02d82f0 100644
--- a/doc/pub/Regression/html/._Regression-bs002.html
+++ b/doc/pub/Regression/html/._Regression-bs002.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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               ('Housing data, the code', 2, None, '___sec28'),
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-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
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-              ('Mathematical Properties', 2, None, '___sec40'),
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-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
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-              ('Statistics', 2, None, '___sec50'),
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-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
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-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
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+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
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-              ('Statistics, more technicalities', 2, None, '___sec66'),
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-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
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+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs003.html b/doc/pub/Regression/html/._Regression-bs003.html
index 761ee7817..9cac48f39 100644
--- a/doc/pub/Regression/html/._Regression-bs003.html
+++ b/doc/pub/Regression/html/._Regression-bs003.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs004.html b/doc/pub/Regression/html/._Regression-bs004.html
index a8191158c..c6d617952 100644
--- a/doc/pub/Regression/html/._Regression-bs004.html
+++ b/doc/pub/Regression/html/._Regression-bs004.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
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+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
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-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
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               ('Resampling methods: Jackknife and Bootstrap',
                2,
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-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs005.html b/doc/pub/Regression/html/._Regression-bs005.html
index e77c3c692..7f18e57d8 100644
--- a/doc/pub/Regression/html/._Regression-bs005.html
+++ b/doc/pub/Regression/html/._Regression-bs005.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs006.html b/doc/pub/Regression/html/._Regression-bs006.html
index 90eb666d9..69a2cfe84 100644
--- a/doc/pub/Regression/html/._Regression-bs006.html
+++ b/doc/pub/Regression/html/._Regression-bs006.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
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+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
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-              ('Statistics and sample variables', 2, None, '___sec60'),
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+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
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-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs007.html b/doc/pub/Regression/html/._Regression-bs007.html
index 5634ba805..19f0763a0 100644
--- a/doc/pub/Regression/html/._Regression-bs007.html
+++ b/doc/pub/Regression/html/._Regression-bs007.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
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-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
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-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs008.html b/doc/pub/Regression/html/._Regression-bs008.html
index 2274ff3e2..5ab9ebacf 100644
--- a/doc/pub/Regression/html/._Regression-bs008.html
+++ b/doc/pub/Regression/html/._Regression-bs008.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs009.html b/doc/pub/Regression/html/._Regression-bs009.html
index fe8fa5429..d12c98bd8 100644
--- a/doc/pub/Regression/html/._Regression-bs009.html
+++ b/doc/pub/Regression/html/._Regression-bs009.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs010.html b/doc/pub/Regression/html/._Regression-bs010.html
index bf9ce1fed..3f8a9bb59 100644
--- a/doc/pub/Regression/html/._Regression-bs010.html
+++ b/doc/pub/Regression/html/._Regression-bs010.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs011.html b/doc/pub/Regression/html/._Regression-bs011.html
index 4fd62faf4..9dcc70156 100644
--- a/doc/pub/Regression/html/._Regression-bs011.html
+++ b/doc/pub/Regression/html/._Regression-bs011.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
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-              ('Linear Regression Problems', 2, None, '___sec34'),
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-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
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-              ('Mathematical Properties', 2, None, '___sec40'),
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-              ('More on Ridge Regression', 2, None, '___sec42'),
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-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
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-              ('Statistics', 2, None, '___sec50'),
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-              ('Statistics, central moments', 2, None, '___sec52'),
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-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
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-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
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-              ('Statistics, more technicalities', 2, None, '___sec66'),
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-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
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+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs012.html b/doc/pub/Regression/html/._Regression-bs012.html
index 775438c9b..5dde47771 100644
--- a/doc/pub/Regression/html/._Regression-bs012.html
+++ b/doc/pub/Regression/html/._Regression-bs012.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs013.html b/doc/pub/Regression/html/._Regression-bs013.html
index 45de12e02..e8d0a6a5b 100644
--- a/doc/pub/Regression/html/._Regression-bs013.html
+++ b/doc/pub/Regression/html/._Regression-bs013.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs014.html b/doc/pub/Regression/html/._Regression-bs014.html
index 15e5125f6..367ccd0e0 100644
--- a/doc/pub/Regression/html/._Regression-bs014.html
+++ b/doc/pub/Regression/html/._Regression-bs014.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs015.html b/doc/pub/Regression/html/._Regression-bs015.html
index a519c661a..5ac0d6272 100644
--- a/doc/pub/Regression/html/._Regression-bs015.html
+++ b/doc/pub/Regression/html/._Regression-bs015.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
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+              ('More on Ridge Regression', 2, None, '___sec38'),
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+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
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+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs016.html b/doc/pub/Regression/html/._Regression-bs016.html
index e533de6e2..ae0a00af2 100644
--- a/doc/pub/Regression/html/._Regression-bs016.html
+++ b/doc/pub/Regression/html/._Regression-bs016.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
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-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs017.html b/doc/pub/Regression/html/._Regression-bs017.html
index 91bd675bb..f2ae569c8 100644
--- a/doc/pub/Regression/html/._Regression-bs017.html
+++ b/doc/pub/Regression/html/._Regression-bs017.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs018.html b/doc/pub/Regression/html/._Regression-bs018.html
index b5ddefa4e..866b3bf5f 100644
--- a/doc/pub/Regression/html/._Regression-bs018.html
+++ b/doc/pub/Regression/html/._Regression-bs018.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs019.html b/doc/pub/Regression/html/._Regression-bs019.html
index cd94396dc..59099e694 100644
--- a/doc/pub/Regression/html/._Regression-bs019.html
+++ b/doc/pub/Regression/html/._Regression-bs019.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs020.html b/doc/pub/Regression/html/._Regression-bs020.html
index ced9e0bb6..af8d65e55 100644
--- a/doc/pub/Regression/html/._Regression-bs020.html
+++ b/doc/pub/Regression/html/._Regression-bs020.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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               ('Housing data, the code', 2, None, '___sec28'),
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-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
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-              ('Mathematical Properties', 2, None, '___sec40'),
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-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
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-              ('Covariance example', 2, None, '___sec55'),
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-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
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-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs021.html b/doc/pub/Regression/html/._Regression-bs021.html
index 0b4be1765..46ebf3b80 100644
--- a/doc/pub/Regression/html/._Regression-bs021.html
+++ b/doc/pub/Regression/html/._Regression-bs021.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs022.html b/doc/pub/Regression/html/._Regression-bs022.html
index 33fd33d3a..0cc67fc3c 100644
--- a/doc/pub/Regression/html/._Regression-bs022.html
+++ b/doc/pub/Regression/html/._Regression-bs022.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs023.html b/doc/pub/Regression/html/._Regression-bs023.html
index f091d893a..ce7e28538 100644
--- a/doc/pub/Regression/html/._Regression-bs023.html
+++ b/doc/pub/Regression/html/._Regression-bs023.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs024.html b/doc/pub/Regression/html/._Regression-bs024.html
index 73311fad7..726f3a2b9 100644
--- a/doc/pub/Regression/html/._Regression-bs024.html
+++ b/doc/pub/Regression/html/._Regression-bs024.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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-              ('Resampling methods', 2, None, '___sec46'),
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+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
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-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
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+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
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+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
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-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs025.html b/doc/pub/Regression/html/._Regression-bs025.html
index 37f881698..5c4573a72 100644
--- a/doc/pub/Regression/html/._Regression-bs025.html
+++ b/doc/pub/Regression/html/._Regression-bs025.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs026.html b/doc/pub/Regression/html/._Regression-bs026.html
index 2c346d461..effe8cf1b 100644
--- a/doc/pub/Regression/html/._Regression-bs026.html
+++ b/doc/pub/Regression/html/._Regression-bs026.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs027.html b/doc/pub/Regression/html/._Regression-bs027.html
index 26c600e40..1b61aa69f 100644
--- a/doc/pub/Regression/html/._Regression-bs027.html
+++ b/doc/pub/Regression/html/._Regression-bs027.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs028.html b/doc/pub/Regression/html/._Regression-bs028.html
index b1a3a2b1a..e754349d2 100644
--- a/doc/pub/Regression/html/._Regression-bs028.html
+++ b/doc/pub/Regression/html/._Regression-bs028.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
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                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
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+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs029.html b/doc/pub/Regression/html/._Regression-bs029.html
index 938ec03ad..a470c974a 100644
--- a/doc/pub/Regression/html/._Regression-bs029.html
+++ b/doc/pub/Regression/html/._Regression-bs029.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
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-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
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-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
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-              ('Mathematical Properties', 2, None, '___sec40'),
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-              ('More on Ridge Regression', 2, None, '___sec42'),
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-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
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-              ('Statistics', 2, None, '___sec50'),
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-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
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-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
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-              ('Statistics, law of large numbers', 2, None, '___sec62'),
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-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
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+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
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               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs030.html b/doc/pub/Regression/html/._Regression-bs030.html
index 51f31975b..0d4a702bc 100644
--- a/doc/pub/Regression/html/._Regression-bs030.html
+++ b/doc/pub/Regression/html/._Regression-bs030.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -395,8 +395,7 @@ The examples we have looked at so far are cases where we normally can
 invert the matrix \( \boldsymbol{X}^T\boldsymbol{X} \). Using a polynomial expansion as we
 did both for the masses and the fitting of the equation of state,
 leads to row vectors of the design matrix which are essentially
-orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. 
-More material to come here.
+orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition.
 
 <p>
 This may
@@ -417,9 +416,6 @@ inversion algorithm. Thereafter we dive into the math of the SVD.
 </div>
 
 
-<p>
-<!-- todo: change model here. -->
-
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs031.html b/doc/pub/Regression/html/._Regression-bs031.html
index a2bf95958..0c1e209c9 100644
--- a/doc/pub/Regression/html/._Regression-bs031.html
+++ b/doc/pub/Regression/html/._Regression-bs031.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,59 +383,53 @@ MathJax.Hub.Config({
 <a name="part0031"></a>
 <!-- !split -->
 
-<h2 id="___sec30" class="anchor">The Ising model </h2>
+<h2 id="___sec30" class="anchor">Linear Regression Problems </h2>
 
 <p>
-The one-dimensional Ising model with nearest neighbor interaction, no
-external field and a constant coupling constant \( J \) is given by
-
+One of the typical problems we encounter with linear regression, in particular 
+when the matrix \( \boldsymbol{X} \) (our so-called design matrix) is high-dimensional, 
+are problems with near singular or singular matrices. The column vectors of \( \boldsymbol{X} \) 
+may be linearly dependent, normally referred to as super-collinearity.  
+This means that the matrix may be rank deficient and it is basically impossible to 
+to model the data using linear regression. As an example, consider the matrix
 $$
-\begin{align}
-    H = -J \sum_{k}^L s_k s_{k + 1},
-\tag{1}
-\end{align}
+\begin{align*}
+\mathbf{X} & =  \left[
+\begin{array}{rrr}
+1 & -1 & 2
+\\
+1 & 0 & 1
+\\
+1 & 2  & -1
+\\
+1 & 1  & 0
+\end{array} \right]
+\end{align*}
 $$
 
 <p>
-where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
-in the system is determined by \( L \). For the one-dimensional system
-there is no phase transition.
+The columns of \( \boldsymbol{X} \) are linearly dependent. We see this easily since the 
+the first column is the row-wise sum of the other two columns. The rank (more correct,
+the column rank) of a matrix is the dimension of the space spanned by the
+column vectors. Hence, the rank of \( \mathbf{X} \) is equal to the number
+of linearly independent columns. In this particular case the matrix has rank 2.
 
 <p>
-We will look at a system of \( L = 40 \) spins with a coupling constant of
-\( J = 1 \). To get enough training data we will generate 10000 states
-with their respective energies.
+Super-collinearity of an \( (n \times p) \)-dimensional design matrix \( \mathbf{X} \) implies
+that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
+$$
+\begin{align*}
+\boldsymbol{X} & =  \left[
+\begin{array}{rr}
+1 & -1
+\\
+1 & -1
+\end{array} \right].
+\end{align*}
+$$
 
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
-sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000">True</span>)
-cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&#39;seismic&#39;</span>)
-
-L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
-n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
-
-spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
-J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
-
-energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
-
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
-    energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
-</pre></div>
-<p>
-Here we use ordinary least squares
-regression to predict the energy for the nearest neighbor
-one-dimensional Ising model on a ring, i.e., the endpoints wrap
-around. We will use linear regression to fit a value for
-the coupling constant to achieve this.
+We see easily that  \( \mbox{det}(\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0 \). Hence, \( \mathbf{X} \) is singular and its inverse is undefined.
+This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least an eigenvalue which is zero.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs032.html b/doc/pub/Regression/html/._Regression-bs032.html
index 83c49de0e..3450418ad 100644
--- a/doc/pub/Regression/html/._Regression-bs032.html
+++ b/doc/pub/Regression/html/._Regression-bs032.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,53 +383,31 @@ MathJax.Hub.Config({
 <a name="part0032"></a>
 <!-- !split -->
 
-<h2 id="___sec31" class="anchor">Reformulating the problem to suit regression </h2>
+<h2 id="___sec31" class="anchor">Fixing the singularity </h2>
 
 <p>
-A more general form for the one-dimensional Ising model is
-
+If our design matrix \( \boldsymbol{X} \) which enters the linear regression problem
 $$
 \begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\tag{2}
+\boldsymbol{\beta} & =  (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y},
+\tag{1}
 \end{align}
 $$
 
-<p>
-Here we allow for interactions beyond the nearest neighbors and a state dependent
-coupling constant. This latter expression can be formulated as
-a matrix-product
-$$
-\begin{align}
-    \boldsymbol{H} = \boldsymbol{X} J,
-\tag{3}
-\end{align}
-$$
+has linearly dependent column vectors, we will not be able to compute the inverse
+of \( \boldsymbol{X}^T\boldsymbol{X} \) and we cannot find the parameters (estimators) \( \beta_i \). 
+The estimators are only well-defined if \( (\boldsymbol{X}^{T}\boldsymbol{X})^{-1} \) exits. 
+This is more likely to happen when the matrix \( \boldsymbol{X} \) is high-dimensional. In this case it is likely to encounter a situation where 
+the regression parameters \( \beta_i \) cannot be estimated.
 
 <p>
-where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
-elements \( -J_{jk} \). This form of writing the energy fits perfectly
-with the form utilized in linear regression, that is
-
+A cheap  <em>ad hoc</em> approach is  simply to add a small diagonal component to the matrix to invert, that is we change
 $$
-\begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
-\tag{4}
-\end{align}
+\boldsymbol{X}^{T} \boldsymbol{X} \rightarrow \boldsymbol{X}^{T} \boldsymbol{X}+\lambda \boldsymbol{I},
 $$
 
-<p>
-We split the data in training and test data as discussed in the previous example
+where \( \boldsymbol{I} \) is the identity matrix.  When we discuss <b>Ridge</b> regression this is actually what we end up evaluating. The parameter \( \lambda \) is called a hyperparameter. More about this later.
 
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
-    X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
-y <span style="color: #666666">=</span> energies
-X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
-</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs033.html b/doc/pub/Regression/html/._Regression-bs033.html
index 64b27e207..f9723f764 100644
--- a/doc/pub/Regression/html/._Regression-bs033.html
+++ b/doc/pub/Regression/html/._Regression-bs033.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,51 +383,42 @@ MathJax.Hub.Config({
 <a name="part0033"></a>
 <!-- !split -->
 
-<h2 id="___sec32" class="anchor">Linear regression </h2>
+<h2 id="___sec32" class="anchor">Basic math of the SVD </h2>
 
 <p>
-In the ordinary least squares method we choose the cost function
+From standard linear algebra we know that a square matrix \( \boldsymbol{X} \) can be diagonalized if and only it is 
+a so-called <a href="https://en.wikipedia.org/wiki/Normal_matrix" target="_self">normal matrix</a>, that is if \( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \)
+we have \( \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X} \) or if \( \boldsymbol{X}\in {\mathbb{C}}^{n\times n} \) we have \( \boldsymbol{X}\boldsymbol{X}^{\dagger}=\boldsymbol{X}^{\dagger}\boldsymbol{X} \).
+The matrix has then a set of eigenpairs 
 
 $$
-\begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
-\tag{5}
-\end{align}
+(\lambda_1,\boldsymbol{u}_1),\dots, (\lambda_n,\boldsymbol{u}_n),
 $$
 
-<p>
-We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
-This yields the expression for \( \boldsymbol{\beta} \) to be
-
+and the eigenvalues are given by the diagonal matrix
 $$
-    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+\boldsymbol{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n).
 $$
 
-<p>
-which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
-an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
-intercept, i.e., a constant term, we must make sure that the
-first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
+The matrix \( \boldsymbol{X} \) can be written in terms of an orthogonal/unitary transformation \( \boldsymbol{U} \)
+$$
+\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
+$$
+
+with \( \boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) or \( \boldsymbol{U}\boldsymbol{U}^{\dagger}=\boldsymbol{I} \).
 
 <p>
+Not all square matrices are diagonalizable. A matrix like the one discussed above
+$$
+\boldsymbol{X} = \begin{bmatrix} 
+1&  -1 \\
+1& -1\\
+\end{bmatrix} 
+$$
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
-    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
-    axis<span style="color: #666666">=1</span>
-)
-X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
-    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
-    axis<span style="color: #666666">=1</span>
-)
-</pre></div>
-<p>
+is not diagonalizable, it is a so-called <a href="https://en.wikipedia.org/wiki/Defective_matrix" target="_self">defective matrix</a>. It is easy to see that the condition
+\( \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X} \) is not fulfilled.
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_inv</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
-    <span style="color: #008000; font-weight: bold">return</span> scl<span style="color: #666666">.</span>inv(x<span style="color: #666666">.</span>T @ x) @ (x<span style="color: #666666">.</span>T @ y)
-beta <span style="color: #666666">=</span> ols_inv(X_train_own, y_train)
-</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs034.html b/doc/pub/Regression/html/._Regression-bs034.html
index 45d40727a..6c36399ff 100644
--- a/doc/pub/Regression/html/._Regression-bs034.html
+++ b/doc/pub/Regression/html/._Regression-bs034.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,89 +383,33 @@ MathJax.Hub.Config({
 <a name="part0034"></a>
 <!-- !split -->
 
-<h2 id="___sec33" class="anchor">Singular Value decomposition </h2>
+<h2 id="___sec33" class="anchor">The SVD, a Fantastic Algorithm </h2>
 
 <p>
-Doing the inversion directly turns out to be a bad idea since the matrix
-\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
-value decomposition</b>. Using the definition of the Moore-Penrose
-pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+However, and this is the strength of the SVD algorithm, any general
+matrix \( \boldsymbol{X} \) can be decomposed in terms of a diagonal matrix and
+two orthogonal/unitary matrices.  The <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition" target="_self">Singular Value Decompostion
+(SVD) theorem</a>
+states that a general \( m\times n \) matrix \( \boldsymbol{X} \) can be written in
+terms of a diagonal matrix \( \boldsymbol{\Sigma} \) of dimensionality \( n\times n \)
+and two orthognal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \), where the first has
+dimensionality \( m \times m \) and the last dimensionality \( n\times n \).
+We have then
 
-$$
-    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$ 
+\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T 
 $$
 
 <p>
-where the pseudoinverse of \( \boldsymbol{X} \) is given by
+As an example, the above defective matrix can be decomposed as
 
 $$
-    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix}  2&  0 \\ 0& 0\\ \end{bmatrix}    \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, 
 $$
 
 <p>
-Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
-where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
-where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
-\( \omega \) to
-$$
-\begin{align}
-    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
-\tag{6}
-\end{align}
-$$
-
-<p>
-Note that solving this equation by actually doing the pseudoinverse
-(which is what we will do) is not a good idea as this operation scales
-as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
-general matrix. Instead, doing \( QR \)-factorization and solving the
-linear system as an equation would reduce this down to
-\( \mathcal{O}(n^2) \) operations.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_svd</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
-    u, s, v <span style="color: #666666">=</span> scl<span style="color: #666666">.</span>svd(x)
-    <span style="color: #008000; font-weight: bold">return</span> v<span style="color: #666666">.</span>T @ scl<span style="color: #666666">.</span>pinv(scl<span style="color: #666666">.</span>diagsvd(s, u<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], v<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])) @ u<span style="color: #666666">.</span>T @ y
-</pre></div>
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>beta <span style="color: #666666">=</span> ols_svd(X_train_own,y_train)
-</pre></div>
-<p>
-When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J <span style="color: #666666">=</span> beta[<span style="color: #666666">1</span>:]<span style="color: #666666">.</span>reshape(L, L)
-</pre></div>
-<p>
-A way of looking at the coefficients in \( J \) is to plot the matrices as images.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
-im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J, <span style="color: #666666">**</span>cmap_args)
-plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;OLS&quot;</span>, fontsize<span style="color: #666666">=18</span>)
-plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
-plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
-cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
-cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
-plt<span style="color: #666666">.</span>show()
-</pre></div>
-<p>
-It is interesting to note that OLS
-considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
-valid matrix elements for \( J \).
-In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
-this problem can be removed, partly and only with Lasso regression.
-
-<p>
-In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+with eigenvalues \( \sigma_1=2 \) and \( \sigma_2=0 \). 
+The SVD exits always!
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs035.html b/doc/pub/Regression/html/._Regression-bs035.html
index 23a45041a..78c5abab6 100644
--- a/doc/pub/Regression/html/._Regression-bs035.html
+++ b/doc/pub/Regression/html/._Regression-bs035.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,53 +383,40 @@ MathJax.Hub.Config({
 <a name="part0035"></a>
 <!-- !split -->
 
-<h2 id="___sec34" class="anchor">Linear Regression Problems </h2>
+<h2 id="___sec34" class="anchor">Another Example </h2>
 
 <p>
-One of the typical problems we encounter with linear regression, in particular 
-when the matrix \( \boldsymbol{X} \) (our so-called design matrix) is high-dimensional, 
-are problems with near singular or singular matrices. The column vectors of \( \boldsymbol{X} \) 
-may be linearly dependent, normally referred to as super-collinearity.  
-This means that the matrix may be rank deficient and it is basically impossible to 
-to model the data using linear regression. As an example, consider the matrix
+Consider the following matrix which can be SVD decomposed as
+
 $$
-\begin{align*}
-\mathbf{X} & =  \left[
-\begin{array}{rrr}
-1 & -1 & 2
-\\
-1 & 0 & 1
-\\
-1 & 2  & -1
-\\
-1 & 1  & 0
-\end{array} \right]
-\end{align*}
+\boldsymbol{X} = \frac{1}{15}\begin{bmatrix} 14 & 2\\ 4 & 22\\ 16 & 13\end{bmatrix}=\frac{1}{3}\begin{bmatrix}  1&  2 & 2 \\ 2& -1 & 1\\ 2 & 1& -2\end{bmatrix} \begin{bmatrix}  2&  0 \\ 0& 1\\ 0 & 0\end{bmatrix}\frac{1}{5}\begin{bmatrix}  3&  4 \\ 4& -3\end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T. 
 $$
 
 <p>
-The columns of \( \boldsymbol{X} \) are linearly dependent. We see this easily since the 
-the first column is the row-wise sum of the other two columns. The rank (more correct,
-the column rank) of a matrix is the dimension of the space spanned by the
-column vectors. Hence, the rank of \( \mathbf{X} \) is equal to the number
-of linearly independent columns. In this particular case the matrix has rank 2.
+This is a \( 3\times 2 \) matrix which is decomposed in terms of a
+\( 3\times 3 \) matrix \( \boldsymbol{U} \), and a \( 2\times 2 \) matrix \( \boldsymbol{V} \). It is easy to see
+that \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal (how?).
 
 <p>
-Super-collinearity of an \( (n \times p) \)-dimensional design matrix \( \mathbf{X} \) implies
-that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{x} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
-$$
-\begin{align*}
-\boldsymbol{X} & =  \left[
-\begin{array}{rr}
-1 & -1
-\\
-1 & -1
-\end{array} \right].
-\end{align*}
-$$
+And the SVD
+decomposition (singular values) gives eigenvalues 
+\( \sigma_i\geq\sigma_{i+1} \) for all \( i \) and for dimensions larger than \( i=2 \), the
+eigenvalues (singular values) are zero.
 
-We see easily that  \( \mbox{det}(\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0 \). Hence, \( \mathbf{X} \) is singular and its inverse is undefined.
-This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least an eigenvalue which is zero.
+<p>
+In the general case, where our design matrix \( \boldsymbol{X} \) has dimension
+\( n\times p \), the matrix is thus decomposed into an \( n\times n \)
+orthogonal matrix \( \boldsymbol{U} \), a \( p\times p \) orthogonal matrix \( \boldsymbol{V} \)
+and a diagonal matrix \( \boldsymbol{\Sigma} \) with \( r=\mathrm{min}(n,p) \)
+singular values \( \sigma_i\geq 0 \) on the main diagonal and zeros filling
+the rest of the matrix.  There are at most \( p \) singular values
+assuming that \( n > p \). In our regression examples for the nuclear
+masses and the equation of state this is indeed the case, while for
+the Ising model we have \( p > n \). These are often cases that lead to
+near singular or singular matrices.
+
+<p>
+The columns of \( \boldsymbol{U} \) are called the left singular vectors while the columns of \( \boldsymbol{V} \) are the right singular vectors.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs036.html b/doc/pub/Regression/html/._Regression-bs036.html
index df2e0707a..73ca49000 100644
--- a/doc/pub/Regression/html/._Regression-bs036.html
+++ b/doc/pub/Regression/html/._Regression-bs036.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,30 +383,27 @@ MathJax.Hub.Config({
 <a name="part0036"></a>
 <!-- !split -->
 
-<h2 id="___sec35" class="anchor">Fixing the singularity </h2>
+<h2 id="___sec35" class="anchor">Economy-size SVD </h2>
 
 <p>
-If our design matrix \( \boldsymbol{X} \) which enters the linear regression problem
-$$
-\begin{align}
-\boldsymbol{\beta} & =  (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y},
-\tag{7}
-\end{align}
-$$
-
-has linearly dependent column vectors, we will not be able to compute the inverse
-of \( \boldsymbol{X}^T\boldsymbol{X} \) and we cannot find the parameters (estimators) \( \beta_i \). 
-The estimators are only well-defined if \( (\boldsymbol{X}^{T}\boldsymbol{X})^{-1} \) exits. 
-This is more likely to happen when the matrix \( \boldsymbol{X} \) is high-dimensional. In this case it is likely to encounter a situation where 
-the regression parameters \( \beta_i \) cannot be estimated.
+If we assume that \( n > p \), then our matrix \( \boldsymbol{U} \) has dimension \( n
+\times n \). The last \( n-p \) columns of \( \boldsymbol{U} \) become however
+irrelevant in our calculations since they are multiplied with the
+zeros in \( \boldsymbol{\Sigma} \).
 
 <p>
-A cheap  <em>ad hoc</em> approach is  simply to add a small diagonal component to the matrix to invert, that is we change
-$$
-\boldsymbol{X}^{T} \boldsymbol{X} \rightarrow \boldsymbol{X}^{T} \boldsymbol{X}+\lambda \boldsymbol{I},
-$$
+The economy-size decomposition removes extra rows or columns of zeros
+from the diagonal matrix of singular values, \( \boldsymbol{\Sigma} \), along with the columns
+in either \( \boldsymbol{U} \) or \( \boldsymbol{V} \) that multiply those zeros in the expression. 
+Removing these zeros and columns can improve execution time
+and reduce storage requirements without compromising the accuracy of
+the decomposition.
 
-where \( \boldsymbol{I} \) is the identity matrix.  When we discuss <b>Ridge</b> regression this is actually what we end up evaluating. The parameter \( \lambda \) is called a hyperparameter. More about this later.
+<p>
+If \( n > p \), we keep only the first \( p \) columns of \( \boldsymbol{U} \) and \( \boldsymbol{\Sigma} \) has dimension \( p\times p \). 
+If \( p > n \), then only the first \( n \) columns of \( \boldsymbol{V} \) are computed and \( \boldsymbol{\Sigma} \) has dimension \( n\times n \).
+The \( n=p \) case is obvious, we retain the full SVD. 
+In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs037.html b/doc/pub/Regression/html/._Regression-bs037.html
index 400c5aebf..6142e651b 100644
--- a/doc/pub/Regression/html/._Regression-bs037.html
+++ b/doc/pub/Regression/html/._Regression-bs037.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,41 +383,54 @@ MathJax.Hub.Config({
 <a name="part0037"></a>
 <!-- !split -->
 
-<h2 id="___sec36" class="anchor">Basic math of the SVD </h2>
+<h2 id="___sec36" class="anchor">Mathematical Properties </h2>
 
 <p>
-From standard linear algebra we know that a square matrix \( \boldsymbol{X} \) can be diagonalized if and only it is 
-a so-called <a href="https://en.wikipedia.org/wiki/Normal_matrix" target="_self">normal matrix</a>, that is if \( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \)
-we have \( \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X} \) or if \( \boldsymbol{X}\in {\mathbb{C}}^{n\times n} \) we have \( \boldsymbol{X}\boldsymbol{X}^{\dagger}=\boldsymbol{X}^{\dagger}\boldsymbol{X} \).
-The matrix has then a set of eigenpairs 
-
-$$
-(\lambda_1,\boldsymbol{u}_1),\dots, (\lambda_n,\boldsymbol{u}_n),
-$$
-
-and the eigenvalues are given by the diagonal matrix
-$$
-\boldsymbol{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n).
-$$
-
-The matrix \( \boldsymbol{X} \) can be written in terms of an orthogonal/unitary transformation \( \boldsymbol{U} \)
-$$
-\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
-$$
-
-with \( \boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) or \( \boldsymbol{U}\boldsymbol{U}^{\dagger}=\boldsymbol{I} \).
+There are several interesting mathematical properties which will be
+relevant when we are going to discuss the differences between say
+ordinary least squares (OLS) and <b>Ridge</b> regression.
 
 <p>
-Not all square matrices are diagonalizable. A matrix like the one discussed above
+We have from OLS that the parameters of the linear approximation are given by
 $$
-\boldsymbol{X} = \begin{bmatrix} 
-1&  -1 \\
-1& -1\\
-\end{bmatrix} 
+\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. 
 $$
 
-is not diagonalizable, it is a so-called <a href="https://en.wikipedia.org/wiki/Defective_matrix" target="_self">defective matrix</a>. It is easy to see that the condition
-\( \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X} \) is not fulfilled.
+<p>
+The matrix to invert can be rewritten in terms of our SVD decomposition as
+
+$$
+\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T.
+$$
+
+Using the orthogonality properties of \( \boldsymbol{U} \) we have
+
+$$
+\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T =  \boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T,
+$$
+
+with \( \boldsymbol{D} \) being a diagonal matrix with values along the diagonal given by the singular values squared.
+
+<p>
+This means that
+$$
+(\boldsymbol{X}^T\boldsymbol{X})\boldsymbol{V} = \boldsymbol{V}\boldsymbol{D},
+$$
+
+that is the eigenvectors of \( (\boldsymbol{X}^T\boldsymbol{X}) \) are given by the columns of the right singular matrix of \( \boldsymbol{X} \) and the eigenvalues are the squared singular values.  It is easy to show (show this) that
+$$
+(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D},
+$$
+
+that is, the eigenvectors of \( (\boldsymbol{X}\boldsymbol{X})^T \) are the columns of the left singular matrix and the eigenvalues are the same.
+
+<p>
+Going back to our OLS equation we have
+$$
+\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}.
+$$
+
+We will come back to this expression when we discuss Ridge regression.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs038.html b/doc/pub/Regression/html/._Regression-bs038.html
index 13c587454..5b604d134 100644
--- a/doc/pub/Regression/html/._Regression-bs038.html
+++ b/doc/pub/Regression/html/._Regression-bs038.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,33 +383,60 @@ MathJax.Hub.Config({
 <a name="part0038"></a>
 <!-- !split -->
 
-<h2 id="___sec37" class="anchor">The SVD, a Fantastic Algorithm </h2>
+<h2 id="___sec37" class="anchor">Ridge and LASSO Regression </h2>
 
 <p>
-However, and this is the strength of the SVD algorithm, any general
-matrix \( \boldsymbol{X} \) can be decomposed in terms of a diagonal matrix and
-two orthogonal/unitary matrices.  The <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition" target="_self">Singular Value Decompostion
-(SVD) theorem</a>
-states that a general \( m\times n \) matrix \( \boldsymbol{X} \) can be written in
-terms of a diagonal matrix \( \boldsymbol{\Sigma} \) of dimensionality \( n\times n \)
-and two orthognal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \), where the first has
-dimensionality \( m \times m \) and the last dimensionality \( n\times n \).
-We have then
+Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
+our optimization problem is
+$$
+{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
+$$
 
-$$ 
-\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T 
+or we can state it as
+$$
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
+$$
+
+where we have used the definition of  a norm-2 vector, that is
+$$
+\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. 
 $$
 
 <p>
-As an example, the above defective matrix can be decomposed as
+By minimizing the above equation with respect to the parameters
+\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the
+parameters \( \boldsymbol{\beta} \).  We can add a regularization parameter \( \lambda \) by
+defining a new cost function to be optimized, that is
 
 $$
-\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix}  2&  0 \\ 0& 0\\ \end{bmatrix}    \frac{1}{\sqrt{2}}\begin{bmatrix}  1&  -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, 
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
 $$
 
 <p>
-with eigenvalues \( \sigma_1=2 \) and \( \sigma_2=0 \). 
-The SVD exits always!
+which leads to the Ridge regression minimization problem where we
+require that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is
+a finite number larger than zero. By defining
+
+$$
+C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,
+$$
+
+<p>
+we have a new optimization equation
+$$
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1
+$$
+
+which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.
+
+<p>
+Here we have defined the norm-1 as 
+$$
+\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert. 
+$$
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs039.html b/doc/pub/Regression/html/._Regression-bs039.html
index d01066aec..c63c5f3ef 100644
--- a/doc/pub/Regression/html/._Regression-bs039.html
+++ b/doc/pub/Regression/html/._Regression-bs039.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,40 +383,61 @@ MathJax.Hub.Config({
 <a name="part0039"></a>
 <!-- !split -->
 
-<h2 id="___sec38" class="anchor">Another Example </h2>
+<h2 id="___sec38" class="anchor">More on Ridge Regression </h2>
 
 <p>
-Consider the following matrix which can be SVD decomposed as
+Using the matrix-vector expression for Ridge regression,
 
 $$
-\boldsymbol{X} = \frac{1}{15}\begin{bmatrix} 14 & 2\\ 4 & 22\\ 16 & 13\end{bmatrix}=\frac{1}{3}\begin{bmatrix}  1&  2 & 2 \\ 2& -1 & 1\\ 2 & 1& -2\end{bmatrix} \begin{bmatrix}  2&  0 \\ 0& 1\\ 0 & 0\end{bmatrix}\frac{1}{5}\begin{bmatrix}  3&  4 \\ 4& -3\end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T. 
+C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta},
 $$
 
 <p>
-This is a \( 3\times 2 \) matrix which is decomposed in terms of a
-\( 3\times 3 \) matrix \( \boldsymbol{U} \), and a \( 2\times 2 \) matrix \( \boldsymbol{V} \). It is easy to see
-that \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal (how?).
+by taking the derivatives with respect to \( \boldsymbol{\beta} \) we obtain then
+a slightly modified matrix inversion problem which for finite values
+of \( \lambda \) does not suffer from singularity problems. We obtain
+
+$$
+\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
+$$
 
 <p>
-And the SVD
-decomposition (singular values) gives eigenvalues 
-\( \sigma_i\geq\sigma_{i+1} \) for all \( i \) and for dimensions larger than \( i=2 \), the
-eigenvalues (singular values) are zero.
+with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that
+
+$$
+\sum_{i=0}^{p-1} \beta_i^2 \leq t,
+$$
 
 <p>
-In the general case, where our design matrix \( \boldsymbol{X} \) has dimension
-\( n\times p \), the matrix is thus decomposed into an \( n\times n \)
-orthogonal matrix \( \boldsymbol{U} \), a \( p\times p \) orthogonal matrix \( \boldsymbol{V} \)
-and a diagonal matrix \( \boldsymbol{\Sigma} \) with \( r=\mathrm{min}(n,p) \)
-singular values \( \sigma_i\lg 0 \) on the main diagonal and zeros filling
-the rest of the matrix.  There are at most \( p \) singular values
-assuming that \( n > p \). In our regression examples for the nuclear
-masses and the equation of state this is indeed the case, while for
-the Ising model we have \( p > n \). These are often cases that lead to
-near singular or singular matrices.
+with \( t \) a finite positive number.
 
 <p>
-The columns of \( \boldsymbol{U} \) are called the left singular vectors while the columns of \( \boldsymbol{V} \) are the right singular vectors.
+We see that Ridge regression is nothing but the standard
+OLS with a modified diagonal term added to \( \boldsymbol{X}^T\boldsymbol{X} \). The
+consequences, in particular for our discussion of the bias-variance
+are rather interesting.
+
+<p>
+Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had
+$$
+(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}.
+$$
+
+<p>
+We can  analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \( \boldsymbol{U} \) as
+$$
+\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}
+$$
+
+<p>
+For Ridge regression this becomes
+
+$$
+\boldsymbol{X}\boldsymbol{\beta}^{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
+$$
+
+<p>
+with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs040.html b/doc/pub/Regression/html/._Regression-bs040.html
index d1ac113a7..7fd678478 100644
--- a/doc/pub/Regression/html/._Regression-bs040.html
+++ b/doc/pub/Regression/html/._Regression-bs040.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,27 +383,26 @@ MathJax.Hub.Config({
 <a name="part0040"></a>
 <!-- !split -->
 
-<h2 id="___sec39" class="anchor">Economy-size SVD </h2>
+<h2 id="___sec39" class="anchor">Interpreting the Ridge results </h2>
 
 <p>
-If we assume that \( n > p \), then our matrix \( \boldsymbol{U} \) has dimension \( n
-\times n \). The last \( n-p \) columns of \( \boldsymbol{U} \) become however
-irrelevant in our calculations since they are multiplied with the
-zeros in \( \boldsymbol{\Sigma} \).
+Since \( \lambda \geq 0 \), it means that compared to OLS, we have 
+
+$$
+\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. 
+$$
 
 <p>
-The economy-size decomposition removes extra rows or columns of zeros
-from the diagonal matrix of singular values, \( \boldsymbol{\Sigma} \), along with the columns
-in either \( \boldsymbol{U} \) or \( \boldsymbol{V} \) that multiply those zeros in the expression. 
-Removing these zeros and columns can improve execution time
-and reduce storage requirements without compromising the accuracy of
-the decomposition.
+Ridge regression finds the coordinates of \( \boldsymbol{y} \) with respect to the
+orthonormal basis \( \boldsymbol{U} \), it then shrinks the coordinates by
+\( \frac{\sigma_j^2}{\sigma_j^2+\lambda} \). Recall that the SVD has
+eigenvalues ordered in a descending way, that is \( \sigma_i \geq
+\sigma_{i+1} \).
 
 <p>
-If \( n > p \), we keep only the first \( p \) columns of \( \boldsymbol{U} \) and \( \boldsymbol{\Sigma} \) has dimension \( p\times p \). 
-If \( p > n \), then only the first \( n \) columns of \( \boldsymbol{V} \) are computed and \( \boldsymbol{\Sigma} \) has dimension \( n\times n \).
-The \( n=p \) case is obvious, we retain the full SVD. 
-In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.
+For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
+Actually, calculating the variance of \( \boldsymbol{X}\boldsymbol{v}_j \) shows that this quantity is equal to \( \sigma_j^2/n \).
+With a parameter \( \lambda \) we can thus shrink the role of specific parameters.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs041.html b/doc/pub/Regression/html/._Regression-bs041.html
index 2f58af5f0..ab77a840d 100644
--- a/doc/pub/Regression/html/._Regression-bs041.html
+++ b/doc/pub/Regression/html/._Regression-bs041.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,54 +383,39 @@ MathJax.Hub.Config({
 <a name="part0041"></a>
 <!-- !split -->
 
-<h2 id="___sec40" class="anchor">Mathematical Properties </h2>
+<h2 id="___sec40" class="anchor">More interpretations </h2>
 
 <p>
-There are several interesting mathematical properties which will be
-relevant when we are going to discuss the differences between say
-ordinary least squares (OLS) and <b>Ridge</b> regression.
+For the sake of simplicity, let us assume that the design matrix is orthonormal, that is 
 
-<p>
-We have from OLS that the parameters of the linear approximation are given by
 $$
-\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. 
+\boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}. 
 $$
 
 <p>
-The matrix to invert can be rewritten in terms of our SVD decomposition as
-
+In this case the standard OLS results in 
 $$
-\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T.
+\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y},
 $$
 
-Using the orthogonality properties of \( \boldsymbol{U} \) we have
-
-$$
-\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T =  \boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T,
-$$
-
-with \( \boldsymbol{D} \) being a diagonal matrix with values along the diagonal given by the singular values squared.
-
 <p>
-This means that
-$$
-(\boldsymbol{X}^T\boldsymbol{X})\boldsymbol{V} = \boldsymbol{V}\boldsymbol{D},
-$$
+and
 
-that is the eigenvectors of \( (\boldsymbol{X}^T\boldsymbol{X}) \) are given by the columns of the right singular matrix of \( \boldsymbol{X} \) and the eigenvalues are the squared singular values.  It is easy to show (show this) that
 $$
-(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D},
+\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}},
 $$
 
-that is, the eigenvectors of \( (\boldsymbol{X}\boldsymbol{X})^T \) are the columns of the left singular matrix and the eigenvalues are the same.
-
 <p>
-Going back to our OLS equation we have
-$$
-\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}.
-$$
+that is the Ridge estimator scales the OLS estimator by the inverse of a factor \( 1+\lambda \), and
+the Ridge estimator converges to zero when the hyperparameter goes to
+infinity.
 
-We will come back to this expression when we discuss Ridge regression.
+<p>
+We will come back to more interpreations after we have gone through some of the statistical analysis part.
+
+<p>
+For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_self">Wessel van Wieringen's</a> article is highly recommended.
+Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_self">Mehta et al's article</a> is also recommended.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs042.html b/doc/pub/Regression/html/._Regression-bs042.html
index 49f2db109..8d2b5faee 100644
--- a/doc/pub/Regression/html/._Regression-bs042.html
+++ b/doc/pub/Regression/html/._Regression-bs042.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,60 +383,19 @@ MathJax.Hub.Config({
 <a name="part0042"></a>
 <!-- !split -->
 
-<h2 id="___sec41" class="anchor">Ridge and LASSO Regression </h2>
+<h2 id="___sec41" class="anchor">Where are we going? </h2>
 
 <p>
-Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
-our optimization problem is
-$$
-{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
-$$
+Before we proceed, we need to rethink what we have been doing. In our
+eager to fit the data, we have omitted several important elements in
+our regression analysis. In what follows we will
 
-or we can state it as
-$$
-{\displaystyle \min_{\boldsymbol{\beta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
-$$
+<ol>
+<li> look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff</li>
+<li> introduce resampling techniques like cross-validation, bootstrapping and jackknife and more</li>
+</ol>
 
-where we have used the definition of  a norm-2 vector, that is
-$$
-\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. 
-$$
-
-<p>
-By minimizing the above equation with respect to the parameters
-\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the
-parameters \( \boldsymbol{\beta} \).  We can add a regularization parameter \( \lambda \) by
-defining a new cost function to be optimized, that is
-
-$$
-{\displaystyle \min_{\boldsymbol{\beta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
-$$
-
-<p>
-which leads to the Ridge regression minimization problem where we
-require that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is
-a finite number larger than zero. By defining
-
-$$
-C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,
-$$
-
-<p>
-we have a new optimization equation
-$$
-{\displaystyle \min_{\boldsymbol{\beta}\in
-{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1
-$$
-
-which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.
-
-<p>
-Here we have defined the norm-1 as 
-$$
-\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert. 
-$$
+This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs043.html b/doc/pub/Regression/html/._Regression-bs043.html
index 857579026..416da079f 100644
--- a/doc/pub/Regression/html/._Regression-bs043.html
+++ b/doc/pub/Regression/html/._Regression-bs043.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
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-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,61 +383,23 @@ MathJax.Hub.Config({
 <a name="part0043"></a>
 <!-- !split -->
 
-<h2 id="___sec42" class="anchor">More on Ridge Regression </h2>
+<h2 id="___sec42" class="anchor">Resampling methods </h2>
+<div class="panel panel-default">
+<div class="panel-body">
+<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+Resampling methods are an indispensable tool in modern
+statistics. They involve repeatedly drawing samples from a training
+set and refitting a model of interest on each sample in order to
+obtain additional information about the fitted model. For example, in
+order to estimate the variability of a linear regression fit, we can
+repeatedly draw different samples from the training data, fit a linear
+regression to each new sample, and then examine the extent to which
+the resulting fits differ. Such an approach may allow us to obtain
+information that would not be available from fitting the model only
+once using the original training sample.
+</div>
+</div>
 
-<p>
-Using the matrix-vector expression for Ridge regression,
-
-$$
-C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta},
-$$
-
-<p>
-by taking the derivatives with respect to \( \boldsymbol{\beta} \) we obtain then
-a slightly modified matrix inversion problem which for finite values
-of \( \lambda \) does not suffer from singularity problems. We obtain
-
-$$
-\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
-$$
-
-<p>
-with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that
-
-$$
-\sum_{i=0}^{p-1} \beta_i^2 \leq t,
-$$
-
-<p>
-with \( t \) a finite positive number.
-
-<p>
-We see that Ridge regression is nothing but the standard
-OLS with a modified diagonal term added to \( \boldsymbol{X}^T\boldsymbol{X} \). The
-consequences, in particular for our discussion of the bias-variance
-are rather interesting.
-
-<p>
-Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had
-$$
-(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}.
-$$
-
-<p>
-We can  analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \( \boldsymbol{U} \) as
-$$
-\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}
-$$
-
-<p>
-For Ridge regression this becomes
-
-$$
-\boldsymbol{X}\boldsymbol{\beta}^{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
-$$
-
-<p>
-with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs044.html b/doc/pub/Regression/html/._Regression-bs044.html
index 564661628..d3532c4de 100644
--- a/doc/pub/Regression/html/._Regression-bs044.html
+++ b/doc/pub/Regression/html/._Regression-bs044.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,26 +383,32 @@ MathJax.Hub.Config({
 <a name="part0044"></a>
 <!-- !split -->
 
-<h2 id="___sec43" class="anchor">Interpreting the Ridge results </h2>
+<h2 id="___sec43" class="anchor">Resampling approaches can be computationally expensive </h2>
+<div class="panel panel-default">
+<div class="panel-body">
+<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
 
 <p>
-Since \( \lambda \geq 0 \), it means that compared to OLS, we have 
-
-$$
-\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. 
-$$
+Resampling approaches can be computationally expensive, because they
+involve fitting the same statistical method multiple times using
+different subsets of the training data. However, due to recent
+advances in computing power, the computational requirements of
+resampling methods generally are not prohibitive. In this chapter, we
+discuss two of the most commonly used resampling methods,
+cross-validation and the bootstrap. Both methods are important tools
+in the practical application of many statistical learning
+procedures. For example, cross-validation can be used to estimate the
+test error associated with a given statistical learning method in
+order to evaluate its performance, or to select the appropriate level
+of flexibility. The process of evaluating a model&#8217;s performance is
+known as model assessment, whereas the process of selecting the proper
+level of flexibility for a model is known as model selection. The
+bootstrap is widely used.
 
 <p>
-Ridge regression finds the coordinates of \( \boldsymbol{y} \) with respect to the
-orthonormal basis \( \boldsymbol{U} \), it then shrinks the coordinates by
-\( \frac{\sigma_j^2}{\sigma_j^2+\lambda} \). Recall that the SVD has
-eigenvalues ordered in a descending way, that is \( \sigma_i \geq
-\sigma_{i+1} \).
+</div>
+</div>
 
-<p>
-For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
-Actually, calculating the variance of \( \boldsymbol{X}\boldsymbol{v}_j \) shows that this quantity is equal to \( \sigma_j^2/n \).
-With a parameter \( \lambda \) we can thus shrink the role of specific parameters.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs045.html b/doc/pub/Regression/html/._Regression-bs045.html
index c8b667d8b..a2680f2fe 100644
--- a/doc/pub/Regression/html/._Regression-bs045.html
+++ b/doc/pub/Regression/html/._Regression-bs045.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,39 +383,19 @@ MathJax.Hub.Config({
 <a name="part0045"></a>
 <!-- !split -->
 
-<h2 id="___sec44" class="anchor">More interpretations </h2>
+<h2 id="___sec44" class="anchor">Why resampling methods ? </h2>
+<div class="panel panel-default">
+<div class="panel-body">
+<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
 
-<p>
-For the sake of simplicity, let us assume that the design matrix is orthonormal, that is 
+<ul>
+<li> Our simulations can be treated as <em>computer experiments</em>. This is particularly the case for Monte Carlo methods</li>
+<li> The results can be analysed with the same statistical tools as we would use analysing experimental data.</li>
+<li> As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.</li>
+</ul>
+</div>
+</div>
 
-$$
-\boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}. 
-$$
-
-<p>
-In this case the standard OLS results in 
-$$
-\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y},
-$$
-
-<p>
-and
-
-$$
-\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}},
-$$
-
-<p>
-that is the Ridge estimator scales the OLS estimator by the inverse of a factor \( 1+\lambda \), and
-the Ridge estimator converges to zero when the hyperparameter goes to
-infinity.
-
-<p>
-We will come back to more interpreations after we have gone through some of the statistical analysis part.
-
-<p>
-For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_self">Wessel van Wieringen's</a> article is highly recommended.
-Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_self">Mehta et al's article</a> is also recommended.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs046.html b/doc/pub/Regression/html/._Regression-bs046.html
index 4ae502f5b..05142a991 100644
--- a/doc/pub/Regression/html/._Regression-bs046.html
+++ b/doc/pub/Regression/html/._Regression-bs046.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
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-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
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               ('Statistics, sample variance and covariance',
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                None,
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-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
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               ('Statistics, effective number of correlations',
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                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
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+              ('Assumptions made', 2, None, '___sec72'),
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               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
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                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
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+              ('Jackknife code example', 2, None, '___sec81'),
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               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
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               ('Code Example for Cross-validation and $k$-fold '
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                2,
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-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
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+               2,
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+              ('Linear regression', 2, None, '___sec96'),
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               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,19 +383,25 @@ MathJax.Hub.Config({
 <a name="part0046"></a>
 <!-- !split -->
 
-<h2 id="___sec45" class="anchor">Where are we going? </h2>
+<h2 id="___sec45" class="anchor">Statistical analysis </h2>
+<div class="panel panel-default">
+<div class="panel-body">
+<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
 
-<p>
-Before we proceed, we need to rethink what we have been doing. In our
-eager to fit the data, we have omitted several important elements in
-our regression analysis. In what follows we will
+<ul>
+<li> As in other experiments, many numerical  experiments have two classes of errors:</li>
 
-<ol>
-<li> look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff</li>
-<li> introduce resampling techniques like cross-validation, bootstrapping and jackknife and more</li>
-</ol>
+<ul>
+  <li> Statistical errors</li>
+  <li> Systematical errors</li>
+</ul>
+
+<li> Statistical errors can be estimated using standard tools from statistics</li>
+<li> Systematical errors are method specific and must be treated differently from case to case.</li> 
+</ul>
+</div>
+</div>
 
-This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs047.html b/doc/pub/Regression/html/._Regression-bs047.html
index a8564ed23..7fa3a3ae5 100644
--- a/doc/pub/Regression/html/._Regression-bs047.html
+++ b/doc/pub/Regression/html/._Regression-bs047.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,20 +383,31 @@ MathJax.Hub.Config({
 <a name="part0047"></a>
 <!-- !split -->
 
-<h2 id="___sec46" class="anchor">Resampling methods </h2>
+<h2 id="___sec46" class="anchor">Statistics  </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Resampling methods are an indispensable tool in modern
-statistics. They involve repeatedly drawing samples from a training
-set and refitting a model of interest on each sample in order to
-obtain additional information about the fitted model. For example, in
-order to estimate the variability of a linear regression fit, we can
-repeatedly draw different samples from the training data, fit a linear
-regression to each new sample, and then examine the extent to which
-the resulting fits differ. Such an approach may allow us to obtain
-information that would not be available from fitting the model only
-once using the original training sample.
+The <em>probability distribution function (PDF)</em> is a function
+\( p(x) \) on the domain which, in the discrete case, gives us the
+probability or relative frequency with which these values of \( X \) occur:
+$$
+p(x) = \mathrm{prob}(X=x)
+$$
+
+In the continuous case, the PDF does not directly depict the
+actual probability. Instead we define the probability for the
+stochastic variable to assume any value on an infinitesimal interval
+around \( x \) to be \( p(x)dx \). The continuous function \( p(x) \) then gives us
+the <em>density</em> of the probability rather than the probability
+itself. The probability for a stochastic variable to assume any value
+on a non-infinitesimal interval \( [a,\,b] \) is then just the integral:
+$$
+\mathrm{prob}(a\leq X\leq b) = \int_a^b p(x)dx
+$$
+
+Qualitatively speaking, a stochastic variable represents the values of
+numbers chosen as if by chance from some specified PDF so that the
+selection of a large set of these numbers reproduces this PDF.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs048.html b/doc/pub/Regression/html/._Regression-bs048.html
index 0ed4f8137..9d1aeebbf 100644
--- a/doc/pub/Regression/html/._Regression-bs048.html
+++ b/doc/pub/Regression/html/._Regression-bs048.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,29 +383,23 @@ MathJax.Hub.Config({
 <a name="part0048"></a>
 <!-- !split -->
 
-<h2 id="___sec47" class="anchor">Resampling approaches can be computationally expensive </h2>
+<h2 id="___sec47" class="anchor">Statistics, moments </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+A particularly useful class of special expectation values are the
+<em>moments</em>. The \( n \)-th moment of the PDF \( p \) is defined as
+follows:
+$$
+\langle x^n\rangle \equiv \int\! x^n p(x)\,dx
+$$
 
-<p>
-Resampling approaches can be computationally expensive, because they
-involve fitting the same statistical method multiple times using
-different subsets of the training data. However, due to recent
-advances in computing power, the computational requirements of
-resampling methods generally are not prohibitive. In this chapter, we
-discuss two of the most commonly used resampling methods,
-cross-validation and the bootstrap. Both methods are important tools
-in the practical application of many statistical learning
-procedures. For example, cross-validation can be used to estimate the
-test error associated with a given statistical learning method in
-order to evaluate its performance, or to select the appropriate level
-of flexibility. The process of evaluating a model&#8217;s performance is
-known as model assessment, whereas the process of selecting the proper
-level of flexibility for a model is known as model selection. The
-bootstrap is widely used.
-
-<p>
+The zero-th moment \( \langle 1\rangle \) is just the normalization condition of
+\( p \). The first moment, \( \langle x\rangle \), is called the <em>mean</em> of \( p \)
+and often denoted by the letter \( \mu \):
+$$
+\langle x\rangle = \mu \equiv \int\! x p(x)\,dx
+$$
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs049.html b/doc/pub/Regression/html/._Regression-bs049.html
index 98dd9b01a..b24a901cb 100644
--- a/doc/pub/Regression/html/._Regression-bs049.html
+++ b/doc/pub/Regression/html/._Regression-bs049.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,16 +383,38 @@ MathJax.Hub.Config({
 <a name="part0049"></a>
 <!-- !split -->
 
-<h2 id="___sec48" class="anchor">Why resampling methods ? </h2>
+<h2 id="___sec48" class="anchor">Statistics, central moments </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+A special version of the moments is the set of <em>central moments</em>,
+the n-th central moment defined as:
+$$
+\langle (x-\langle x \rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx
+$$
 
-<ul>
-<li> Our simulations can be treated as <em>computer experiments</em>. This is particularly the case for Monte Carlo methods</li>
-<li> The results can be analysed with the same statistical tools as we would use analysing experimental data.</li>
-<li> As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.</li>
-</ul>
+The zero-th and first central moments are both trivial, equal \( 1 \) and
+\( 0 \), respectively. But the second central moment, known as the
+<em>variance</em> of \( p \), is of particular interest. For the stochastic
+variable \( X \), the variance is denoted as \( \sigma^2_X \) or \( \mathrm{var}(X) \):
+$$
+\begin{align}
+\sigma^2_X\ \ =\ \ \mathrm{var}(X) & =  \langle (x-\langle x\rangle)^2\rangle =
+\int\! (x-\langle x\rangle)^2 p(x)\,dx
+\tag{2}\\
+& =  \int\! \left(x^2 - 2 x \langle x\rangle^{2} +
+  \langle x\rangle^2\right)p(x)\,dx
+\tag{3}\\
+& =  \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2
+\tag{4}\\
+& =  \langle x^2\rangle - \langle x\rangle^2
+\tag{5}
+\end{align}
+$$
+
+The square root of the variance, \( \sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle} \) is called the <em>standard deviation</em> of \( p \). It is clearly just the RMS (root-mean-square)
+value of the deviation of the PDF from its mean value, interpreted
+qualitatively as the <em>spread</em> of \( p \) around its mean.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs050.html b/doc/pub/Regression/html/._Regression-bs050.html
index 1b93786c3..74018c6ae 100644
--- a/doc/pub/Regression/html/._Regression-bs050.html
+++ b/doc/pub/Regression/html/._Regression-bs050.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,22 +383,31 @@ MathJax.Hub.Config({
 <a name="part0050"></a>
 <!-- !split -->
 
-<h2 id="___sec49" class="anchor">Statistical analysis </h2>
+<h2 id="___sec49" class="anchor">Statistics, covariance </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+Another important quantity is the so called covariance, a variant of
+the above defined variance. Consider again the set \( \{X_i\} \) of \( n \)
+stochastic variables (not necessarily uncorrelated) with the
+multivariate PDF \( P(x_1,\dots,x_n) \). The <em>covariance</em> of two
+of the stochastic variables, \( X_i \) and \( X_j \), is defined as follows:
+$$
+\begin{align}
+\mathrm{cov}(X_i,\,X_j) &\equiv \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
+\nonumber\\
+&=
+\int\!\cdots\!\int\!(x_i-\langle x_i \rangle)(x_j-\langle x_j \rangle)\,
+P(x_1,\dots,x_n)\,dx_1\dots dx_n
+\tag{6}
+\end{align}
+$$
 
-<ul>
-<li> As in other experiments, many numerical  experiments have two classes of errors:</li>
-
-<ul>
-  <li> Statistical errors</li>
-  <li> Systematical errors</li>
-</ul>
-
-<li> Statistical errors can be estimated using standard tools from statistics</li>
-<li> Systematical errors are method specific and must be treated differently from case to case.</li> 
-</ul>
+with
+$$
+\langle x_i\rangle =
+\int\!\cdots\!\int\!x_i\,P(x_1,\dots,x_n)\,dx_1\dots dx_n
+$$
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs051.html b/doc/pub/Regression/html/._Regression-bs051.html
index 9edf6eddb..8a4c80b93 100644
--- a/doc/pub/Regression/html/._Regression-bs051.html
+++ b/doc/pub/Regression/html/._Regression-bs051.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,31 +383,32 @@ MathJax.Hub.Config({
 <a name="part0051"></a>
 <!-- !split -->
 
-<h2 id="___sec50" class="anchor">Statistics  </h2>
+<h2 id="___sec50" class="anchor">Statistics, more covariance  </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-The <em>probability distribution function (PDF)</em> is a function
-\( p(x) \) on the domain which, in the discrete case, gives us the
-probability or relative frequency with which these values of \( X \) occur:
+If we consider the above covariance as a matrix \( C_{ij}=\mathrm{cov}(X_i,\,X_j) \), then the diagonal elements are just the familiar
+variances, \( C_{ii} = \mathrm{cov}(X_i,\,X_i) = \mathrm{var}(X_i) \). It turns out that
+all the off-diagonal elements are zero if the stochastic variables are
+uncorrelated. This is easy to show, keeping in mind the linearity of
+the expectation value. Consider the stochastic variables \( X_i \) and
+\( X_j \), (\( i\neq j \)):
 $$
-p(x) = \mathrm{prob}(X=x)
+\begin{align}
+\mathrm{cov}(X_i,\,X_j) &= \langle(x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
+\tag{7}\\
+&=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle 
+\tag{8}\\
+&=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j\rangle +
+\langle \langle x_i\rangle\langle x_j\rangle\rangle
+\tag{9}\\
+&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle +
+\langle x_i\rangle\langle x_j\rangle
+\tag{10}\\
+&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle
+\tag{11}
+\end{align}
 $$
-
-In the continuous case, the PDF does not directly depict the
-actual probability. Instead we define the probability for the
-stochastic variable to assume any value on an infinitesimal interval
-around \( x \) to be \( p(x)dx \). The continuous function \( p(x) \) then gives us
-the <em>density</em> of the probability rather than the probability
-itself. The probability for a stochastic variable to assume any value
-on a non-infinitesimal interval \( [a,\,b] \) is then just the integral:
-$$
-\mathrm{prob}(a\leq X\leq b) = \int_a^b p(x)dx
-$$
-
-Qualitatively speaking, a stochastic variable represents the values of
-numbers chosen as if by chance from some specified PDF so that the
-selection of a large set of these numbers reproduces this PDF.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs052.html b/doc/pub/Regression/html/._Regression-bs052.html
index 78ef192c7..715ea3015 100644
--- a/doc/pub/Regression/html/._Regression-bs052.html
+++ b/doc/pub/Regression/html/._Regression-bs052.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,26 +383,51 @@ MathJax.Hub.Config({
 <a name="part0052"></a>
 <!-- !split -->
 
-<h2 id="___sec51" class="anchor">Statistics, moments </h2>
-<div class="panel panel-default">
-<div class="panel-body">
-<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-A particularly useful class of special expectation values are the
-<em>moments</em>. The \( n \)-th moment of the PDF \( p \) is defined as
-follows:
+<h2 id="___sec51" class="anchor">Covariance example </h2>
+
+<p>
+Suppose we have defined three vectors \( \hat{x}, \hat{y}, \hat{z} \) with
+\( n \) elements each. The covariance matrix is defined as
+
 $$
-\langle x^n\rangle \equiv \int\! x^n p(x)\,dx
+\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
+                              \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
+                              \sigma_{zx} & \sigma_{zy} & \sigma_{zz}
+             \end{bmatrix},
 $$
 
-The zero-th moment \( \langle 1\rangle \) is just the normalization condition of
-\( p \). The first moment, \( \langle x\rangle \), is called the <em>mean</em> of \( p \)
-and often denoted by the letter \( \mu \):
+where for example
 $$
-\langle x\rangle = \mu \equiv \int\! x p(x)\,dx
+\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
 $$
-</div>
-</div>
 
+<p>
+The Numpy function <b>np.cov</b> calculates the covariance elements using
+the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have
+the exact mean valu\ es.
+
+<p>
+The following simple function uses the <b>np.vstack</b> function which
+takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \)
+matrix \( \hat{W} \)
+
+$$
+\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
+                          x_1 & y_1 & z_1 \\
+                          x_2 & y_2 & z_2 \\
+                          \dots & \dots & \dots \\
+                          x_{n-2} & y_{n-2} & z_{n-2} \\
+                          x_{n-1} & y_{n-1} & z_{n-1}
+             \end{bmatrix},
+$$
+
+<p>
+which in turn is converted into into the \( 3\times 3 \) covariance matrix
+\( \hat{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can
+also calculate the mean value of each set of samples \( \hat{x} \) etc
+using the Numpy function <b>np.mean(x)</b>. We can also extract the
+eigenvalues of the covariance matrix through the <b>np.linalg.eig()</b>
+function.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs053.html b/doc/pub/Regression/html/._Regression-bs053.html
index 597c51e87..108f03bca 100644
--- a/doc/pub/Regression/html/._Regression-bs053.html
+++ b/doc/pub/Regression/html/._Regression-bs053.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,42 +383,42 @@ MathJax.Hub.Config({
 <a name="part0053"></a>
 <!-- !split -->
 
-<h2 id="___sec52" class="anchor">Statistics, central moments </h2>
-<div class="panel panel-default">
-<div class="panel-body">
-<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-A special version of the moments is the set of <em>central moments</em>,
-the n-th central moment defined as:
-$$
-\langle (x-\langle x \rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx
-$$
+<h2 id="___sec52" class="anchor">Covariance in numpy </h2>
 
-The zero-th and first central moments are both trivial, equal \( 1 \) and
-\( 0 \), respectively. But the second central moment, known as the
-<em>variance</em> of \( p \), is of particular interest. For the stochastic
-variable \( X \), the variance is denoted as \( \sigma^2_X \) or \( \mathrm{var}(X) \):
-$$
-\begin{align}
-\sigma^2_X\ \ =\ \ \mathrm{var}(X) & =  \langle (x-\langle x\rangle)^2\rangle =
-\int\! (x-\langle x\rangle)^2 p(x)\,dx
-\tag{8}\\
-& =  \int\! \left(x^2 - 2 x \langle x\rangle^{2} +
-  \langle x\rangle^2\right)p(x)\,dx
-\tag{9}\\
-& =  \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2
-\tag{10}\\
-& =  \langle x^2\rangle - \langle x\rangle^2
-\tag{11}
-\end{align}
-$$
+<p>
 
-The square root of the variance, \( \sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle} \) is called the <em>standard deviation</em> of \( p \). It is clearly just the RMS (root-mean-square)
-value of the deviation of the PDF from its mean value, interpreted
-qualitatively as the <em>spread</em> of \( p \) around its mean.
-</div>
-</div>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
 
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
+<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(x))
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
+<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(y))
+z <span style="color: #666666">=</span> x<span style="color: #666666">**3+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
+<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(z))
+W <span style="color: #666666">=</span> np<span style="color: #666666">.</span>vstack((x, y, z))
+Sigma <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cov(W)
+<span style="color: #008000; font-weight: bold">print</span>(Sigma)
+Eigvals, Eigvecs <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(Sigma)
+<span style="color: #008000; font-weight: bold">print</span>(Eigvals)
+</pre></div>
+<p>
 
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">scipy</span> <span style="color: #008000; font-weight: bold">import</span> sparse
+eye <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">4</span>)
+<span style="color: #008000; font-weight: bold">print</span>(eye)
+sparse_mtx <span style="color: #666666">=</span> sparse<span style="color: #666666">.</span>csr_matrix(eye)
+<span style="color: #008000; font-weight: bold">print</span>(sparse_mtx)
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-10</span>,<span style="color: #666666">10</span>,<span style="color: #666666">100</span>)
+y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sin(x)
+plt<span style="color: #666666">.</span>plot(x,y,marker<span style="color: #666666">=</span><span style="color: #BA2121">&#39;x&#39;</span>)
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs054.html b/doc/pub/Regression/html/._Regression-bs054.html
index 2ca729a79..deb8b2308 100644
--- a/doc/pub/Regression/html/._Regression-bs054.html
+++ b/doc/pub/Regression/html/._Regression-bs054.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,30 +383,25 @@ MathJax.Hub.Config({
 <a name="part0054"></a>
 <!-- !split -->
 
-<h2 id="___sec53" class="anchor">Statistics, covariance </h2>
+<h2 id="___sec53" class="anchor">Statistics, independent variables  </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Another important quantity is the so called covariance, a variant of
-the above defined variance. Consider again the set \( \{X_i\} \) of \( n \)
-stochastic variables (not necessarily uncorrelated) with the
-multivariate PDF \( P(x_1,\dots,x_n) \). The <em>covariance</em> of two
-of the stochastic variables, \( X_i \) and \( X_j \), is defined as follows:
+If \( X_i \) and \( X_j \) are independent, we get 
+\( \langle x_i x_j\rangle =\langle x_i\rangle\langle x_j\rangle \), resulting in \( \mathrm{cov}(X_i, X_j) = 0\ \ (i\neq j) \).
+
+<p>
+Also useful for us is the covariance of linear combinations of
+stochastic variables. Let \( \{X_i\} \) and \( \{Y_i\} \) be two sets of
+stochastic variables. Let also \( \{a_i\} \) and \( \{b_i\} \) be two sets of
+scalars. Consider the linear combination:
 $$
-\begin{align}
-\mathrm{cov}(X_i,\,X_j) &\equiv \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
-\nonumber\\
-&=
-\int\!\cdots\!\int\!(x_i-\langle x_i \rangle)(x_j-\langle x_j \rangle)\,
-P(x_1,\dots,x_n)\,dx_1\dots dx_n
-\tag{12}
-\end{align}
+U = \sum_i a_i X_i \qquad V = \sum_j b_j Y_j
 $$
 
-with
+By the linearity of the expectation value
 $$
-\langle x_i\rangle =
-\int\!\cdots\!\int\!x_i\,P(x_1,\dots,x_n)\,dx_1\dots dx_n
+\mathrm{cov}(U, V) = \sum_{i,j}a_i b_j \mathrm{cov}(X_i, Y_j)
 $$
 </div>
 </div>
diff --git a/doc/pub/Regression/html/._Regression-bs055.html b/doc/pub/Regression/html/._Regression-bs055.html
index d011d3b3b..09eddaf4f 100644
--- a/doc/pub/Regression/html/._Regression-bs055.html
+++ b/doc/pub/Regression/html/._Regression-bs055.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,32 +383,32 @@ MathJax.Hub.Config({
 <a name="part0055"></a>
 <!-- !split -->
 
-<h2 id="___sec54" class="anchor">Statistics, more covariance  </h2>
+<h2 id="___sec54" class="anchor">Statistics, more variance  </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-If we consider the above covariance as a matrix \( C_{ij}=\mathrm{cov}(X_i,\,X_j) \), then the diagonal elements are just the familiar
-variances, \( C_{ii} = \mathrm{cov}(X_i,\,X_i) = \mathrm{var}(X_i) \). It turns out that
-all the off-diagonal elements are zero if the stochastic variables are
-uncorrelated. This is easy to show, keeping in mind the linearity of
-the expectation value. Consider the stochastic variables \( X_i \) and
-\( X_j \), (\( i\neq j \)):
+Now, since the variance is just \( \mathrm{var}(X_i) = \mathrm{cov}(X_i, X_i) \), we get
+the variance of the linear combination \( U = \sum_i a_i X_i \):
 $$
-\begin{align}
-\mathrm{cov}(X_i,\,X_j) &= \langle(x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
-\tag{13}\\
-&=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle 
-\tag{14}\\
-&=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j\rangle +
-\langle \langle x_i\rangle\langle x_j\rangle\rangle
-\tag{15}\\
-&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle +
-\langle x_i\rangle\langle x_j\rangle
-\tag{16}\\
-&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle
-\tag{17}
-\end{align}
+\begin{equation}
+\mathrm{var}(U) = \sum_{i,j}a_i a_j \mathrm{cov}(X_i, X_j)
+\tag{12}
+\end{equation}
 $$
+
+And in the special case when the stochastic variables are
+uncorrelated, the off-diagonal elements of the covariance are as we
+know zero, resulting in:
+$$
+\mathrm{var}(U) = \sum_i a_i^2 \mathrm{cov}(X_i, X_i) = \sum_i a_i^2 \mathrm{var}(X_i)
+$$
+
+$$
+\mathrm{var}(\sum_i a_i X_i) = \sum_i a_i^2 \mathrm{var}(X_i)
+$$
+
+which will become very useful in our study of the error in the mean
+value of a set of measurements.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs056.html b/doc/pub/Regression/html/._Regression-bs056.html
index acbd35880..5ece4bfd0 100644
--- a/doc/pub/Regression/html/._Regression-bs056.html
+++ b/doc/pub/Regression/html/._Regression-bs056.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,51 +383,31 @@ MathJax.Hub.Config({
 <a name="part0056"></a>
 <!-- !split -->
 
-<h2 id="___sec55" class="anchor">Covariance example </h2>
-
-<p>
-Suppose we have defined three vectors \( \hat{x}, \hat{y}, \hat{z} \) with
-\( n \) elements each. The covariance matrix is defined as
-
+<h2 id="___sec55" class="anchor">Statistics and stochastic processes  </h2>
+<div class="panel panel-default">
+<div class="panel-body">
+<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+A <em>stochastic process</em> is a process that produces sequentially a
+chain of values:
 $$
-\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
-                              \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
-                              \sigma_{zx} & \sigma_{zy} & \sigma_{zz}
-             \end{bmatrix},
+\{x_1, x_2,\dots\,x_k,\dots\}.
 $$
 
-where for example
-$$
-\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
-$$
+We will call these
+values our <em>measurements</em> and the entire set as our measured
+<em>sample</em>.  The action of measuring all the elements of a sample
+we will call a stochastic <em>experiment</em> since, operationally,
+they are often associated with results of empirical observation of
+some physical or mathematical phenomena; precisely an experiment. We
+assume that these values are distributed according to some 
+PDF \( p_X^{\phantom X}(x) \), where \( X \) is just the formal symbol for the
+stochastic variable whose PDF is \( p_X^{\phantom X}(x) \). Instead of
+trying to determine the full distribution \( p \) we are often only
+interested in finding the few lowest moments, like the mean
+\( \mu_X^{\phantom X} \) and the variance \( \sigma_X^{\phantom X} \).
+</div>
+</div>
 
-<p>
-The Numpy function <b>np.cov</b> calculates the covariance elements using
-the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have
-the exact mean valu\ es.
-
-<p>
-The following simple function uses the <b>np.vstack</b> function which
-takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \)
-matrix \( \hat{W} \)
-
-$$
-\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
-                          x_1 & y_1 & z_1 \\
-                          x_2 & y_2 & z_2 \\
-                          \dots & \dots & \dots \\
-                          x_{n-2} & y_{n-2} & z_{n-2} \\
-                          x_{n-1} & y_{n-1} & z_{n-1}
-             \end{bmatrix},
-$$
-
-<p>
-which in turn is converted into into the \( 3\times 3 \) covariance matrix
-\( \hat{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can
-also calculate the mean value of each set of samples \( \hat{x} \) etc
-using the Numpy function <b>np.mean(x)</b>. We can also extract the
-eigenvalues of the covariance matrix through the <b>np.linalg.eig()</b>
-function.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs057.html b/doc/pub/Regression/html/._Regression-bs057.html
index e969faf61..01bd4d7b1 100644
--- a/doc/pub/Regression/html/._Regression-bs057.html
+++ b/doc/pub/Regression/html/._Regression-bs057.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,44 +381,32 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0057"></a>
-<!-- !split -->
+<!-- !split  -->
 
-<h2 id="___sec56" class="anchor">Covariance in numpy </h2>
+<h2 id="___sec56" class="anchor">Statistics and sample variables  </h2>
+<div class="panel panel-default">
+<div class="panel-body">
+<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+In practical situations a sample is always of finite size. Let that
+size be \( n \). The expectation value of a sample, the <em>sample mean</em>, is then defined as follows:
+$$
+\bar{x}_n \equiv \frac{1}{n}\sum_{k=1}^n x_k
+$$
 
-<p>
+The <em>sample variance</em> is:
+$$
+\mathrm{var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_k - \bar{x}_n)^2
+$$
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+its square root being the <em>standard deviation of the sample</em>. The
+<em>sample covariance</em> is:
+$$
+\mathrm{cov}(x)\equiv\frac{1}{n}\sum_{kl}(x_k - \bar{x}_n)(x_l - \bar{x}_n)
+$$
+</div>
+</div>
 
-n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
-x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
-<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(x))
-y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
-<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(y))
-z <span style="color: #666666">=</span> x<span style="color: #666666">**3+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
-<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(z))
-W <span style="color: #666666">=</span> np<span style="color: #666666">.</span>vstack((x, y, z))
-Sigma <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cov(W)
-<span style="color: #008000; font-weight: bold">print</span>(Sigma)
-Eigvals, Eigvecs <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(Sigma)
-<span style="color: #008000; font-weight: bold">print</span>(Eigvals)
-</pre></div>
-<p>
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">scipy</span> <span style="color: #008000; font-weight: bold">import</span> sparse
-eye <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">4</span>)
-<span style="color: #008000; font-weight: bold">print</span>(eye)
-sparse_mtx <span style="color: #666666">=</span> sparse<span style="color: #666666">.</span>csr_matrix(eye)
-<span style="color: #008000; font-weight: bold">print</span>(sparse_mtx)
-x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-10</span>,<span style="color: #666666">10</span>,<span style="color: #666666">100</span>)
-y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sin(x)
-plt<span style="color: #666666">.</span>plot(x,y,marker<span style="color: #666666">=</span><span style="color: #BA2121">&#39;x&#39;</span>)
-plt<span style="color: #666666">.</span>show()
-</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs058.html b/doc/pub/Regression/html/._Regression-bs058.html
index b4a0d26a7..280ac6b3d 100644
--- a/doc/pub/Regression/html/._Regression-bs058.html
+++ b/doc/pub/Regression/html/._Regression-bs058.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
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-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
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+              ('Statistics, more variance', 2, None, '___sec54'),
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               ('Statistics, sample variance and covariance',
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+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
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               ('Statistics, more on computations of errors',
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                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
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+               '___sec67'),
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               ('Statistics, effective number of correlations',
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-               '___sec74'),
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                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
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+              ('Assumptions made', 2, None, '___sec72'),
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               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
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               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
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+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
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               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
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+               2,
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+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
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               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,26 +383,21 @@ MathJax.Hub.Config({
 <a name="part0058"></a>
 <!-- !split -->
 
-<h2 id="___sec57" class="anchor">Statistics, independent variables  </h2>
+<h2 id="___sec57" class="anchor">Statistics, sample variance and covariance </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-If \( X_i \) and \( X_j \) are independent, we get 
-\( \langle x_i x_j\rangle =\langle x_i\rangle\langle x_j\rangle \), resulting in \( \mathrm{cov}(X_i, X_j) = 0\ \ (i\neq j) \).
+Note that the sample variance is the sample covariance without the
+cross terms. In a similar manner as the covariance in Eq.&nbsp;<a href="._Regression-bs050.html#mjx-eqn-6">(6)</a> is a measure of the correlation between
+two stochastic variables, the above defined sample covariance is a
+measure of the sequential correlation between succeeding measurements
+of a sample.
 
 <p>
-Also useful for us is the covariance of linear combinations of
-stochastic variables. Let \( \{X_i\} \) and \( \{Y_i\} \) be two sets of
-stochastic variables. Let also \( \{a_i\} \) and \( \{b_i\} \) be two sets of
-scalars. Consider the linear combination:
-$$
-U = \sum_i a_i X_i \qquad V = \sum_j b_j Y_j
-$$
-
-By the linearity of the expectation value
-$$
-\mathrm{cov}(U, V) = \sum_{i,j}a_i b_j \mathrm{cov}(X_i, Y_j)
-$$
+These quantities, being known experimental values, differ
+significantly from and must not be confused with the similarly named
+quantities for stochastic variables, mean \( \mu_X \), variance \( \mathrm{var}(X) \)
+and covariance \( \mathrm{cov}(X,Y) \).
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs059.html b/doc/pub/Regression/html/._Regression-bs059.html
index 8382b0968..bd53f2d50 100644
--- a/doc/pub/Regression/html/._Regression-bs059.html
+++ b/doc/pub/Regression/html/._Regression-bs059.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,32 +383,32 @@ MathJax.Hub.Config({
 <a name="part0059"></a>
 <!-- !split -->
 
-<h2 id="___sec58" class="anchor">Statistics, more variance  </h2>
+<h2 id="___sec58" class="anchor">Statistics, law of large numbers </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Now, since the variance is just \( \mathrm{var}(X_i) = \mathrm{cov}(X_i, X_i) \), we get
-the variance of the linear combination \( U = \sum_i a_i X_i \):
+The law of large numbers
+states that as the size of our sample grows to infinity, the sample
+mean approaches the true mean \( \mu_X^{\phantom X} \) of the chosen PDF:
 $$
-\begin{equation}
-\mathrm{var}(U) = \sum_{i,j}a_i a_j \mathrm{cov}(X_i, X_j)
-\tag{18}
-\end{equation}
+\lim_{n\to\infty}\bar{x}_n = \mu_X^{\phantom X}
 $$
 
-And in the special case when the stochastic variables are
-uncorrelated, the off-diagonal elements of the covariance are as we
-know zero, resulting in:
-$$
-\mathrm{var}(U) = \sum_i a_i^2 \mathrm{cov}(X_i, X_i) = \sum_i a_i^2 \mathrm{var}(X_i)
-$$
+The sample mean \( \bar{x}_n \) works therefore as an estimate of the true
+mean \( \mu_X^{\phantom X} \).
 
-$$
-\mathrm{var}(\sum_i a_i X_i) = \sum_i a_i^2 \mathrm{var}(X_i)
-$$
-
-which will become very useful in our study of the error in the mean
-value of a set of measurements.
+<p>
+What we need to find out is how good an approximation \( \bar{x}_n \) is to
+\( \mu_X^{\phantom X} \). In any stochastic measurement, an estimated
+mean is of no use to us without a measure of its error. A quantity
+that tells us how well we can reproduce it in another experiment. We
+are therefore interested in the PDF of the sample mean itself. Its
+standard deviation will be a measure of the spread of sample means,
+and we will simply call it the <em>error</em> of the sample mean, or
+just sample error, and denote it by \( \mathrm{err}_X^{\phantom X} \). In
+practice, we will only be able to produce an <em>estimate</em> of the
+sample error since the exact value would require the knowledge of the
+true PDFs behind, which we usually do not have.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs060.html b/doc/pub/Regression/html/._Regression-bs060.html
index e91dc22f6..db17155ca 100644
--- a/doc/pub/Regression/html/._Regression-bs060.html
+++ b/doc/pub/Regression/html/._Regression-bs060.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,28 +383,22 @@ MathJax.Hub.Config({
 <a name="part0060"></a>
 <!-- !split -->
 
-<h2 id="___sec59" class="anchor">Statistics and stochastic processes  </h2>
+<h2 id="___sec59" class="anchor">Statistics, more on sample error </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-A <em>stochastic process</em> is a process that produces sequentially a
-chain of values:
+Let us first take a look at what happens to the sample error as the
+size of the sample grows. In a sample, each of the measurements \( x_i \)
+can be associated with its own stochastic variable \( X_i \). The
+stochastic variable \( \overline X_n \) for the sample mean \( \bar{x}_n \) is
+then just a linear combination, already familiar to us:
 $$
-\{x_1, x_2,\dots\,x_k,\dots\}.
+\overline X_n = \frac{1}{n}\sum_{i=1}^n X_i
 $$
 
-We will call these
-values our <em>measurements</em> and the entire set as our measured
-<em>sample</em>.  The action of measuring all the elements of a sample
-we will call a stochastic <em>experiment</em> since, operationally,
-they are often associated with results of empirical observation of
-some physical or mathematical phenomena; precisely an experiment. We
-assume that these values are distributed according to some 
-PDF \( p_X^{\phantom X}(x) \), where \( X \) is just the formal symbol for the
-stochastic variable whose PDF is \( p_X^{\phantom X}(x) \). Instead of
-trying to determine the full distribution \( p \) we are often only
-interested in finding the few lowest moments, like the mean
-\( \mu_X^{\phantom X} \) and the variance \( \sigma_X^{\phantom X} \).
+All the coefficients are just equal \( 1/n \). The PDF of \( \overline X_n \),
+denoted by \( p_{\overline X_n}(x) \) is the desired PDF of the sample
+means.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs061.html b/doc/pub/Regression/html/._Regression-bs061.html
index e820b9387..b7395a27f 100644
--- a/doc/pub/Regression/html/._Regression-bs061.html
+++ b/doc/pub/Regression/html/._Regression-bs061.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,28 +381,23 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0061"></a>
-<!-- !split  -->
+<!-- !split -->
 
-<h2 id="___sec60" class="anchor">Statistics and sample variables  </h2>
+<h2 id="___sec60" class="anchor">Statistics  </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-In practical situations a sample is always of finite size. Let that
-size be \( n \). The expectation value of a sample, the <em>sample mean</em>, is then defined as follows:
+The probability density of obtaining a sample mean \( \bar x_n \)
+is the product of probabilities of obtaining arbitrary values \( x_1,
+x_2,\dots,x_n \) with the constraint that the mean of the set \( \{x_i\} \)
+is \( \bar x_n \):
 $$
-\bar{x}_n \equiv \frac{1}{n}\sum_{k=1}^n x_k
+p_{\overline X_n}(x) = \int p_X^{\phantom X}(x_1)\cdots
+\int p_X^{\phantom X}(x_n)\ 
+\delta\!\left(x - \frac{x_1+x_2+\dots+x_n}{n}\right)dx_n \cdots dx_1
 $$
 
-The <em>sample variance</em> is:
-$$
-\mathrm{var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_k - \bar{x}_n)^2
-$$
-
-its square root being the <em>standard deviation of the sample</em>. The
-<em>sample covariance</em> is:
-$$
-\mathrm{cov}(x)\equiv\frac{1}{n}\sum_{kl}(x_k - \bar{x}_n)(x_l - \bar{x}_n)
-$$
+And in particular we are interested in its variance \( \mathrm{var}(\overline X_n) \).
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs062.html b/doc/pub/Regression/html/._Regression-bs062.html
index 4fda2d83b..34493f2e5 100644
--- a/doc/pub/Regression/html/._Regression-bs062.html
+++ b/doc/pub/Regression/html/._Regression-bs062.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,21 +383,25 @@ MathJax.Hub.Config({
 <a name="part0062"></a>
 <!-- !split -->
 
-<h2 id="___sec61" class="anchor">Statistics, sample variance and covariance </h2>
+<h2 id="___sec61" class="anchor">Statistics, central limit theorem </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Note that the sample variance is the sample covariance without the
-cross terms. In a similar manner as the covariance in Eq.&nbsp;<a href="._Regression-bs054.html#mjx-eqn-12">(12)</a> is a measure of the correlation between
-two stochastic variables, the above defined sample covariance is a
-measure of the sequential correlation between succeeding measurements
-of a sample.
-
-<p>
-These quantities, being known experimental values, differ
-significantly from and must not be confused with the similarly named
-quantities for stochastic variables, mean \( \mu_X \), variance \( \mathrm{var}(X) \)
-and covariance \( \mathrm{cov}(X,Y) \).
+It is generally not possible to express \( p_{\overline X_n}(x) \) in a
+closed form given an arbitrary PDF \( p_X^{\phantom X} \) and a number
+\( n \). But for the limit \( n\to\infty \) it is possible to make an
+approximation. The very important result is called <em>the central limit theorem</em>. It tells us that as \( n \) goes to infinity,
+\( p_{\overline X_n}(x) \) approaches a Gaussian distribution whose mean
+and variance equal the true mean and variance, \( \mu_{X}^{\phantom X} \)
+and \( \sigma_{X}^{2} \), respectively:
+$$
+\begin{equation}
+\lim_{n\to\infty} p_{\overline X_n}(x) =
+\left(\frac{n}{2\pi\mathrm{var}(X)}\right)^{1/2}
+e^{-\frac{n(x-\bar x_n)^2}{2\mathrm{var}(X)}}
+\tag{13}
+\end{equation}
+$$
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs063.html b/doc/pub/Regression/html/._Regression-bs063.html
index 4248aec4f..533907a24 100644
--- a/doc/pub/Regression/html/._Regression-bs063.html
+++ b/doc/pub/Regression/html/._Regression-bs063.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,32 +383,29 @@ MathJax.Hub.Config({
 <a name="part0063"></a>
 <!-- !split -->
 
-<h2 id="___sec62" class="anchor">Statistics, law of large numbers </h2>
+<h2 id="___sec62" class="anchor">Statistics, more technicalities  </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-The law of large numbers
-states that as the size of our sample grows to infinity, the sample
-mean approaches the true mean \( \mu_X^{\phantom X} \) of the chosen PDF:
+The desired variance
+\( \mathrm{var}(\overline X_n) \), i.e. the sample error squared
+\( \mathrm{err}_X^2 \), is given by:
 $$
-\lim_{n\to\infty}\bar{x}_n = \mu_X^{\phantom X}
+\begin{equation}
+\mathrm{err}_X^2 = \mathrm{var}(\overline X_n) = \frac{1}{n^2}
+\sum_{ij} \mathrm{cov}(X_i, X_j)
+\tag{14}
+\end{equation}
 $$
 
-The sample mean \( \bar{x}_n \) works therefore as an estimate of the true
-mean \( \mu_X^{\phantom X} \).
-
-<p>
-What we need to find out is how good an approximation \( \bar{x}_n \) is to
-\( \mu_X^{\phantom X} \). In any stochastic measurement, an estimated
-mean is of no use to us without a measure of its error. A quantity
-that tells us how well we can reproduce it in another experiment. We
-are therefore interested in the PDF of the sample mean itself. Its
-standard deviation will be a measure of the spread of sample means,
-and we will simply call it the <em>error</em> of the sample mean, or
-just sample error, and denote it by \( \mathrm{err}_X^{\phantom X} \). In
-practice, we will only be able to produce an <em>estimate</em> of the
-sample error since the exact value would require the knowledge of the
-true PDFs behind, which we usually do not have.
+We see now that in order to calculate the exact error of the sample
+with the above expression, we would need the true means
+\( \mu_{X_i}^{\phantom X} \) of the stochastic variables \( X_i \). To
+calculate these requires that we know the true multivariate PDF of all
+the \( X_i \). But this PDF is unknown to us, we have only got the measurements of
+one sample. The best we can do is to let the sample itself be an
+estimate of the PDF of each of the \( X_i \), estimating all properties of
+\( X_i \) through the measurements of the sample.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs064.html b/doc/pub/Regression/html/._Regression-bs064.html
index 1c1b07735..71b4a66b1 100644
--- a/doc/pub/Regression/html/._Regression-bs064.html
+++ b/doc/pub/Regression/html/._Regression-bs064.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,22 +383,27 @@ MathJax.Hub.Config({
 <a name="part0064"></a>
 <!-- !split -->
 
-<h2 id="___sec63" class="anchor">Statistics, more on sample error </h2>
+<h2 id="___sec63" class="anchor">Statistics </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Let us first take a look at what happens to the sample error as the
-size of the sample grows. In a sample, each of the measurements \( x_i \)
-can be associated with its own stochastic variable \( X_i \). The
-stochastic variable \( \overline X_n \) for the sample mean \( \bar{x}_n \) is
-then just a linear combination, already familiar to us:
+Our estimate of \( \mu_{X_i}^{\phantom X} \) is then the sample mean \( \bar x \)
+itself, in accordance with the the central limit theorem:
 $$
-\overline X_n = \frac{1}{n}\sum_{i=1}^n X_i
+\mu_{X_i}^{\phantom X} = \langle x_i\rangle \approx \frac{1}{n}\sum_{k=1}^n x_k = \bar x
 $$
 
-All the coefficients are just equal \( 1/n \). The PDF of \( \overline X_n \),
-denoted by \( p_{\overline X_n}(x) \) is the desired PDF of the sample
-means.
+Using \( \bar x \) in place of \( \mu_{X_i}^{\phantom X} \) we can give an
+<em>estimate</em> of the covariance in Eq.&nbsp;<a href="._Regression-bs063.html#mjx-eqn-14">(14)</a>
+$$
+\mathrm{cov}(X_i, X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
+\approx\langle (x_i - \bar x)(x_j - \bar{x})\rangle,
+$$
+
+resulting in
+$$ 
+\frac{1}{n} \sum_{l}^n \left(\frac{1}{n}\sum_{k}^n (x_k -\bar x_n)(x_l - \bar x_n)\right)=\frac{1}{n}\frac{1}{n} \sum_{kl} (x_k -\bar x_n)(x_l - \bar x_n)=\frac{1}{n}\mathrm{cov}(x)
+$$
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs065.html b/doc/pub/Regression/html/._Regression-bs065.html
index f779adecb..552e00de3 100644
--- a/doc/pub/Regression/html/._Regression-bs065.html
+++ b/doc/pub/Regression/html/._Regression-bs065.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,21 +383,39 @@ MathJax.Hub.Config({
 <a name="part0065"></a>
 <!-- !split -->
 
-<h2 id="___sec64" class="anchor">Statistics  </h2>
+<h2 id="___sec64" class="anchor">Statistics and sample variance  </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-The probability density of obtaining a sample mean \( \bar x_n \)
-is the product of probabilities of obtaining arbitrary values \( x_1,
-x_2,\dots,x_n \) with the constraint that the mean of the set \( \{x_i\} \)
-is \( \bar x_n \):
+By the same procedure we can use the sample variance as an
+estimate of the variance of any of the stochastic variables \( X_i \)
 $$
-p_{\overline X_n}(x) = \int p_X^{\phantom X}(x_1)\cdots
-\int p_X^{\phantom X}(x_n)\ 
-\delta\!\left(x - \frac{x_1+x_2+\dots+x_n}{n}\right)dx_n \cdots dx_1
+\mathrm{var}(X_i)=\langle x_i - \langle x_i\rangle\rangle \approx \langle x_i - \bar x_n\rangle\nonumber,
 $$
 
-And in particular we are interested in its variance \( \mathrm{var}(\overline X_n) \).
+which is approximated as 
+$$
+\begin{equation}
+\mathrm{var}(X_i)\approx \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)=\mathrm{var}(x)
+\tag{15}
+\end{equation}
+$$
+
+<p>
+Now we can calculate an estimate of the error
+\( \mathrm{err}_X^{\phantom X} \) of the sample mean \( \bar x_n \):
+$$
+\begin{align}
+\mathrm{err}_X^2
+&=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) \nonumber \\
+&\approx&\frac{1}{n^2}\sum_{ij}\frac{1}{n}\mathrm{cov}(x) =\frac{1}{n^2}n^2\frac{1}{n}\mathrm{cov}(x)\nonumber\\
+&=\frac{1}{n}\mathrm{cov}(x)
+\tag{16}
+\end{align}
+$$
+
+which is nothing but the sample covariance divided by the number of
+measurements in the sample.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs066.html b/doc/pub/Regression/html/._Regression-bs066.html
index d12754001..89800e2fa 100644
--- a/doc/pub/Regression/html/._Regression-bs066.html
+++ b/doc/pub/Regression/html/._Regression-bs066.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,25 +383,35 @@ MathJax.Hub.Config({
 <a name="part0066"></a>
 <!-- !split -->
 
-<h2 id="___sec65" class="anchor">Statistics, central limit theorem </h2>
+<h2 id="___sec65" class="anchor">Statistics, uncorrelated results </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-It is generally not possible to express \( p_{\overline X_n}(x) \) in a
-closed form given an arbitrary PDF \( p_X^{\phantom X} \) and a number
-\( n \). But for the limit \( n\to\infty \) it is possible to make an
-approximation. The very important result is called <em>the central limit theorem</em>. It tells us that as \( n \) goes to infinity,
-\( p_{\overline X_n}(x) \) approaches a Gaussian distribution whose mean
-and variance equal the true mean and variance, \( \mu_{X}^{\phantom X} \)
-and \( \sigma_{X}^{2} \), respectively:
+
+<p>
+In the special case that the measurements of the sample are
+uncorrelated (equivalently the stochastic variables \( X_i \) are
+uncorrelated) we have that the off-diagonal elements of the covariance
+are zero. This gives the following estimate of the sample error:
+$$
+\mathrm{err}_X^2=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) =
+\frac{1}{n^2} \sum_i \mathrm{var}(X_i),
+$$
+
+resulting in
 $$
 \begin{equation}
-\lim_{n\to\infty} p_{\overline X_n}(x) =
-\left(\frac{n}{2\pi\mathrm{var}(X)}\right)^{1/2}
-e^{-\frac{n(x-\bar x_n)^2}{2\mathrm{var}(X)}}
-\tag{19}
+\mathrm{err}_X^2\approx \frac{1}{n^2} \sum_i \mathrm{var}(x)= \frac{1}{n}\mathrm{var}(x)
+\tag{17}
 \end{equation}
 $$
+
+where in the second step we have used Eq.&nbsp;<a href="._Regression-bs065.html#mjx-eqn-15">(15)</a>.
+The error of the sample is then just its standard deviation divided by
+the square root of the number of measurements the sample contains.
+This is a very useful formula which is easy to compute. It acts as a
+first approximation to the error, but in numerical experiments, we
+cannot overlook the always present correlations.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs067.html b/doc/pub/Regression/html/._Regression-bs067.html
index ba2cdc20b..f94ac7265 100644
--- a/doc/pub/Regression/html/._Regression-bs067.html
+++ b/doc/pub/Regression/html/._Regression-bs067.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,29 +383,29 @@ MathJax.Hub.Config({
 <a name="part0067"></a>
 <!-- !split -->
 
-<h2 id="___sec66" class="anchor">Statistics, more technicalities  </h2>
+<h2 id="___sec66" class="anchor">Statistics, computations </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-The desired variance
-\( \mathrm{var}(\overline X_n) \), i.e. the sample error squared
-\( \mathrm{err}_X^2 \), is given by:
+For computational purposes one usually splits up the estimate of
+\( \mathrm{err}_X^2 \), given by Eq.&nbsp;<a href="._Regression-bs065.html#mjx-eqn-16">(16)</a>, into two
+parts
+$$
+\mathrm{err}_X^2 = \frac{1}{n}\mathrm{var}(x) + \frac{1}{n}(\mathrm{cov}(x)-\mathrm{var}(x)),
+$$
+
+which equals
 $$
 \begin{equation}
-\mathrm{err}_X^2 = \mathrm{var}(\overline X_n) = \frac{1}{n^2}
-\sum_{ij} \mathrm{cov}(X_i, X_j)
-\tag{20}
+\frac{1}{n^2}\sum_{k=1}^n (x_k - \bar x_n)^2 +\frac{2}{n^2}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n)
+\tag{18}
 \end{equation}
 $$
 
-We see now that in order to calculate the exact error of the sample
-with the above expression, we would need the true means
-\( \mu_{X_i}^{\phantom X} \) of the stochastic variables \( X_i \). To
-calculate these requires that we know the true multivariate PDF of all
-the \( X_i \). But this PDF is unknown to us, we have only got the measurements of
-one sample. The best we can do is to let the sample itself be an
-estimate of the PDF of each of the \( X_i \), estimating all properties of
-\( X_i \) through the measurements of the sample.
+The first term is the same as the error in the uncorrelated case,
+Eq.&nbsp;<a href="._Regression-bs066.html#mjx-eqn-17">(17)</a>. This means that the second
+term accounts for the error correction due to correlation between the
+measurements. For uncorrelated measurements this second term is zero.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs068.html b/doc/pub/Regression/html/._Regression-bs068.html
index a53e75292..97ff76be5 100644
--- a/doc/pub/Regression/html/._Regression-bs068.html
+++ b/doc/pub/Regression/html/._Regression-bs068.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,27 +383,22 @@ MathJax.Hub.Config({
 <a name="part0068"></a>
 <!-- !split -->
 
-<h2 id="___sec67" class="anchor">Statistics </h2>
+<h2 id="___sec67" class="anchor">Statistics, more on computations of errors </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Our estimate of \( \mu_{X_i}^{\phantom X} \) is then the sample mean \( \bar x \)
-itself, in accordance with the the central limit theorem:
+Computationally the uncorrelated first term is much easier to treat
+efficiently than the second.
 $$
-\mu_{X_i}^{\phantom X} = \langle x_i\rangle \approx \frac{1}{n}\sum_{k=1}^n x_k = \bar x
+\mathrm{var}(x) = \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)^2 =
+\left(\frac{1}{n}\sum_{k=1}^n x_k^2\right) - \bar x_n^2
 $$
 
-Using \( \bar x \) in place of \( \mu_{X_i}^{\phantom X} \) we can give an
-<em>estimate</em> of the covariance in Eq.&nbsp;<a href="._Regression-bs067.html#mjx-eqn-20">(20)</a>
-$$
-\mathrm{cov}(X_i, X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
-\approx\langle (x_i - \bar x)(x_j - \bar{x})\rangle,
-$$
-
-resulting in
-$$ 
-\frac{1}{n} \sum_{l}^n \left(\frac{1}{n}\sum_{k}^n (x_k -\bar x_n)(x_l - \bar x_n)\right)=\frac{1}{n}\frac{1}{n} \sum_{kl} (x_k -\bar x_n)(x_l - \bar x_n)=\frac{1}{n}\mathrm{cov}(x)
-$$
+We just accumulate separately the values \( x^2 \) and \( x \) for every
+measurement \( x \) we receive. The correlation term, though, has to be
+calculated at the end of the experiment since we need all the
+measurements to calculate the cross terms. Therefore, all measurements
+have to be stored throughout the experiment.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs069.html b/doc/pub/Regression/html/._Regression-bs069.html
index 45900b23e..f7e050e27 100644
--- a/doc/pub/Regression/html/._Regression-bs069.html
+++ b/doc/pub/Regression/html/._Regression-bs069.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,39 +383,33 @@ MathJax.Hub.Config({
 <a name="part0069"></a>
 <!-- !split -->
 
-<h2 id="___sec68" class="anchor">Statistics and sample variance  </h2>
+<h2 id="___sec68" class="anchor">Statistics, wrapping up 1 </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-By the same procedure we can use the sample variance as an
-estimate of the variance of any of the stochastic variables \( X_i \)
+Let us analyze the problem by splitting up the correlation term into
+partial sums of the form:
 $$
-\mathrm{var}(X_i)=\langle x_i - \langle x_i\rangle\rangle \approx \langle x_i - \bar x_n\rangle\nonumber,
+f_d = \frac{1}{n-d}\sum_{k=1}^{n-d}(x_k - \bar x_n)(x_{k+d} - \bar x_n)
 $$
 
-which is approximated as 
+The correlation term of the error can now be rewritten in terms of
+\( f_d \)
 $$
-\begin{equation}
-\mathrm{var}(X_i)\approx \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)=\mathrm{var}(x)
-\tag{21}
-\end{equation}
+\frac{2}{n}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n) =
+2\sum_{d=1}^{n-1} f_d
 $$
 
-<p>
-Now we can calculate an estimate of the error
-\( \mathrm{err}_X^{\phantom X} \) of the sample mean \( \bar x_n \):
+The value of \( f_d \) reflects the correlation between measurements
+separated by the distance \( d \) in the sample samples.  Notice that for
+\( d=0 \), \( f \) is just the sample variance, \( \mathrm{var}(x) \). If we divide \( f_d \)
+by \( \mathrm{var}(x) \), we arrive at the so called <em>autocorrelation function</em>
 $$
-\begin{align}
-\mathrm{err}_X^2
-&=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) \nonumber \\
-&\approx&\frac{1}{n^2}\sum_{ij}\frac{1}{n}\mathrm{cov}(x) =\frac{1}{n^2}n^2\frac{1}{n}\mathrm{cov}(x)\nonumber\\
-&=\frac{1}{n}\mathrm{cov}(x)
-\tag{22}
-\end{align}
+\kappa_d = \frac{f_d}{\mathrm{var}(x)}
 $$
 
-which is nothing but the sample covariance divided by the number of
-measurements in the sample.
+which gives us a useful measure of pairwise correlations
+starting always at \( 1 \) for \( d=0 \).
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs070.html b/doc/pub/Regression/html/._Regression-bs070.html
index b5c9f9812..bd0dcd1f0 100644
--- a/doc/pub/Regression/html/._Regression-bs070.html
+++ b/doc/pub/Regression/html/._Regression-bs070.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,35 +383,33 @@ MathJax.Hub.Config({
 <a name="part0070"></a>
 <!-- !split -->
 
-<h2 id="___sec69" class="anchor">Statistics, uncorrelated results </h2>
+<h2 id="___sec69" class="anchor">Statistics, final expression </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-
-<p>
-In the special case that the measurements of the sample are
-uncorrelated (equivalently the stochastic variables \( X_i \) are
-uncorrelated) we have that the off-diagonal elements of the covariance
-are zero. This gives the following estimate of the sample error:
+The sample error (see eq.&nbsp;<a href="._Regression-bs067.html#mjx-eqn-18">(18)</a>) can now be
+written in terms of the autocorrelation function:
 $$
-\mathrm{err}_X^2=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) =
-\frac{1}{n^2} \sum_i \mathrm{var}(X_i),
+\begin{align}
+\mathrm{err}_X^2 &=
+\frac{1}{n}\mathrm{var}(x)+\frac{2}{n}\cdot\mathrm{var}(x)\sum_{d=1}^{n-1}
+\frac{f_d}{\mathrm{var}(x)}\nonumber\\ &=&
+\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\mathrm{var}(x)\nonumber\\
+&=\frac{\tau}{n}\cdot\mathrm{var}(x)
+\tag{19}
+\end{align}
 $$
 
-resulting in
+and we see that \( \mathrm{err}_X \) can be expressed in terms the
+uncorrelated sample variance times a correction factor \( \tau \) which
+accounts for the correlation between measurements. We call this
+correction factor the <em>autocorrelation time</em>:
 $$
 \begin{equation}
-\mathrm{err}_X^2\approx \frac{1}{n^2} \sum_i \mathrm{var}(x)= \frac{1}{n}\mathrm{var}(x)
-\tag{23}
+\tau = 1+2\sum_{d=1}^{n-1}\kappa_d
+\tag{20}
 \end{equation}
 $$
-
-where in the second step we have used Eq.&nbsp;<a href="._Regression-bs069.html#mjx-eqn-21">(21)</a>.
-The error of the sample is then just its standard deviation divided by
-the square root of the number of measurements the sample contains.
-This is a very useful formula which is easy to compute. It acts as a
-first approximation to the error, but in numerical experiments, we
-cannot overlook the always present correlations.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs071.html b/doc/pub/Regression/html/._Regression-bs071.html
index adf7712e5..c8cada5d7 100644
--- a/doc/pub/Regression/html/._Regression-bs071.html
+++ b/doc/pub/Regression/html/._Regression-bs071.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
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-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
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+              ('Statistics', 2, None, '___sec46'),
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+              ('Statistics, central moments', 2, None, '___sec48'),
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+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
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+              ('Statistics, more variance', 2, None, '___sec54'),
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               ('Statistics, sample variance and covariance',
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-              ('Statistics, law of large numbers', 2, None, '___sec62'),
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+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
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               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
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               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,29 +383,25 @@ MathJax.Hub.Config({
 <a name="part0071"></a>
 <!-- !split -->
 
-<h2 id="___sec70" class="anchor">Statistics, computations </h2>
+<h2 id="___sec70" class="anchor">Statistics, effective number of correlations </h2>
 <div class="panel panel-default">
 <div class="panel-body">
 <p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-For computational purposes one usually splits up the estimate of
-\( \mathrm{err}_X^2 \), given by Eq.&nbsp;<a href="._Regression-bs069.html#mjx-eqn-22">(22)</a>, into two
-parts
+For a correlation free experiment, \( \tau \)
+equals 1. From the point of view of
+eq.&nbsp;<a href="._Regression-bs070.html#mjx-eqn-19">(19)</a> we can interpret a sequential
+correlation as an effective reduction of the number of measurements by
+a factor \( \tau \). The effective number of measurements becomes:
 $$
-\mathrm{err}_X^2 = \frac{1}{n}\mathrm{var}(x) + \frac{1}{n}(\mathrm{cov}(x)-\mathrm{var}(x)),
+n_\mathrm{eff} = \frac{n}{\tau}
 $$
 
-which equals
-$$
-\begin{equation}
-\frac{1}{n^2}\sum_{k=1}^n (x_k - \bar x_n)^2 +\frac{2}{n^2}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n)
-\tag{24}
-\end{equation}
-$$
-
-The first term is the same as the error in the uncorrelated case,
-Eq.&nbsp;<a href="._Regression-bs070.html#mjx-eqn-23">(23)</a>. This means that the second
-term accounts for the error correction due to correlation between the
-measurements. For uncorrelated measurements this second term is zero.
+To neglect the autocorrelation time \( \tau \) will always cause our
+simple uncorrelated estimate of \( \mathrm{err}_X^2\approx \mathrm{var}(x)/n \) to
+be less than the true sample error. The estimate of the error will be
+too <em>good</em>. On the other hand, the calculation of the full
+autocorrelation time poses an efficiency problem if the set of
+measurements is very large.
 </div>
 </div>
 
diff --git a/doc/pub/Regression/html/._Regression-bs072.html b/doc/pub/Regression/html/._Regression-bs072.html
index aa8990750..07713ece8 100644
--- a/doc/pub/Regression/html/._Regression-bs072.html
+++ b/doc/pub/Regression/html/._Regression-bs072.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,27 +381,42 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0072"></a>
-<!-- !split -->
+<!-- !split  -->
 
-<h2 id="___sec71" class="anchor">Statistics, more on computations of errors </h2>
-<div class="panel panel-default">
-<div class="panel-body">
-<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Computationally the uncorrelated first term is much easier to treat
-efficiently than the second.
+<h2 id="___sec71" class="anchor">Linking the regression analysis with a statistical interpretation </h2>
+
+<p>
+Finally, we are going to discuss several statistical properties which can be obtained in terms of analytical expressions. 
+The
+advantage of doing linear regression is that we actually end up with
+analytical expressions for several statistical quantities.  
+Standard least squares and Ridge regression  allow us to
+derive quantities like the variance and other expectation values in a
+rather straightforward way.
+
+<p>
+It is assumed that \( \varepsilon_i
+\sim \mathcal{N}(0, \sigma^2) \) and the \( \varepsilon_{i} \) are
+independent, i.e.: 
 $$
-\mathrm{var}(x) = \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)^2 =
-\left(\frac{1}{n}\sum_{k=1}^n x_k^2\right) - \bar x_n^2
+\begin{align*} 
+\mbox{Cov}(\varepsilon_{i_1},
+\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
+& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2.  \end{array} \right.
+\end{align*} 
 $$
 
-We just accumulate separately the values \( x^2 \) and \( x \) for every
-measurement \( x \) we receive. The correlation term, though, has to be
-calculated at the end of the experiment since we need all the
-measurements to calculate the cross terms. Therefore, all measurements
-have to be stored throughout the experiment.
-</div>
-</div>
+The randomness of \( \varepsilon_i \) implies that
+\( \mathbf{y}_i \) is also a random variable. In particular,
+\( \mathbf{y}_i \) is normally distributed, because \( \varepsilon_i \sim
+\mathcal{N}(0, \sigma^2) \) and \( \mathbf{X}_{i,\ast} \, \boldsymbol{\beta} \) is a
+non-random scalar. To specify the parameters of the distribution of
+\( \mathbf{y}_i \) we need to calculate its first two moments.
 
+<p>
+Recall that \( \boldsymbol{X} \) is a matrix of dimensionality \( n\times p \). The
+notation above \( \mathbf{X}_{i,\ast} \) means that we are looking at the
+row number \( i \) and perform a sum over all values \( p \).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs073.html b/doc/pub/Regression/html/._Regression-bs073.html
index 541bac035..ebcb6cf6c 100644
--- a/doc/pub/Regression/html/._Regression-bs073.html
+++ b/doc/pub/Regression/html/._Regression-bs073.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,37 +383,23 @@ MathJax.Hub.Config({
 <a name="part0073"></a>
 <!-- !split -->
 
-<h2 id="___sec72" class="anchor">Statistics, wrapping up 1 </h2>
-<div class="panel panel-default">
-<div class="panel-body">
-<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Let us analyze the problem by splitting up the correlation term into
-partial sums of the form:
+<h2 id="___sec72" class="anchor">Assumptions made </h2>
+
+<p>
+The assumption we have made here can be summarized as (and this is going to useful when we discuss the bias-variance trade off)
+that there exists a function \( f(\boldsymbol{x}) \) and  a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \)
+which describes our data
 $$
-f_d = \frac{1}{n-d}\sum_{k=1}^{n-d}(x_k - \bar x_n)(x_{k+d} - \bar x_n)
+\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
 $$
 
-The correlation term of the error can now be rewritten in terms of
-\( f_d \)
+<p>
+We approximate this function with our model from the solution of the linear regression equations, that is our
+function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we want to minimize \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), our MSE, with
 $$
-\frac{2}{n}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n) =
-2\sum_{d=1}^{n-1} f_d
+\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
 $$
 
-The value of \( f_d \) reflects the correlation between measurements
-separated by the distance \( d \) in the sample samples.  Notice that for
-\( d=0 \), \( f \) is just the sample variance, \( \mathrm{var}(x) \). If we divide \( f_d \)
-by \( \mathrm{var}(x) \), we arrive at the so called <em>autocorrelation function</em>
-$$
-\kappa_d = \frac{f_d}{\mathrm{var}(x)}
-$$
-
-which gives us a useful measure of pairwise correlations
-starting always at \( 1 \) for \( d=0 \).
-</div>
-</div>
-
-
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs074.html b/doc/pub/Regression/html/._Regression-bs074.html
index 4fe97e6f0..0732b2c11 100644
--- a/doc/pub/Regression/html/._Regression-bs074.html
+++ b/doc/pub/Regression/html/._Regression-bs074.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,36 +383,37 @@ MathJax.Hub.Config({
 <a name="part0074"></a>
 <!-- !split -->
 
-<h2 id="___sec73" class="anchor">Statistics, final expression </h2>
-<div class="panel panel-default">
-<div class="panel-body">
-<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-The sample error (see eq.&nbsp;<a href="._Regression-bs071.html#mjx-eqn-24">(24)</a>) can now be
-written in terms of the autocorrelation function:
+<h2 id="___sec73" class="anchor">Expectation value and variance </h2>
+
+<p>
+We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) 
 $$
-\begin{align}
-\mathrm{err}_X^2 &=
-\frac{1}{n}\mathrm{var}(x)+\frac{2}{n}\cdot\mathrm{var}(x)\sum_{d=1}^{n-1}
-\frac{f_d}{\mathrm{var}(x)}\nonumber\\ &=&
-\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\mathrm{var}(x)\nonumber\\
-&=\frac{\tau}{n}\cdot\mathrm{var}(x)
-\tag{25}
-\end{align}
+\begin{align*} 
+\mathbb{E}(y_i) & =
+\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
+\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, 
+\end{align*} 
 $$
 
-and we see that \( \mathrm{err}_X \) can be expressed in terms the
-uncorrelated sample variance times a correction factor \( \tau \) which
-accounts for the correlation between measurements. We call this
-correction factor the <em>autocorrelation time</em>:
+while
+its variance is 
 $$
-\begin{equation}
-\tau = 1+2\sum_{d=1}^{n-1}\kappa_d
-\tag{26}
-\end{equation}
+\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i
+- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
+[\mathbb{E}(y_i)]^2  \\  & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
+\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &
+= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
+\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
+\ast} \, \beta)^2 \\  & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
+\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
+\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 
+\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
+\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.  
+\end{align*}
 $$
-</div>
-</div>
 
+Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with 
+mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (not be confused with the singular values of the SVD).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs075.html b/doc/pub/Regression/html/._Regression-bs075.html
index faf8babc4..0af6f2aa2 100644
--- a/doc/pub/Regression/html/._Regression-bs075.html
+++ b/doc/pub/Regression/html/._Regression-bs075.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,28 +383,87 @@ MathJax.Hub.Config({
 <a name="part0075"></a>
 <!-- !split -->
 
-<h2 id="___sec74" class="anchor">Statistics, effective number of correlations </h2>
-<div class="panel panel-default">
-<div class="panel-body">
-<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-For a correlation free experiment, \( \tau \)
-equals 1. From the point of view of
-eq.&nbsp;<a href="._Regression-bs074.html#mjx-eqn-25">(25)</a> we can interpret a sequential
-correlation as an effective reduction of the number of measurements by
-a factor \( \tau \). The effective number of measurements becomes:
+<h2 id="___sec74" class="anchor">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
+
+<p>
+With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
 $$
-n_\mathrm{eff} = \frac{n}{\tau}
+\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
 $$
 
-To neglect the autocorrelation time \( \tau \) will always cause our
-simple uncorrelated estimate of \( \mathrm{err}_X^2\approx \mathrm{var}(x)/n \) to
-be less than the true sample error. The estimate of the error will be
-too <em>good</em>. On the other hand, the calculation of the full
-autocorrelation time poses an efficiency problem if the set of
-measurements is very large.
-</div>
-</div>
+This means that the estimator of the regression parameters is unbiased.
 
+<p>
+We can also calculate the variance
+
+<p>
+The variance of \( \boldsymbol{\beta} \) is
+$$
+\begin{eqnarray*}
+\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
+\\
+& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \}
+\\
+% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1}  \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+\\
+& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \,  \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+% \\
+% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \,  \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
+% \\
+% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T  \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
+\\
+& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}  + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
+\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
+\end{eqnarray*}
+$$
+
+<p>
+where we have used  that \( \mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) =
+\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} +
+\sigma^2 \, \mathbf{I}_{nn} \). From \( \mbox{Var}(\boldsymbol{\beta}) = \sigma^2
+\, (\mathbf{X}^{T} \mathbf{X})^{-1} \), one obtains an estimate of the
+variance of the estimate of the \( j \)-th regression coefficient:
+\( \hat{\sigma}^2 (\hat{\beta}_j ) = \hat{\sigma}^2 \sqrt{
+[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} } \). This may be used to
+construct a confidence interval for the estimates.
+
+<p>
+In a similar way, we cna obtain analytical expressions for say the
+expectation values of the parameters \( \boldsymbol{\beta} \) and their variance
+when we employ Ridge regression, and thereby a confidence interval.
+
+<p>
+It is rather straightforward to show that
+$$
+\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}.
+$$
+
+We see clearly that 
+\( \mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}} \) for any \( \lambda > 0 \). We say then that the ridge estimator is biased.
+
+<p>
+We can also compute the variance as 
+
+$$
+\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
+$$
+
+and it is easy to see that if the parameter \( \lambda \) goes to infinity then the variance of Ridge parameters \( \boldsymbol{\beta} \) goes to zero.
+
+<p>
+With this, we can compute the difference 
+
+$$
+\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
+$$
+
+The difference is non-negative definite since each component of the
+matrix product is non-negative definite. 
+This means the variance we obtain with the standard OLS will always for \( \lambda > 0 \) be larger than the variance of \( \boldsymbol{\beta} \) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs076.html b/doc/pub/Regression/html/._Regression-bs076.html
index a2a888cab..435769825 100644
--- a/doc/pub/Regression/html/._Regression-bs076.html
+++ b/doc/pub/Regression/html/._Regression-bs076.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
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-              ('Another Example', 2, None, '___sec38'),
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-              ('Mathematical Properties', 2, None, '___sec40'),
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-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
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                None,
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-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
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+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
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+              ('Statistics and sample variables', 2, None, '___sec56'),
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+              ('Statistics, more on sample error', 2, None, '___sec59'),
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+              ('Assumptions made', 2, None, '___sec72'),
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+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,40 +383,36 @@ MathJax.Hub.Config({
 <a name="part0076"></a>
 <!-- !split  -->
 
-<h2 id="___sec75" class="anchor">Linking the regression analysis with a statistical interpretation </h2>
+<h2 id="___sec75" class="anchor">Cross-validation </h2>
 
 <p>
-Finally, we are going to discuss several statistical properties which can be obtained in terms of analytical expressions. 
-The
-advantage of doing linear regression is that we actually end up with
-analytical expressions for several statistical quantities.  
-Standard least squares and Ridge regression  allow us to
-derive quantities like the variance and other expectation values in a
-rather straightforward way.
+Instead of choosing the penalty parameter to balance model fit with
+model complexity, cross-validation requires it (i.e. the penalty
+parameter) to yield a model with good prediction
+performance. Commonly, this performance is evaluated on novel
+data. Novel data need not be easy to come by and one has to make do
+with the data at hand.
 
 <p>
-It is assumed that \( \varepsilon_i
-\sim \mathcal{N}(0, \sigma^2) \) and the \( \varepsilon_{i} \) are
-independent, i.e.: 
-$$
-\begin{align*} 
-\mbox{Cov}(\varepsilon_{i_1},
-\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
-& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2.  \end{array} \right.
-\end{align*} 
-$$
-
-The randomness of \( \varepsilon_i \) implies that
-\( \mathbf{y}_i \) is also a random variable. In particular,
-\( \mathbf{y}_i \) is normally distributed, because \( \varepsilon_i \sim
-\mathcal{N}(0, \sigma^2) \) and \( \mathbf{X}_{i,\ast} \, \boldsymbol{\beta} \) is a
-non-random scalar. To specify the parameters of the distribution of
-\( \mathbf{y}_i \) we need to calculate its first two moments.
-
-<p>
-Recall that \( \boldsymbol{X} \) is a matrix of dimensionality \( n\times p \). The
-notation above \( \mathbf{X}_{i,\ast} \) means that we are looking at the
-row number \( i \) and perform a sum over all values \( p \).
+The setting of <b>original</b> and novel data is
+then mimicked by sample splitting: the data set is divided into two
+(groups of samples). One of these two data sets, called the 
+<em>training set</em>, plays the role of <b>original</b> data on which the model is
+built. The second of these data sets, called the <em>test set</em>, plays the
+role of the <b>novel</b> data and is used to evaluate the prediction
+performance (often operationalized as the log-likelihood or the
+prediction error or its square or the R2 score) of the model built on the training data set. This
+procedure (model building and prediction evaluation on training and
+test set, respectively) is done for a collection of possible penalty
+parameter choices. The penalty parameter that yields the model with
+the best prediction performance is to be preferred. The thus obtained
+performance evaluation depends on the actual split of the data set. To
+remove this dependence the data set is split many times into a
+training and test set. For each split the model parameters are
+estimated for all choices of \( \lambda \) using the training data and
+estimated parameters are evaluated on the corresponding test set. The
+penalty parameter that on average over the test sets performs best (in
+some sense) is then selected.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs077.html b/doc/pub/Regression/html/._Regression-bs077.html
index 1a192e2bd..61ce3a51d 100644
--- a/doc/pub/Regression/html/._Regression-bs077.html
+++ b/doc/pub/Regression/html/._Regression-bs077.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,24 +383,16 @@ MathJax.Hub.Config({
 <a name="part0077"></a>
 <!-- !split -->
 
-<h2 id="___sec76" class="anchor">Assumptions made </h2>
+<h2 id="___sec76" class="anchor">Computationally expensive </h2>
 
 <p>
-The assumption we have made here can be summarized as (and this is going to useful when we discuss the bias-variance trade off)
-that there exists a function \( f(\boldsymbol{x}) \) and  a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \)
-which describes our data
-$$
-\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
-$$
+The validation set approach is conceptually simple and is easy to implement. But it has two potential drawbacks:
 
-<p>
-We approximate this function with our model from the solution of the linear regression equations, that is our
-function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we want to minimize \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), our MSE, with
-$$
-\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
-$$
+<ul>
+<li> The validation estimate of the test error rate can be highly variable, depending on precisely which observations are included in the training set and which observations are included in the validation set.</li>
+<li> In the validation approach, only a subset of the observations, those that are included in the training set rather than in the validation set are used to fit the model. Since statistical methods tend to perform worse when trained on fewer observations, this suggests that the validation set error rate may tend to overestimate the test error rate for the model fit on the entire data set.</li>
+</ul>
 
-<p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
diff --git a/doc/pub/Regression/html/._Regression-bs078.html b/doc/pub/Regression/html/._Regression-bs078.html
index f44de4268..355b00f66 100644
--- a/doc/pub/Regression/html/._Regression-bs078.html
+++ b/doc/pub/Regression/html/._Regression-bs078.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,39 +381,25 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0078"></a>
-<!-- !split -->
+<!-- !split  -->
 
-<h2 id="___sec77" class="anchor">Expectation value and variance </h2>
+<h2 id="___sec77" class="anchor">Various steps in cross-validation </h2>
 
 <p>
-We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) 
-$$
-\begin{align*} 
-\mathbb{E}(y_i) & =
-\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i)
-\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, 
-\end{align*} 
-$$
-
-while
-its variance is 
-$$
-\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i
-- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
-[\mathbb{E}(y_i)]^2  \\  & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
-\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ &
-= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i
-\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
-\ast} \, \beta)^2 \\  & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2
-\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} +
-\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 
-\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
-\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.  
-\end{align*}
-$$
-
-Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with 
-mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (not be confused with the singular values of the SVD).
+When the repetitive splitting of the data set is done randomly,
+samples may accidently end up in a fast majority of the splits in
+either training or test set. Such samples may have an unbalanced
+influence on either model building or prediction evaluation. To avoid
+this \( k \)-fold cross-validation structures the data splitting. The
+samples are divided into \( k \) more or less equally sized exhaustive and
+mutually exclusive subsets. In turn (at each split) one of these
+subsets plays the role of the test set while the union of the
+remaining subsets constitutes the training set. Such a splitting
+warrants a balanced representation of each sample in both training and
+test set over the splits. Still the division into the \( k \) subsets
+involves a degree of randomness. This may be fully excluded when
+choosing \( k=n \). This particular case is referred to as leave-one-out
+cross-validation (LOOCV).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs079.html b/doc/pub/Regression/html/._Regression-bs079.html
index a725bb2f4..b0555066c 100644
--- a/doc/pub/Regression/html/._Regression-bs079.html
+++ b/doc/pub/Regression/html/._Regression-bs079.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,91 +381,42 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0079"></a>
-<!-- !split -->
+<!-- !split  -->
 
-<h2 id="___sec78" class="anchor">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
+<h2 id="___sec78" class="anchor">How to set up the cross-validation for Ridge and/or Lasso </h2>
 
-<p>
-With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
-$$
-\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
-$$
-
-This means that the estimator of the regression parameters is unbiased.
-
-<p>
-We can also calculate the variance
-
-<p>
-The variance of \( \boldsymbol{\beta} \) is
-$$
-\begin{eqnarray*}
-\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
-\\
-& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \}
-\\
-% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-% \\
-% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1}  \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-% \\
-& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-\\
-& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \,  \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-% \\
-% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \,  \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
-% \\
-% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T  \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
-\\
-& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}  + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
-\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
-\end{eqnarray*}
-$$
-
-<p>
-where we have used  that \( \mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) =
-\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} +
-\sigma^2 \, \mathbf{I}_{nn} \). From \( \mbox{Var}(\boldsymbol{\beta}) = \sigma^2
-\, (\mathbf{X}^{T} \mathbf{X})^{-1} \), one obtains an estimate of the
-variance of the estimate of the \( j \)-th regression coefficient:
-\( \hat{\sigma}^2 (\hat{\beta}_j ) = \hat{\sigma}^2 \sqrt{
-[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} } \). This may be used to
-construct a confidence interval for the estimates.
-
-<p>
-In a similar way, we cna obtain analytical expressions for say the
-expectation values of the parameters \( \boldsymbol{\beta} \) and their variance
-when we employ Ridge regression, and thereby a confidence interval.
-
-<p>
-It is rather straightforward to show that
-$$
-\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}.
-$$
-
-We see clearly that 
-\( \mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}} \) for any \( \lambda > 0 \). We say then that the ridge estimator is biased.
-
-<p>
-We can also compute the variance as 
+<ul>
+<li> Define a range of interest for the penalty parameter.</li>
+<li> Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.</li>
+<li> Fit the linear regression model by means of ridge estimation  for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as</li>
+</ul>
 
 $$
-\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T} \mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
+\begin{align*}
+\boldsymbol{\beta}_{-i}(\lambda) & =  ( \boldsymbol{X}_{-i, \ast}^{T}
+\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1}
+\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i}
+\end{align*}
 $$
 
-and it is easy to see that if the parameter \( \lambda \) goes to infinity then the variance of Ridge parameters \( \boldsymbol{\beta} \) goes to zero.
 
-<p>
-With this, we can compute the difference 
+<ul>
+<li> Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
+<li> Repeat the first three steps  such that each sample plays the role of the test set once.</li>
+<li> Average the prediction performances of the test sets at each grid point of the penalty bias/parameter by computing the <em>cross-validated log-likelihood</em>. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</li>
+</ul>
 
 $$
-\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
+\begin{align*}
+\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}.
+\end{align*}
 $$
 
-The difference is non-negative definite since each component of the
-matrix product is non-negative definite. 
-This means the variance we obtain with the standard OLS will always for \( \lambda > 0 \) be larger than the variance of \( \boldsymbol{\beta} \) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.
 
-<p>
+<ul>
+<li> The value of the penalty parameter that maximizes the cross-validated log-likelihood is the value of choice. Or we can use the MSE or the R2 score functions.</li>
+</ul>
+
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
diff --git a/doc/pub/Regression/html/._Regression-bs080.html b/doc/pub/Regression/html/._Regression-bs080.html
index 05ce4d4a0..96c416585 100644
--- a/doc/pub/Regression/html/._Regression-bs080.html
+++ b/doc/pub/Regression/html/._Regression-bs080.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,38 +381,27 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0080"></a>
-<!-- !split  -->
+<!-- !split -->
 
-<h2 id="___sec79" class="anchor">Cross-validation </h2>
+<h2 id="___sec79" class="anchor">Resampling methods: Jackknife and Bootstrap </h2>
 
 <p>
-Instead of choosing the penalty parameter to balance model fit with
-model complexity, cross-validation requires it (i.e. the penalty
-parameter) to yield a model with good prediction
-performance. Commonly, this performance is evaluated on novel
-data. Novel data need not be easy to come by and one has to make do
-with the data at hand.
+Two famous
+resampling methods are the <b>independent bootstrap</b> and <b>the jackknife</b>.
 
 <p>
-The setting of <b>original</b> and novel data is
-then mimicked by sample splitting: the data set is divided into two
-(groups of samples). One of these two data sets, called the 
-<em>training set</em>, plays the role of <b>original</b> data on which the model is
-built. The second of these data sets, called the <em>test set</em>, plays the
-role of the <b>novel</b> data and is used to evaluate the prediction
-performance (often operationalized as the log-likelihood or the
-prediction error or its square or the R2 score) of the model built on the training data set. This
-procedure (model building and prediction evaluation on training and
-test set, respectively) is done for a collection of possible penalty
-parameter choices. The penalty parameter that yields the model with
-the best prediction performance is to be preferred. The thus obtained
-performance evaluation depends on the actual split of the data set. To
-remove this dependence the data set is split many times into a
-training and test set. For each split the model parameters are
-estimated for all choices of \( \lambda \) using the training data and
-estimated parameters are evaluated on the corresponding test set. The
-penalty parameter that on average over the test sets performs best (in
-some sense) is then selected.
+The jackknife is a special case of the independent bootstrap. Still, the jackknife was made
+popular prior to the independent bootstrap. And as the popularity of
+the independent bootstrap soared, new variants, such as <b>the dependent bootstrap</b>.
+
+<p>
+The Jackknife and independent bootstrap work for
+independent, identically distributed random variables.
+If these conditions are not
+satisfied, the methods will fail.  Yet, it should be said that if the data are
+independent, identically distributed, and we only want to estimate the
+variance of \( \overline{X} \) (which often is the case), then there is no
+need for bootstrapping.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs081.html b/doc/pub/Regression/html/._Regression-bs081.html
index 271a6f60a..c2ff2409d 100644
--- a/doc/pub/Regression/html/._Regression-bs081.html
+++ b/doc/pub/Regression/html/._Regression-bs081.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,16 +383,23 @@ MathJax.Hub.Config({
 <a name="part0081"></a>
 <!-- !split -->
 
-<h2 id="___sec80" class="anchor">Computationally expensive </h2>
+<h2 id="___sec80" class="anchor">Resampling methods: Jackknife </h2>
 
 <p>
-The validation set approach is conceptually simple and is easy to implement. But it has two potential drawbacks:
+The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). 
+The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values \( \boldsymbol{x} = (x_1,x_2,\cdots,X_n) \). 
+Let \( \boldsymbol{x}_i \) denote the vector
+$$
+\boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n),
+$$
 
-<ul>
-<li> The validation estimate of the test error rate can be highly variable, depending on precisely which observations are included in the training set and which observations are included in the validation set.</li>
-<li> In the validation approach, only a subset of the observations, those that are included in the training set rather than in the validation set are used to fit the model. Since statistical methods tend to perform worse when trained on fewer observations, this suggests that the validation set error rate may tend to overestimate the test error rate for the model fit on the entire data set.</li>
-</ul>
+<p>
+which equals the vector \( \boldsymbol{x} \) with the exception that observation
+number \( i \) is left out. Using this notation, define
+\( \widehat{\theta}_i \) to be the estimator
+\( \widehat{\theta} \) computed using \( \vec{X}_i \).
 
+<p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
diff --git a/doc/pub/Regression/html/._Regression-bs082.html b/doc/pub/Regression/html/._Regression-bs082.html
index 708bf2566..c6510e35b 100644
--- a/doc/pub/Regression/html/._Regression-bs082.html
+++ b/doc/pub/Regression/html/._Regression-bs082.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,26 +381,41 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0082"></a>
-<!-- !split  -->
-
-<h2 id="___sec81" class="anchor">Various steps in cross-validation </h2>
+<!-- !split -->
 
+<h2 id="___sec81" class="anchor">Jackknife code example </h2>
 <p>
-When the repetitive splitting of the data set is done randomly,
-samples may accidently end up in a fast majority of the splits in
-either training or test set. Such samples may have an unbalanced
-influence on either model building or prediction evaluation. To avoid
-this \( k \)-fold cross-validation structures the data splitting. The
-samples are divided into \( k \) more or less equally sized exhaustive and
-mutually exclusive subsets. In turn (at each split) one of these
-subsets plays the role of the test set while the union of the
-remaining subsets constitutes the training set. Such a splitting
-warrants a balanced representation of each sample in both training and
-test set over the splits. Still the division into the \( k \) subsets
-involves a degree of randomness. This may be fully excluded when
-choosing \( k=n \). This particular case is referred to as leave-one-out
-cross-validation (LOOCV).
 
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">jackknife</span>(data, stat):
+    n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data);t <span style="color: #666666">=</span> zeros(n); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
+    <span style="color: #408080; font-style: italic">## &#39;jackknifing&#39; by leaving out an observation for each i                                                                                                                      </span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
+        t[i] <span style="color: #666666">=</span> stat(delete(data,i) )
+
+    <span style="color: #408080; font-style: italic"># analysis                                                                                                                                                                     </span>
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec&quot;</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Jackknife Statistics :&quot;</span>)
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;original           bias      std. error&quot;</span>)
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (stat(data),(n<span style="color: #666666">-1</span>)<span style="color: #666666">*</span>mean(t)<span style="color: #666666">/</span>n, (n<span style="color: #666666">*</span>var(t))<span style="color: #666666">**.5</span>))
+
+    <span style="color: #008000; font-weight: bold">return</span> t
+
+
+<span style="color: #408080; font-style: italic"># Returns mean of data samples                                                                                                                                                     </span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
+    <span style="color: #008000; font-weight: bold">return</span> mean(data)
+
+
+mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
+datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
+x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
+<span style="color: #408080; font-style: italic"># jackknife returns the data sample                                                                                                                                                </span>
+t <span style="color: #666666">=</span> jackknife(x, stat)
+</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs083.html b/doc/pub/Regression/html/._Regression-bs083.html
index efc0e7d3c..45bf09ab0 100644
--- a/doc/pub/Regression/html/._Regression-bs083.html
+++ b/doc/pub/Regression/html/._Regression-bs083.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,42 +381,28 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0083"></a>
-<!-- !split  -->
+<!-- !split -->
 
-<h2 id="___sec82" class="anchor">How to set up the cross-validation for Ridge and/or Lasso </h2>
+<h2 id="___sec82" class="anchor">Resampling methods: Bootstrap </h2>
+<div class="panel panel-default">
+<div class="panel-body">
+<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+Bootstrapping is a nonparametric approach to statistical inference
+that substitutes computation for more traditional distributional
+assumptions and asymptotic results. Bootstrapping offers a number of
+advantages: 
 
-<ul>
-<li> Define a range of interest for the penalty parameter.</li>
-<li> Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.</li>
-<li> Fit the linear regression model by means of ridge estimation  for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as</li>
-</ul>
-
-$$
-\begin{align*}
-\boldsymbol{\beta}_{-i}(\lambda) & =  ( \boldsymbol{X}_{-i, \ast}^{T}
-\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1}
-\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i}
-\end{align*}
-$$
+<ol>
+<li> The bootstrap is quite general, although there are some cases in which it fails.</li>  
+<li> Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.</li>  
+<li> It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.</li> 
+<li> It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).</li>
+</ol>
+</div>
+</div>
 
 
-<ul>
-<li> Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
-<li> Repeat the first three steps  such that each sample plays the role of the test set once.</li>
-<li> Average the prediction performances of the test sets at each grid point of the penalty bias/parameter by computing the <em>cross-validated log-likelihood</em>. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</li>
-</ul>
-
-$$
-\begin{align*}
-\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}.
-\end{align*}
-$$
-
-
-<ul>
-<li> The value of the penalty parameter that maximizes the cross-validated log-likelihood is the value of choice. Or we can use the MSE or the R2 score functions.</li>
-</ul>
-
+<p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
diff --git a/doc/pub/Regression/html/._Regression-bs084.html b/doc/pub/Regression/html/._Regression-bs084.html
index df2c47478..9ca496c44 100644
--- a/doc/pub/Regression/html/._Regression-bs084.html
+++ b/doc/pub/Regression/html/._Regression-bs084.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,25 +383,18 @@ MathJax.Hub.Config({
 <a name="part0084"></a>
 <!-- !split -->
 
-<h2 id="___sec83" class="anchor">Resampling methods: Jackknife and Bootstrap </h2>
+<h2 id="___sec83" class="anchor">Resampling methods: Bootstrap background </h2>
 
 <p>
-Two famous
-resampling methods are the <b>independent bootstrap</b> and <b>the jackknife</b>.
-
-<p>
-The jackknife is a special case of the independent bootstrap. Still, the jackknife was made
-popular prior to the independent bootstrap. And as the popularity of
-the independent bootstrap soared, new variants, such as <b>the dependent bootstrap</b>.
-
-<p>
-The Jackknife and independent bootstrap work for
-independent, identically distributed random variables.
-If these conditions are not
-satisfied, the methods will fail.  Yet, it should be said that if the data are
-independent, identically distributed, and we only want to estimate the
-variance of \( \overline{X} \) (which often is the case), then there is no
-need for bootstrapping.
+Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
+\( \widehat{\theta} \) itself must be a random variable. Thus it has
+a pdf, call this function \( p(\boldsymbol{t}) \). The aim of the bootstrap is to
+estimate \( p(\boldsymbol{t}) \) by the relative frequency of
+\( \widehat{\theta} \). You can think of this as using a histogram
+in the place of \( p(\boldsymbol{t}) \). If the relative frequency closely
+resembles \( p(\vec{t}) \), then using numerics, it is straight forward to
+estimate all the interesting parameters of \( p(\boldsymbol{t}) \) using point
+estimators.
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs085.html b/doc/pub/Regression/html/._Regression-bs085.html
index 832ab47e8..603c58d33 100644
--- a/doc/pub/Regression/html/._Regression-bs085.html
+++ b/doc/pub/Regression/html/._Regression-bs085.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,21 +383,24 @@ MathJax.Hub.Config({
 <a name="part0085"></a>
 <!-- !split -->
 
-<h2 id="___sec84" class="anchor">Resampling methods: Jackknife </h2>
+<h2 id="___sec84" class="anchor">Resampling methods: More Bootstrap background </h2>
 
 <p>
-The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). 
-The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values \( \boldsymbol{x} = (x_1,x_2,\cdots,X_n) \). 
-Let \( \boldsymbol{x}_i \) denote the vector
-$$
-\boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n),
-$$
+In the case that \( \widehat{\theta} \) has
+more than one component, and the components are independent, we use the
+same estimator on each component separately.  If the probability
+density function of \( X_i \), \( p(x) \), had been known, then it would have
+been straight forward to do this by: 
 
-<p>
-which equals the vector \( \boldsymbol{x} \) with the exception that observation
-number \( i \) is left out. Using this notation, define
-\( \widehat{\theta}_i \) to be the estimator
-\( \widehat{\theta} \) computed using \( \vec{X}_i \).
+<ol>
+<li> Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).</li> 
+<li> Then using these numbers, we could compute a replica of \( \widehat{\theta} \) called \( \widehat{\theta}^* \).</li> 
+</ol>
+
+By repeated use of (1) and (2), many
+estimates of \( \widehat{\theta} \) could have been obtained. The
+idea is to use the relative frequency of \( \widehat{\theta}^* \)
+(think of a histogram) as an estimate of \( p(\boldsymbol{t}) \).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs086.html b/doc/pub/Regression/html/._Regression-bs086.html
index c52aa8363..efe41fa8a 100644
--- a/doc/pub/Regression/html/._Regression-bs086.html
+++ b/doc/pub/Regression/html/._Regression-bs086.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,39 +383,24 @@ MathJax.Hub.Config({
 <a name="part0086"></a>
 <!-- !split -->
 
-<h2 id="___sec85" class="anchor">Jackknife code example </h2>
+<h2 id="___sec85" class="anchor">Resampling methods: Bootstrap approach </h2>
+
 <p>
+But
+unless there is enough information available about the process that
+generated \( X_1,X_2,\cdots,X_n \), \( p(x) \) is in general
+unknown. Therefore, <a href="https://projecteuclid.org/euclid.aos/1176344552" target="_self">Efron in 1979</a>  asked the
+question: What if we replace \( p(x) \) by the relative frequency
+of the observation \( X_i \); if we draw observations in accordance with
+the relative frequency of the observations, will we obtain the same
+result in some asymptotic sense? The answer is yes.
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
+<p>
+Instead of generating the histogram for the relative
+frequency of the observation \( X_i \), just draw the values
+\( (X_1^*,X_2^*,\cdots,X_n^*) \) with replacement from the vector
+\( \boldsymbol{X} \).
 
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">jackknife</span>(data, stat):
-    n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data);t <span style="color: #666666">=</span> zeros(n); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
-    <span style="color: #408080; font-style: italic">## &#39;jackknifing&#39; by leaving out an observation for each i                                                                                                                      </span>
-    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
-        t[i] <span style="color: #666666">=</span> stat(delete(data,i) )
-
-    <span style="color: #408080; font-style: italic"># analysis                                                                                                                                                                     </span>
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec&quot;</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Jackknife Statistics :&quot;</span>)
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;original           bias      std. error&quot;</span>)
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (stat(data),(n<span style="color: #666666">-1</span>)<span style="color: #666666">*</span>mean(t)<span style="color: #666666">/</span>n, (n<span style="color: #666666">*</span>var(t))<span style="color: #666666">**.5</span>))
-
-    <span style="color: #008000; font-weight: bold">return</span> t
-
-
-<span style="color: #408080; font-style: italic"># Returns mean of data samples                                                                                                                                                     </span>
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
-    <span style="color: #008000; font-weight: bold">return</span> mean(data)
-
-
-mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
-datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
-x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
-<span style="color: #408080; font-style: italic"># jackknife returns the data sample                                                                                                                                                </span>
-t <span style="color: #666666">=</span> jackknife(x, stat)
-</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs087.html b/doc/pub/Regression/html/._Regression-bs087.html
index c26977d5f..c52711bcb 100644
--- a/doc/pub/Regression/html/._Regression-bs087.html
+++ b/doc/pub/Regression/html/._Regression-bs087.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,24 +383,27 @@ MathJax.Hub.Config({
 <a name="part0087"></a>
 <!-- !split -->
 
-<h2 id="___sec86" class="anchor">Resampling methods: Bootstrap </h2>
-<div class="panel panel-default">
-<div class="panel-body">
-<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-Bootstrapping is a nonparametric approach to statistical inference
-that substitutes computation for more traditional distributional
-assumptions and asymptotic results. Bootstrapping offers a number of
-advantages: 
+<h2 id="___sec86" class="anchor">Resampling methods: Bootstrap steps </h2>
+
+<p>
+The independent bootstrap works like this: 
 
 <ol>
-<li> The bootstrap is quite general, although there are some cases in which it fails.</li>  
-<li> Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.</li>  
-<li> It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.</li> 
-<li> It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).</li>
+<li> Draw with replacement \( n \) numbers for the observed variables \( \boldsymbol{x} = (x_1,x_2,\cdots,x_n) \).</li> 
+<li> Define a vector \( \boldsymbol{x}^* \) containing the values which were drawn from \( \boldsymbol{x} \).</li> 
+<li> Using the vector \( \boldsymbol{x}^* \) compute \( \widehat{\theta}^* \) by evaluating \( \widehat \theta \) under the observations \( \boldsymbol{x}^* \).</li> 
+<li> Repeat this process \( k \) times.</li> 
 </ol>
-</div>
-</div>
 
+When you are done, you can draw a histogram of the relative frequency
+of \( \widehat \theta^* \). This is your estimate of the probability
+distribution \( p(t) \). Using this probability distribution you can
+estimate any statistics thereof. In principle you never draw the
+histogram of the relative frequency of \( \widehat{\theta}^* \). Instead
+you use the estimators corresponding to the statistic of interest. For
+example, if you are interested in estimating the variance of \( \widehat
+\theta \), apply the etsimator \( \widehat \sigma^2 \) to the values
+\( \widehat \theta ^* \).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs088.html b/doc/pub/Regression/html/._Regression-bs088.html
index 9bbc97bcb..24600b562 100644
--- a/doc/pub/Regression/html/._Regression-bs088.html
+++ b/doc/pub/Regression/html/._Regression-bs088.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
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               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,19 +383,67 @@ MathJax.Hub.Config({
 <a name="part0088"></a>
 <!-- !split -->
 
-<h2 id="___sec87" class="anchor">Resampling methods: Bootstrap background </h2>
+<h2 id="___sec87" class="anchor">Code example for the Bootstrap method </h2>
 
 <p>
-Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
-\( \widehat{\theta} \) itself must be a random variable. Thus it has
-a pdf, call this function \( p(\boldsymbol{t}) \). The aim of the bootstrap is to
-estimate \( p(\boldsymbol{t}) \) by the relative frequency of
-\( \widehat{\theta} \). You can think of this as using a histogram
-in the place of \( p(\boldsymbol{t}) \). If the relative frequency closely
-resembles \( p(\vec{t}) \), then using numerics, it is straight forward to
-estimate all the interesting parameters of \( p(\boldsymbol{t}) \) using point
-estimators.
+The following code starts with a Gaussian distribution with mean value
+\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data
+used in the bootstrap analysis. The bootstrap analysis returns a data
+set after a given number of bootstrap operations (as many as we have
+data points). This data set consists of estimated mean values for each
+bootstrap operation. The histogram generated by the bootstrap method
+shows that the distribution for these mean values is also a Gaussian,
+centered around the mean value \( \mu=100 \) but with standard deviation
+\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in
+this case the same as the number of original data points). The value
+of the standard deviation is what we expect from the central limit
+theorem.
 
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+
+<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples                                                                                                                                                </span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
+    <span style="color: #008000; font-weight: bold">return</span> mean(data)
+
+<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
+    t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
+    <span style="color: #408080; font-style: italic"># non-parametric bootstrap         </span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(R):
+        t[i] <span style="color: #666666">=</span> statistic(data[randint(<span style="color: #666666">0</span>,n,n)])
+
+    <span style="color: #408080; font-style: italic"># analysis    </span>
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec&quot;</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Bootstrap Statistics :&quot;</span>)
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;original           bias      std. error&quot;</span>)
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (statistic(data), std(data),mean(t),std(t)))
+    <span style="color: #008000; font-weight: bold">return</span> t
+
+
+mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
+datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
+x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
+<span style="color: #408080; font-style: italic"># bootstrap returns the data sample                                    </span>
+t <span style="color: #666666">=</span> bootstrap(x, stat, datapoints)
+<span style="color: #408080; font-style: italic"># the histogram of the bootstrapped  data                                                                                                    </span>
+n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;red&#39;</span>, alpha<span style="color: #666666">=0.75</span>)
+
+<span style="color: #408080; font-style: italic"># add a &#39;best fit&#39; line  </span>
+y <span style="color: #666666">=</span> mlab<span style="color: #666666">.</span>normpdf( binsboot, mean(t), std(t))
+lt <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>plot(binsboot, y, <span style="color: #BA2121">&#39;r--&#39;</span>, linewidth<span style="color: #666666">=1</span>)
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;Smarts&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;Probability&#39;</span>)
+plt<span style="color: #666666">.</span>axis([<span style="color: #666666">99.5</span>, <span style="color: #666666">100.6</span>, <span style="color: #666666">0</span>, <span style="color: #666666">3.0</span>])
+plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
+
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs089.html b/doc/pub/Regression/html/._Regression-bs089.html
index 183c6e607..8c221d91b 100644
--- a/doc/pub/Regression/html/._Regression-bs089.html
+++ b/doc/pub/Regression/html/._Regression-bs089.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,25 +383,103 @@ MathJax.Hub.Config({
 <a name="part0089"></a>
 <!-- !split -->
 
-<h2 id="___sec88" class="anchor">Resampling methods: More Bootstrap background </h2>
+<h2 id="___sec88" class="anchor">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
 
 <p>
-In the case that \( \widehat{\theta} \) has
-more than one component, and the components are independent, we use the
-same estimator on each component separately.  If the probability
-density function of \( X_i \), \( p(x) \), had been known, then it would have
-been straight forward to do this by: 
+The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
+<p>
 
-<ol>
-<li> Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).</li> 
-<li> Then using these numbers, we could compute a replica of \( \widehat{\theta} \) called \( \widehat{\theta}^* \).</li> 
-</ol>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> KFold
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
 
-By repeated use of (1) and (2), many
-estimates of \( \widehat{\theta} \) could have been obtained. The
-idea is to use the relative frequency of \( \widehat{\theta}^* \)
-(think of a histogram) as an estimate of \( p(\boldsymbol{t}) \).
+<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
+<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
+np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
 
+<span style="color: #408080; font-style: italic"># Generate the data.</span>
+nsamples <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
+y <span style="color: #666666">=</span> <span style="color: #666666">3*</span>x<span style="color: #666666">**2</span> <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
+
+<span style="color: #408080; font-style: italic">## Cross-validation on Ridge regression using KFold only</span>
+
+<span style="color: #408080; font-style: italic"># Decide degree on polynomial to fit</span>
+poly <span style="color: #666666">=</span> PolynomialFeatures(degree <span style="color: #666666">=</span> <span style="color: #666666">6</span>)
+
+<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
+nlambdas <span style="color: #666666">=</span> <span style="color: #666666">500</span>
+lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-3</span>, <span style="color: #666666">5</span>, nlambdas)
+
+<span style="color: #408080; font-style: italic"># Initialize a KFold instance</span>
+k <span style="color: #666666">=</span> <span style="color: #666666">5</span>
+kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
+
+<span style="color: #408080; font-style: italic"># Perform the cross-validation to estimate MSE</span>
+scores_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((nlambdas, k))
+
+i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
+    ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
+    j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+    <span style="color: #008000; font-weight: bold">for</span> train_inds, test_inds <span style="color: #AA22FF; font-weight: bold">in</span> kfold<span style="color: #666666">.</span>split(x):
+        xtrain <span style="color: #666666">=</span> x[train_inds]
+        ytrain <span style="color: #666666">=</span> y[train_inds]
+
+        xtest <span style="color: #666666">=</span> x[test_inds]
+        ytest <span style="color: #666666">=</span> y[test_inds]
+
+        Xtrain <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtrain[:, np<span style="color: #666666">.</span>newaxis])
+        ridge<span style="color: #666666">.</span>fit(Xtrain, ytrain[:, np<span style="color: #666666">.</span>newaxis])
+
+        Xtest <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtest[:, np<span style="color: #666666">.</span>newaxis])
+        ypred <span style="color: #666666">=</span> ridge<span style="color: #666666">.</span>predict(Xtest)
+
+        scores_KFold[i,j] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum((ypred <span style="color: #666666">-</span> ytest[:, np<span style="color: #666666">.</span>newaxis])<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>np<span style="color: #666666">.</span>size(ypred)
+
+        j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+    i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+
+
+estimated_mse_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(scores_KFold, axis <span style="color: #666666">=</span> <span style="color: #666666">1</span>)
+
+<span style="color: #408080; font-style: italic">## Cross-validation using cross_val_score from sklearn along with KFold</span>
+
+<span style="color: #408080; font-style: italic"># kfold is an instance initialized above as:</span>
+<span style="color: #408080; font-style: italic"># kfold = KFold(n_splits = k)</span>
+
+estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
+i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
+    ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
+
+    X <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(x[:, np<span style="color: #666666">.</span>newaxis])
+    estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(ridge, X, y[:, np<span style="color: #666666">.</span>newaxis], scoring<span style="color: #666666">=</span><span style="color: #BA2121">&#39;neg_mean_squared_error&#39;</span>, cv<span style="color: #666666">=</span>kfold)
+
+    <span style="color: #408080; font-style: italic"># cross_val_score return an array containing the estimated negative mse for every fold.</span>
+    <span style="color: #408080; font-style: italic"># we have to the the mean of every array in order to get an estimate of the mse of the model</span>
+    estimated_mse_sklearn[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
+
+    i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+
+<span style="color: #408080; font-style: italic">## Plot and compare the slightly different ways to perform cross-validation</span>
+
+plt<span style="color: #666666">.</span>figure()
+
+plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_sklearn, label <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;cross_val_score&#39;</span>)
+plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_KFold, <span style="color: #BA2121">&#39;r--&#39;</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;KFold&#39;</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;log10(lambda)&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;mse&#39;</span>)
+
+plt<span style="color: #666666">.</span>legend()
+
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs090.html b/doc/pub/Regression/html/._Regression-bs090.html
index 6f523d063..b02a9ad7b 100644
--- a/doc/pub/Regression/html/._Regression-bs090.html
+++ b/doc/pub/Regression/html/._Regression-bs090.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,23 +383,68 @@ MathJax.Hub.Config({
 <a name="part0090"></a>
 <!-- !split -->
 
-<h2 id="___sec89" class="anchor">Resampling methods: Bootstrap approach </h2>
+<h2 id="___sec89" class="anchor">The bias-variance tradeoff </h2>
 
 <p>
-But
-unless there is enough information available about the process that
-generated \( X_1,X_2,\cdots,X_n \), \( p(x) \) is in general
-unknown. Therefore, <a href="https://projecteuclid.org/euclid.aos/1176344552" target="_self">Efron in 1979</a>  asked the
-question: What if we replace \( p(x) \) by the relative frequency
-of the observation \( X_i \); if we draw observations in accordance with
-the relative frequency of the observations, will we obtain the same
-result in some asymptotic sense? The answer is yes.
+We will discuss the bias-variance tradeoff in the context of
+continuous predictions such as regression. However, many of the
+intuitions and ideas discussed here also carry over to classification
+tasks. Consider a dataset \( \mathcal{L} \) consisting of the data
+\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
 
 <p>
-Instead of generating the histogram for the relative
-frequency of the observation \( X_i \), just draw the values
-\( (X_1^*,X_2^*,\cdots,X_n^*) \) with replacement from the vector
-\( \boldsymbol{X} \).
+Let us assume that the true data is generated from a noisy model
+
+$$
+\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}
+$$
+
+<p>
+where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \).
+
+<p>
+In our derivation of the ordinary least squares method we defined then
+an approximation to the function \( f \) in terms of the parameters
+\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model,
+that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \).
+
+<p>
+Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function
+$$
+C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
+$$
+
+<p>
+We can rewrite this as 
+$$
+\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
+$$
+
+<p>
+The three terms represent the square of the bias of the learning
+method, which can be thought of as the error caused by the simplifying
+assumptions built into the method. The second term represents the
+variance of the chosen model and finally the last terms is variance of
+the error \( \boldsymbol{\epsilon} \).
+
+<p>
+To derive this equation, we need to recall that the variance of \( \boldsymbol{y} \) and \( \boldsymbol{\epsilon} \) are both equal to \( \sigma^2 \). The mean value of \( \boldsymbol{\epsilon} \) is by definition equal to zero. Furthermore, the function \( f \) is not a stochastics variable, idem for \( \boldsymbol{\tilde{y}} \).
+We use a more compact notation in terms of the expectation value 
+$$
+\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right],
+$$
+
+and adding and subtracting \( \mathbb{E}\left[\boldsymbol{\tilde{y}}\right] \) we get
+$$
+\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right],
+$$
+
+which, using the abovementioned expectation values can be rewritten as 
+$$
+\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2,
+$$
+
+that is the rewriting in terms of the so-called bias, the variance of the model \( \boldsymbol{\tilde{y}} \) and the variance of \( \boldsymbol{\epsilon} \).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs091.html b/doc/pub/Regression/html/._Regression-bs091.html
index bdb92c3d0..04bca014d 100644
--- a/doc/pub/Regression/html/._Regression-bs091.html
+++ b/doc/pub/Regression/html/._Regression-bs091.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,28 +383,65 @@ MathJax.Hub.Config({
 <a name="part0091"></a>
 <!-- !split -->
 
-<h2 id="___sec90" class="anchor">Resampling methods: Bootstrap steps </h2>
-
+<h2 id="___sec90" class="anchor">Example code for Bias-Variance tradeoff </h2>
 <p>
-The independent bootstrap works like this: 
 
-<ol>
-<li> Draw with replacement \( n \) numbers for the observed variables \( \boldsymbol{x} = (x_1,x_2,\cdots,x_n) \).</li> 
-<li> Define a vector \( \boldsymbol{x}^* \) containing the values which were drawn from \( \boldsymbol{x} \).</li> 
-<li> Using the vector \( \boldsymbol{x}^* \) compute \( \widehat{\theta}^* \) by evaluating \( \widehat \theta \) under the observations \( \boldsymbol{x}^* \).</li> 
-<li> Repeat this process \( k \) times.</li> 
-</ol>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
 
-When you are done, you can draw a histogram of the relative frequency
-of \( \widehat \theta^* \). This is your estimate of the probability
-distribution \( p(t) \). Using this probability distribution you can
-estimate any statistics thereof. In principle you never draw the
-histogram of the relative frequency of \( \widehat{\theta}^* \). Instead
-you use the estimators corresponding to the statistic of interest. For
-example, if you are interested in estimating the variance of \( \widehat
-\theta \), apply the etsimator \( \widehat \sigma^2 \) to the values
-\( \widehat \theta ^* \).
+np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
 
+n <span style="color: #666666">=</span> <span style="color: #666666">500</span>
+n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+degree <span style="color: #666666">=</span> <span style="color: #666666">18</span>  <span style="color: #408080; font-style: italic"># A quite high value, just to show.</span>
+noise <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
+
+<span style="color: #408080; font-style: italic"># Make data set.</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-1</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
+
+<span style="color: #408080; font-style: italic"># Hold out some test data that is never used in training.</span>
+x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
+
+<span style="color: #408080; font-style: italic"># Combine x transformation and model into one operation.</span>
+<span style="color: #408080; font-style: italic"># Not neccesary, but convenient.</span>
+model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000">False</span>))
+
+<span style="color: #408080; font-style: italic"># The following (m x n_bootstraps) matrix holds the column vectors y_pred</span>
+<span style="color: #408080; font-style: italic"># for each bootstrap iteration.</span>
+y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
+    x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
+
+    <span style="color: #408080; font-style: italic"># Evaluate the new model on the same test data each time.</span>
+    y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
+
+<span style="color: #408080; font-style: italic"># Note: Expectations and variances taken w.r.t. different training</span>
+<span style="color: #408080; font-style: italic"># data sets, hence the axis=1. Subsequent means are taken across the test data</span>
+<span style="color: #408080; font-style: italic"># set in order to obtain a total value, but before this we have error/bias/variance</span>
+<span style="color: #408080; font-style: italic"># calculated per data point in the test set.</span>
+<span style="color: #408080; font-style: italic"># Note 2: The use of keepdims=True is important in the calculation of bias as this </span>
+<span style="color: #408080; font-style: italic"># maintains the column vector form. Dropping this yields very unexpected results.</span>
+error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
+bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>))<span style="color: #666666">**2</span> )
+variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
+<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error)
+<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias)
+<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance)
+<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;{} &gt;= {} + {} = {}&#39;</span><span style="color: #666666">.</span>format(error, bias, variance, bias<span style="color: #666666">+</span>variance))
+
+plt<span style="color: #666666">.</span>plot(x[::<span style="color: #666666">5</span>, :], y[::<span style="color: #666666">5</span>, :], label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;f(x)&#39;</span>)
+plt<span style="color: #666666">.</span>scatter(x_test, y_test, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Data points&#39;</span>)
+plt<span style="color: #666666">.</span>scatter(x_test, np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>), label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Pred&#39;</span>)
+plt<span style="color: #666666">.</span>legend()
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs092.html b/doc/pub/Regression/html/._Regression-bs092.html
index bd2d6da70..83aa75bc5 100644
--- a/doc/pub/Regression/html/._Regression-bs092.html
+++ b/doc/pub/Regression/html/._Regression-bs092.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,65 +383,55 @@ MathJax.Hub.Config({
 <a name="part0092"></a>
 <!-- !split -->
 
-<h2 id="___sec91" class="anchor">Code example for the Bootstrap method </h2>
-
-<p>
-The following code starts with a Gaussian distribution with mean value
-\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data
-used in the bootstrap analysis. The bootstrap analysis returns a data
-set after a given number of bootstrap operations (as many as we have
-data points). This data set consists of estimated mean values for each
-bootstrap operation. The histogram generated by the bootstrap method
-shows that the distribution for these mean values is also a Gaussian,
-centered around the mean value \( \mu=100 \) but with standard deviation
-\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in
-this case the same as the number of original data points). The value
-of the standard deviation is what we expect from the central limit
-theorem.
-
+<h2 id="___sec91" class="anchor">Understanding what happens </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
 
-<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples                                                                                                                                                </span>
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
-    <span style="color: #008000; font-weight: bold">return</span> mean(data)
+np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
 
-<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
-    t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
-    <span style="color: #408080; font-style: italic"># non-parametric bootstrap         </span>
-    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(R):
-        t[i] <span style="color: #666666">=</span> statistic(data[randint(<span style="color: #666666">0</span>,n,n)])
-
-    <span style="color: #408080; font-style: italic"># analysis    </span>
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec&quot;</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Bootstrap Statistics :&quot;</span>)
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;original           bias      std. error&quot;</span>)
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (statistic(data), std(data),mean(t),std(t)))
-    <span style="color: #008000; font-weight: bold">return</span> t
+n <span style="color: #666666">=</span> <span style="color: #666666">40</span>
+n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
 
 
-mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
-datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
-x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
-<span style="color: #408080; font-style: italic"># bootstrap returns the data sample                                    </span>
-t <span style="color: #666666">=</span> bootstrap(x, stat, datapoints)
-<span style="color: #408080; font-style: italic"># the histogram of the bootstrapped  data                                                                                                    </span>
-n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;red&#39;</span>, alpha<span style="color: #666666">=0.75</span>)
+<span style="color: #408080; font-style: italic"># Make data set.</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
+error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
+bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
+variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
+polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
+x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
 
-<span style="color: #408080; font-style: italic"># add a &#39;best fit&#39; line  </span>
-y <span style="color: #666666">=</span> mlab<span style="color: #666666">.</span>normpdf( binsboot, mean(t), std(t))
-lt <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>plot(binsboot, y, <span style="color: #BA2121">&#39;r--&#39;</span>, linewidth<span style="color: #666666">=1</span>)
-plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;Smarts&#39;</span>)
-plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;Probability&#39;</span>)
-plt<span style="color: #666666">.</span>axis([<span style="color: #666666">99.5</span>, <span style="color: #666666">100.6</span>, <span style="color: #666666">0</span>, <span style="color: #666666">3.0</span>])
-plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
+<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
+    model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000">False</span>))
+    y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
+        x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
+        y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
 
+    polydegree[degree] <span style="color: #666666">=</span> degree
+    error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
+    bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>))<span style="color: #666666">**2</span> )
+    variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Polynomial degree:&#39;</span>, degree)
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error[degree])
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias[degree])
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance[degree])
+    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;{} &gt;= {} + {} = {}&#39;</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
+
+plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Error&#39;</span>)
+plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;bias&#39;</span>)
+plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Variance&#39;</span>)
+plt<span style="color: #666666">.</span>legend()
 plt<span style="color: #666666">.</span>show()
 </pre></div>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs093.html b/doc/pub/Regression/html/._Regression-bs093.html
index 36cb3cbc2..e97039030 100644
--- a/doc/pub/Regression/html/._Regression-bs093.html
+++ b/doc/pub/Regression/html/._Regression-bs093.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
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+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
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               ('Resampling approaches can be computationally expensive',
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                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
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-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
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-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
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               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
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-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
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+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,105 +381,39 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0093"></a>
-<!-- !split -->
+<!-- !split  -->
 
-<h2 id="___sec92" class="anchor">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
+<h2 id="___sec92" class="anchor">Summing up </h2>
 
 <p>
-The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
+The bias-variance tradeoff summarizes the fundamental tension in
+machine learning, particularly supervised learning, between the
+complexity of a model and the amount of training data needed to train
+it.  Since data is often limited, in practice it is often useful to
+use a less-complex model with higher bias, that is  a model whose asymptotic
+performance is worse than another model because it is easier to
+train and less sensitive to sampling noise arising from having a
+finite-sized training dataset (smaller variance).
+
 <p>
+The above equations tell us that in
+order to minimize the expected test error, we need to select a
+statistical learning method that simultaneously achieves low variance
+and low bias. Note that variance is inherently a nonnegative quantity,
+and squared bias is also nonnegative. Hence, we see that the expected
+test MSE can never lie below \( Var(\epsilon) \), the irreducible error.
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> KFold
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
+<p>
+What do we mean by the variance and bias of a statistical learning
+method? The variance refers to the amount by which our model would change if we
+estimated it using a different training data set. Since the training
+data are used to fit the statistical learning method, different
+training data sets  will result in a different estimate. But ideally the
+estimate for our model should not vary too much between training
+sets. However, if a method has high variance  then small changes in
+the training data can result in large changes in the model. In general, more
+flexible statistical methods have higher variance.
 
-<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
-<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
-np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
-
-<span style="color: #408080; font-style: italic"># Generate the data.</span>
-nsamples <span style="color: #666666">=</span> <span style="color: #666666">100</span>
-x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
-y <span style="color: #666666">=</span> <span style="color: #666666">3*</span>x<span style="color: #666666">**2</span> <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
-
-<span style="color: #408080; font-style: italic">## Cross-validation on Ridge regression using KFold only</span>
-
-<span style="color: #408080; font-style: italic"># Decide degree on polynomial to fit</span>
-poly <span style="color: #666666">=</span> PolynomialFeatures(degree <span style="color: #666666">=</span> <span style="color: #666666">6</span>)
-
-<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
-nlambdas <span style="color: #666666">=</span> <span style="color: #666666">500</span>
-lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-3</span>, <span style="color: #666666">5</span>, nlambdas)
-
-<span style="color: #408080; font-style: italic"># Initialize a KFold instance</span>
-k <span style="color: #666666">=</span> <span style="color: #666666">5</span>
-kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
-
-<span style="color: #408080; font-style: italic"># Perform the cross-validation to estimate MSE</span>
-scores_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((nlambdas, k))
-
-i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
-<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
-    ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
-    j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
-    <span style="color: #008000; font-weight: bold">for</span> train_inds, test_inds <span style="color: #AA22FF; font-weight: bold">in</span> kfold<span style="color: #666666">.</span>split(x):
-        xtrain <span style="color: #666666">=</span> x[train_inds]
-        ytrain <span style="color: #666666">=</span> y[train_inds]
-
-        xtest <span style="color: #666666">=</span> x[test_inds]
-        ytest <span style="color: #666666">=</span> y[test_inds]
-
-        Xtrain <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtrain[:, np<span style="color: #666666">.</span>newaxis])
-        ridge<span style="color: #666666">.</span>fit(Xtrain, ytrain[:, np<span style="color: #666666">.</span>newaxis])
-
-        Xtest <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtest[:, np<span style="color: #666666">.</span>newaxis])
-        ypred <span style="color: #666666">=</span> ridge<span style="color: #666666">.</span>predict(Xtest)
-
-        scores_KFold[i,j] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum((ypred <span style="color: #666666">-</span> ytest[:, np<span style="color: #666666">.</span>newaxis])<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>np<span style="color: #666666">.</span>size(ypred)
-
-        j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
-    i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
-
-
-estimated_mse_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(scores_KFold, axis <span style="color: #666666">=</span> <span style="color: #666666">1</span>)
-
-<span style="color: #408080; font-style: italic">## Cross-validation using cross_val_score from sklearn along with KFold</span>
-
-<span style="color: #408080; font-style: italic"># kfold is an instance initialized above as:</span>
-<span style="color: #408080; font-style: italic"># kfold = KFold(n_splits = k)</span>
-
-estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
-i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
-<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
-    ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
-
-    X <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(x[:, np<span style="color: #666666">.</span>newaxis])
-    estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(ridge, X, y[:, np<span style="color: #666666">.</span>newaxis], scoring<span style="color: #666666">=</span><span style="color: #BA2121">&#39;neg_mean_squared_error&#39;</span>, cv<span style="color: #666666">=</span>kfold)
-
-    <span style="color: #408080; font-style: italic"># cross_val_score return an array containing the estimated negative mse for every fold.</span>
-    <span style="color: #408080; font-style: italic"># we have to the the mean of every array in order to get an estimate of the mse of the model</span>
-    estimated_mse_sklearn[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
-
-    i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
-
-<span style="color: #408080; font-style: italic">## Plot and compare the slightly different ways to perform cross-validation</span>
-
-plt<span style="color: #666666">.</span>figure()
-
-plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_sklearn, label <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;cross_val_score&#39;</span>)
-plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_KFold, <span style="color: #BA2121">&#39;r--&#39;</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;KFold&#39;</span>)
-
-plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;log10(lambda)&#39;</span>)
-plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;mse&#39;</span>)
-
-plt<span style="color: #666666">.</span>legend()
-
-plt<span style="color: #666666">.</span>show()
-</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs094.html b/doc/pub/Regression/html/._Regression-bs094.html
index c9e3650c3..bee332626 100644
--- a/doc/pub/Regression/html/._Regression-bs094.html
+++ b/doc/pub/Regression/html/._Regression-bs094.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,69 +383,82 @@ MathJax.Hub.Config({
 <a name="part0094"></a>
 <!-- !split -->
 
-<h2 id="___sec93" class="anchor">The bias-variance tradeoff </h2>
-
+<h2 id="___sec93" class="anchor">Another Example rom Scikit-Learn's Repository </h2>
 <p>
-We will discuss the bias-variance tradeoff in the context of
-continuous predictions such as regression. However, many of the
-intuitions and ideas discussed here also carry over to classification
-tasks. Consider a dataset \( \mathcal{L} \) consisting of the data
-\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
 
-<p>
-Let us assume that the true data is generated from a noisy model
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
+<span style="color: #BA2121; font-style: italic">============================</span>
+<span style="color: #BA2121; font-style: italic">Underfitting vs. Overfitting</span>
+<span style="color: #BA2121; font-style: italic">============================</span>
 
-$$
-\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}
-$$
+<span style="color: #BA2121; font-style: italic">This example demonstrates the problems of underfitting and overfitting and</span>
+<span style="color: #BA2121; font-style: italic">how we can use linear regression with polynomial features to approximate</span>
+<span style="color: #BA2121; font-style: italic">nonlinear functions. The plot shows the function that we want to approximate,</span>
+<span style="color: #BA2121; font-style: italic">which is a part of the cosine function. In addition, the samples from the</span>
+<span style="color: #BA2121; font-style: italic">real function and the approximations of different models are displayed. The</span>
+<span style="color: #BA2121; font-style: italic">models have polynomial features of different degrees. We can see that a</span>
+<span style="color: #BA2121; font-style: italic">linear function (polynomial with degree 1) is not sufficient to fit the</span>
+<span style="color: #BA2121; font-style: italic">training samples. This is called **underfitting**. A polynomial of degree 4</span>
+<span style="color: #BA2121; font-style: italic">approximates the true function almost perfectly. However, for higher degrees</span>
+<span style="color: #BA2121; font-style: italic">the model will **overfit** the training data, i.e. it learns the noise of the</span>
+<span style="color: #BA2121; font-style: italic">training data.</span>
+<span style="color: #BA2121; font-style: italic">We evaluate quantitatively **overfitting** / **underfitting** by using</span>
+<span style="color: #BA2121; font-style: italic">cross-validation. We calculate the mean squared error (MSE) on the validation</span>
+<span style="color: #BA2121; font-style: italic">set, the higher, the less likely the model generalizes correctly from the</span>
+<span style="color: #BA2121; font-style: italic">training data.</span>
+<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
 
-<p>
-where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \).
+<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #19177C">__doc__</span>)
 
-<p>
-In our derivation of the ordinary least squares method we defined then
-an approximation to the function \( f \) in terms of the parameters
-\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model,
-that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \).
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
 
-<p>
-Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function
-$$
-C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
-$$
 
-<p>
-We can rewrite this as 
-$$
-\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
-$$
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">true_fun</span>(X):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>pi <span style="color: #666666">*</span> X)
 
-<p>
-The three terms represent the square of the bias of the learning
-method, which can be thought of as the error caused by the simplifying
-assumptions built into the method. The second term represents the
-variance of the chosen model and finally the last terms is variance of
-the error \( \boldsymbol{\epsilon} \).
+np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
 
-<p>
-To derive this equation, we need to recall that the variance of \( \boldsymbol{y} \) and \( \boldsymbol{\epsilon} \) are both equal to \( \sigma^2 \). The mean value of \( \boldsymbol{\epsilon} \) is by definition equal to zero. Furthermore, the function \( f \) is not a stochastics variable, idem for \( \boldsymbol{\tilde{y}} \).
-We use a more compact notation in terms of the expectation value 
-$$
-\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right],
-$$
+n_samples <span style="color: #666666">=</span> <span style="color: #666666">30</span>
+degrees <span style="color: #666666">=</span> [<span style="color: #666666">1</span>, <span style="color: #666666">4</span>, <span style="color: #666666">15</span>]
 
-and adding and subtracting \( \mathbb{E}\left[\boldsymbol{\tilde{y}}\right] \) we get
-$$
-\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right],
-$$
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n_samples))
+y <span style="color: #666666">=</span> true_fun(X) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n_samples) <span style="color: #666666">*</span> <span style="color: #666666">0.1</span>
 
-which, using the abovementioned expectation values can be rewritten as 
-$$
-\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2,
-$$
+plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">14</span>, <span style="color: #666666">5</span>))
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(degrees)):
+    ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(degrees), i <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
+    plt<span style="color: #666666">.</span>setp(ax, xticks<span style="color: #666666">=</span>(), yticks<span style="color: #666666">=</span>())
 
-that is the rewriting in terms of the so-called bias, the variance of the model \( \boldsymbol{\tilde{y}} \) and the variance of \( \boldsymbol{\epsilon} \).
+    polynomial_features <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=</span>degrees[i],
+                                             include_bias<span style="color: #666666">=</span><span style="color: #008000">False</span>)
+    linear_regression <span style="color: #666666">=</span> LinearRegression()
+    pipeline <span style="color: #666666">=</span> Pipeline([(<span style="color: #BA2121">&quot;polynomial_features&quot;</span>, polynomial_features),
+                         (<span style="color: #BA2121">&quot;linear_regression&quot;</span>, linear_regression)])
+    pipeline<span style="color: #666666">.</span>fit(X[:, np<span style="color: #666666">.</span>newaxis], y)
 
+    <span style="color: #408080; font-style: italic"># Evaluate the models using crossvalidation</span>
+    scores <span style="color: #666666">=</span> cross_val_score(pipeline, X[:, np<span style="color: #666666">.</span>newaxis], y,
+                             scoring<span style="color: #666666">=</span><span style="color: #BA2121">&quot;neg_mean_squared_error&quot;</span>, cv<span style="color: #666666">=10</span>)
+
+    X_test <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">100</span>)
+    plt<span style="color: #666666">.</span>plot(X_test, pipeline<span style="color: #666666">.</span>predict(X_test[:, np<span style="color: #666666">.</span>newaxis]), label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Model&quot;</span>)
+    plt<span style="color: #666666">.</span>plot(X_test, true_fun(X_test), label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;True function&quot;</span>)
+    plt<span style="color: #666666">.</span>scatter(X, y, edgecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;b&#39;</span>, s<span style="color: #666666">=20</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Samples&quot;</span>)
+    plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;x&quot;</span>)
+    plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;y&quot;</span>)
+    plt<span style="color: #666666">.</span>xlim((<span style="color: #666666">0</span>, <span style="color: #666666">1</span>))
+    plt<span style="color: #666666">.</span>ylim((<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>))
+    plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">&quot;best&quot;</span>)
+    plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Degree {}</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">MSE = {:.2e}(+/- {:.2e})&quot;</span><span style="color: #666666">.</span>format(
+        degrees[i], <span style="color: #666666">-</span>scores<span style="color: #666666">.</span>mean(), scores<span style="color: #666666">.</span>std()))
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs095.html b/doc/pub/Regression/html/._Regression-bs095.html
index 63247350c..66f3b7357 100644
--- a/doc/pub/Regression/html/._Regression-bs095.html
+++ b/doc/pub/Regression/html/._Regression-bs095.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,65 +383,60 @@ MathJax.Hub.Config({
 <a name="part0095"></a>
 <!-- !split -->
 
-<h2 id="___sec94" class="anchor">Example code for Bias-Variance tradeoff </h2>
+<h2 id="___sec94" class="anchor">The Ising model </h2>
+
+<p>
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant \( J \) is given by
+
+$$
+\begin{align}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{21}
+\end{align}
+$$
+
+<p>
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
+in the system is determined by \( L \). For the one-dimensional system
+there is no phase transition.
+
+<p>
+We will look at a system of \( L = 40 \) spins with a coupling constant of
+\( J = 1 \). To get enough training data we will generate 10000 states
+with their respective energies.
+
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</span>
 <span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
+sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000">True</span>)
+cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&#39;seismic&#39;</span>)
 
-np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
+L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
+n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
 
-n <span style="color: #666666">=</span> <span style="color: #666666">500</span>
-n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
-degree <span style="color: #666666">=</span> <span style="color: #666666">18</span>  <span style="color: #408080; font-style: italic"># A quite high value, just to show.</span>
-noise <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
+spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
+J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
 
-<span style="color: #408080; font-style: italic"># Make data set.</span>
-x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-1</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
-y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
+energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
 
-<span style="color: #408080; font-style: italic"># Hold out some test data that is never used in training.</span>
-x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
-
-<span style="color: #408080; font-style: italic"># Combine x transformation and model into one operation.</span>
-<span style="color: #408080; font-style: italic"># Not neccesary, but convenient.</span>
-model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000">False</span>))
-
-<span style="color: #408080; font-style: italic"># The following (m x n_bootstraps) matrix holds the column vectors y_pred</span>
-<span style="color: #408080; font-style: italic"># for each bootstrap iteration.</span>
-y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
-    x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
-
-    <span style="color: #408080; font-style: italic"># Evaluate the new model on the same test data each time.</span>
-    y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
-
-<span style="color: #408080; font-style: italic"># Note: Expectations and variances taken w.r.t. different training</span>
-<span style="color: #408080; font-style: italic"># data sets, hence the axis=1. Subsequent means are taken across the test data</span>
-<span style="color: #408080; font-style: italic"># set in order to obtain a total value, but before this we have error/bias/variance</span>
-<span style="color: #408080; font-style: italic"># calculated per data point in the test set.</span>
-<span style="color: #408080; font-style: italic"># Note 2: The use of keepdims=True is important in the calculation of bias as this </span>
-<span style="color: #408080; font-style: italic"># maintains the column vector form. Dropping this yields very unexpected results.</span>
-error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
-bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>))<span style="color: #666666">**2</span> )
-variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
-<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error)
-<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias)
-<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance)
-<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;{} &gt;= {} + {} = {}&#39;</span><span style="color: #666666">.</span>format(error, bias, variance, bias<span style="color: #666666">+</span>variance))
-
-plt<span style="color: #666666">.</span>plot(x[::<span style="color: #666666">5</span>, :], y[::<span style="color: #666666">5</span>, :], label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;f(x)&#39;</span>)
-plt<span style="color: #666666">.</span>scatter(x_test, y_test, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Data points&#39;</span>)
-plt<span style="color: #666666">.</span>scatter(x_test, np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>), label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Pred&#39;</span>)
-plt<span style="color: #666666">.</span>legend()
-plt<span style="color: #666666">.</span>show()
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
+    energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
 </pre></div>
+<p>
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs096.html b/doc/pub/Regression/html/._Regression-bs096.html
index 06b8f4b93..79bd7416f 100644
--- a/doc/pub/Regression/html/._Regression-bs096.html
+++ b/doc/pub/Regression/html/._Regression-bs096.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,56 +383,52 @@ MathJax.Hub.Config({
 <a name="part0096"></a>
 <!-- !split -->
 
-<h2 id="___sec95" class="anchor">Understanding what happens </h2>
+<h2 id="___sec95" class="anchor">Reformulating the problem to suit regression </h2>
+
+<p>
+A more general form for the one-dimensional Ising model is
+
+$$
+\begin{align}
+    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\tag{22}
+\end{align}
+$$
+
+<p>
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+$$
+\begin{align}
+    \boldsymbol{H} = \boldsymbol{X} J,
+\tag{23}
+\end{align}
+$$
+
+<p>
+where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+$$
+\begin{align}
+    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\tag{24}
+\end{align}
+$$
+
+<p>
+We split the data in training and test data as discussed in the previous example
+
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
-
-np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
-
-n <span style="color: #666666">=</span> <span style="color: #666666">40</span>
-n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
-maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
-
-
-<span style="color: #408080; font-style: italic"># Make data set.</span>
-x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
-y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
-error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
-bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
-variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
-polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
-x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
-
-<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
-    model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000">False</span>))
-    y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
-    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
-        x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
-        y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
-
-    polydegree[degree] <span style="color: #666666">=</span> degree
-    error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
-    bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>))<span style="color: #666666">**2</span> )
-    variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000">True</span>) )
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Polynomial degree:&#39;</span>, degree)
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error[degree])
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias[degree])
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance[degree])
-    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;{} &gt;= {} + {} = {}&#39;</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
-
-plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Error&#39;</span>)
-plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;bias&#39;</span>)
-plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Variance&#39;</span>)
-plt<span style="color: #666666">.</span>legend()
-plt<span style="color: #666666">.</span>show()
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
+    X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
+y <span style="color: #666666">=</span> energies
+X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
 </pre></div>
 <p>
 <p>
diff --git a/doc/pub/Regression/html/._Regression-bs097.html b/doc/pub/Regression/html/._Regression-bs097.html
index 8e9f10846..548459001 100644
--- a/doc/pub/Regression/html/._Regression-bs097.html
+++ b/doc/pub/Regression/html/._Regression-bs097.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -381,39 +381,53 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 
 <a name="part0097"></a>
-<!-- !split  -->
+<!-- !split -->
 
-<h2 id="___sec96" class="anchor">Summing up </h2>
+<h2 id="___sec96" class="anchor">Linear regression </h2>
 
 <p>
-The bias-variance tradeoff summarizes the fundamental tension in
-machine learning, particularly supervised learning, between the
-complexity of a model and the amount of training data needed to train
-it.  Since data is often limited, in practice it is often useful to
-use a less-complex model with higher bias, that is  a model whose asymptotic
-performance is worse than another model because it is easier to
-train and less sensitive to sampling noise arising from having a
-finite-sized training dataset (smaller variance).
+In the ordinary least squares method we choose the cost function
+
+$$
+\begin{align}
+    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\tag{25}
+\end{align}
+$$
 
 <p>
-The above equations tell us that in
-order to minimize the expected test error, we need to select a
-statistical learning method that simultaneously achieves low variance
-and low bias. Note that variance is inherently a nonnegative quantity,
-and squared bias is also nonnegative. Hence, we see that the expected
-test MSE can never lie below \( Var(\epsilon) \), the irreducible error.
+We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
+This yields the expression for \( \boldsymbol{\beta} \) to be
+
+$$
+    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+$$
 
 <p>
-What do we mean by the variance and bias of a statistical learning
-method? The variance refers to the amount by which our model would change if we
-estimated it using a different training data set. Since the training
-data are used to fit the statistical learning method, different
-training data sets  will result in a different estimate. But ideally the
-estimate for our model should not vary too much between training
-sets. However, if a method has high variance  then small changes in
-the training data can result in large changes in the model. In general, more
-flexible statistical methods have higher variance.
+which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
+an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
 
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
+    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
+    axis<span style="color: #666666">=1</span>
+)
+X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
+    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
+    axis<span style="color: #666666">=1</span>
+)
+</pre></div>
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_inv</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
+    <span style="color: #008000; font-weight: bold">return</span> scl<span style="color: #666666">.</span>inv(x<span style="color: #666666">.</span>T @ x) @ (x<span style="color: #666666">.</span>T @ y)
+beta <span style="color: #666666">=</span> ols_inv(X_train_own, y_train)
+</pre></div>
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs098.html b/doc/pub/Regression/html/._Regression-bs098.html
index 8b9924c73..0dd8464db 100644
--- a/doc/pub/Regression/html/._Regression-bs098.html
+++ b/doc/pub/Regression/html/._Regression-bs098.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -383,82 +383,90 @@ MathJax.Hub.Config({
 <a name="part0098"></a>
 <!-- !split -->
 
-<h2 id="___sec97" class="anchor">Another Example rom Scikit-Learn's Repository </h2>
+<h2 id="___sec97" class="anchor">Singular Value decomposition </h2>
+
+<p>
+Doing the inversion directly turns out to be a bad idea since the matrix
+\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
+value decomposition</b>. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+
+$$
+    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+
+<p>
+where the pseudoinverse of \( \boldsymbol{X} \) is given by
+
+$$
+    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+
+<p>
+Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
+where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
+where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
+\( \omega \) to
+$$
+\begin{align}
+    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\tag{26}
+\end{align}
+$$
+
+<p>
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
+general matrix. Instead, doing \( QR \)-factorization and solving the
+linear system as an equation would reduce this down to
+\( \mathcal{O}(n^2) \) operations.
+
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
-<span style="color: #BA2121; font-style: italic">============================</span>
-<span style="color: #BA2121; font-style: italic">Underfitting vs. Overfitting</span>
-<span style="color: #BA2121; font-style: italic">============================</span>
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_svd</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
+    u, s, v <span style="color: #666666">=</span> scl<span style="color: #666666">.</span>svd(x)
+    <span style="color: #008000; font-weight: bold">return</span> v<span style="color: #666666">.</span>T @ scl<span style="color: #666666">.</span>pinv(scl<span style="color: #666666">.</span>diagsvd(s, u<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], v<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])) @ u<span style="color: #666666">.</span>T @ y
+</pre></div>
+<p>
 
-<span style="color: #BA2121; font-style: italic">This example demonstrates the problems of underfitting and overfitting and</span>
-<span style="color: #BA2121; font-style: italic">how we can use linear regression with polynomial features to approximate</span>
-<span style="color: #BA2121; font-style: italic">nonlinear functions. The plot shows the function that we want to approximate,</span>
-<span style="color: #BA2121; font-style: italic">which is a part of the cosine function. In addition, the samples from the</span>
-<span style="color: #BA2121; font-style: italic">real function and the approximations of different models are displayed. The</span>
-<span style="color: #BA2121; font-style: italic">models have polynomial features of different degrees. We can see that a</span>
-<span style="color: #BA2121; font-style: italic">linear function (polynomial with degree 1) is not sufficient to fit the</span>
-<span style="color: #BA2121; font-style: italic">training samples. This is called **underfitting**. A polynomial of degree 4</span>
-<span style="color: #BA2121; font-style: italic">approximates the true function almost perfectly. However, for higher degrees</span>
-<span style="color: #BA2121; font-style: italic">the model will **overfit** the training data, i.e. it learns the noise of the</span>
-<span style="color: #BA2121; font-style: italic">training data.</span>
-<span style="color: #BA2121; font-style: italic">We evaluate quantitatively **overfitting** / **underfitting** by using</span>
-<span style="color: #BA2121; font-style: italic">cross-validation. We calculate the mean squared error (MSE) on the validation</span>
-<span style="color: #BA2121; font-style: italic">set, the higher, the less likely the model generalizes correctly from the</span>
-<span style="color: #BA2121; font-style: italic">training data.</span>
-<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>beta <span style="color: #666666">=</span> ols_svd(X_train_own,y_train)
+</pre></div>
+<p>
+When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
 
-<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #19177C">__doc__</span>)
+<p>
 
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J <span style="color: #666666">=</span> beta[<span style="color: #666666">1</span>:]<span style="color: #666666">.</span>reshape(L, L)
+</pre></div>
+<p>
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
 
+<p>
 
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">true_fun</span>(X):
-    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>pi <span style="color: #666666">*</span> X)
-
-np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
-
-n_samples <span style="color: #666666">=</span> <span style="color: #666666">30</span>
-degrees <span style="color: #666666">=</span> [<span style="color: #666666">1</span>, <span style="color: #666666">4</span>, <span style="color: #666666">15</span>]
-
-X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n_samples))
-y <span style="color: #666666">=</span> true_fun(X) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n_samples) <span style="color: #666666">*</span> <span style="color: #666666">0.1</span>
-
-plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">14</span>, <span style="color: #666666">5</span>))
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(degrees)):
-    ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(degrees), i <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
-    plt<span style="color: #666666">.</span>setp(ax, xticks<span style="color: #666666">=</span>(), yticks<span style="color: #666666">=</span>())
-
-    polynomial_features <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=</span>degrees[i],
-                                             include_bias<span style="color: #666666">=</span><span style="color: #008000">False</span>)
-    linear_regression <span style="color: #666666">=</span> LinearRegression()
-    pipeline <span style="color: #666666">=</span> Pipeline([(<span style="color: #BA2121">&quot;polynomial_features&quot;</span>, polynomial_features),
-                         (<span style="color: #BA2121">&quot;linear_regression&quot;</span>, linear_regression)])
-    pipeline<span style="color: #666666">.</span>fit(X[:, np<span style="color: #666666">.</span>newaxis], y)
-
-    <span style="color: #408080; font-style: italic"># Evaluate the models using crossvalidation</span>
-    scores <span style="color: #666666">=</span> cross_val_score(pipeline, X[:, np<span style="color: #666666">.</span>newaxis], y,
-                             scoring<span style="color: #666666">=</span><span style="color: #BA2121">&quot;neg_mean_squared_error&quot;</span>, cv<span style="color: #666666">=10</span>)
-
-    X_test <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">100</span>)
-    plt<span style="color: #666666">.</span>plot(X_test, pipeline<span style="color: #666666">.</span>predict(X_test[:, np<span style="color: #666666">.</span>newaxis]), label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Model&quot;</span>)
-    plt<span style="color: #666666">.</span>plot(X_test, true_fun(X_test), label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;True function&quot;</span>)
-    plt<span style="color: #666666">.</span>scatter(X, y, edgecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;b&#39;</span>, s<span style="color: #666666">=20</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Samples&quot;</span>)
-    plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;x&quot;</span>)
-    plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;y&quot;</span>)
-    plt<span style="color: #666666">.</span>xlim((<span style="color: #666666">0</span>, <span style="color: #666666">1</span>))
-    plt<span style="color: #666666">.</span>ylim((<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>))
-    plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">&quot;best&quot;</span>)
-    plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Degree {}</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">MSE = {:.2e}(+/- {:.2e})&quot;</span><span style="color: #666666">.</span>format(
-        degrees[i], <span style="color: #666666">-</span>scores<span style="color: #666666">.</span>mean(), scores<span style="color: #666666">.</span>std()))
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
+im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J, <span style="color: #666666">**</span>cmap_args)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;OLS&quot;</span>, fontsize<span style="color: #666666">=18</span>)
+plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
+plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
+cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
+cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
 plt<span style="color: #666666">.</span>show()
 </pre></div>
+<p>
+It is interesting to note that OLS
+considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
+valid matrix elements for \( J \).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+<p>
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/Regression/html/._Regression-bs099.html b/doc/pub/Regression/html/._Regression-bs099.html
index adbfdb03b..fcf2eeaf6 100644
--- a/doc/pub/Regression/html/._Regression-bs099.html
+++ b/doc/pub/Regression/html/._Regression-bs099.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
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+               '___sec70'),
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
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-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
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-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
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                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
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                None,
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-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
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               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs100.html b/doc/pub/Regression/html/._Regression-bs100.html
index 4e5642d4f..c4fd86968 100644
--- a/doc/pub/Regression/html/._Regression-bs100.html
+++ b/doc/pub/Regression/html/._Regression-bs100.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
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                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs101.html b/doc/pub/Regression/html/._Regression-bs101.html
index d45e0422a..4ee0509a5 100644
--- a/doc/pub/Regression/html/._Regression-bs101.html
+++ b/doc/pub/Regression/html/._Regression-bs101.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs102.html b/doc/pub/Regression/html/._Regression-bs102.html
index b7ad177cc..2b93c0821 100644
--- a/doc/pub/Regression/html/._Regression-bs102.html
+++ b/doc/pub/Regression/html/._Regression-bs102.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
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+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
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+              ('More on Ridge Regression', 2, None, '___sec38'),
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+              ('More interpretations', 2, None, '___sec40'),
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-              ('Statistics and sample variables', 2, None, '___sec60'),
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+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
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+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/._Regression-bs103.html b/doc/pub/Regression/html/._Regression-bs103.html
index f5502b8fc..f14984238 100644
--- a/doc/pub/Regression/html/._Regression-bs103.html
+++ b/doc/pub/Regression/html/._Regression-bs103.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
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                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
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-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
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-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
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-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
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               ('Linking the regression analysis with a statistical '
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                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
diff --git a/doc/pub/Regression/html/Regression-bs.html b/doc/pub/Regression/html/Regression-bs.html
index 786ff5b64..b5293106b 100644
--- a/doc/pub/Regression/html/Regression-bs.html
+++ b/doc/pub/Regression/html/Regression-bs.html
@@ -111,112 +111,112 @@ Automatically generated HTML file from DocOnce source
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
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+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -295,74 +295,74 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The singular value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">The Ising model</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linear regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Singular Value decomposition</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Where are we going?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Resampling methods</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Why resampling methods ?</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistical analysis</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Statistics, moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Statistics, central moments</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Covariance example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Covariance in numpy</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, independent variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, more variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics and stochastic processes</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics and sample variables</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, law of large numbers</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, more on sample error</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, central limit theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, more technicalities</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics and sample variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, computations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, final expression</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Assumptions made</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Expectation value and variance</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Computationally expensive</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Various steps in cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Jackknife code example</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">The bias-variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Understanding what happens</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Summing up</a></li>
-     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Linear Regression Problems</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Fixing the singularity</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Basic math of the SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Another Example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Economy-size SVD</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Mathematical Properties</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">More on Ridge Regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Interpreting the Ridge results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">More interpretations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Where are we going?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Resampling methods</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Why resampling methods ?</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Statistical analysis</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Statistics, moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Statistics, central moments</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Statistics, covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Statistics, more covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Covariance example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Covariance in numpy</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Statistics, independent variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Statistics, more variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Statistics and stochastic processes</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Statistics and sample variables</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Statistics, law of large numbers</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Statistics, more on sample error</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics, central limit theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, more technicalities</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics and sample variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Statistics, computations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, final expression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Assumptions made</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Expectation value and variance</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Computationally expensive</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Various steps in cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Jackknife code example</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">The bias-variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Understanding what happens</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Summing up</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Another Example rom Scikit-Learn's Repository</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">The Ising model</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Linear regression</a></li>
+     <!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Singular Value decomposition</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">The one-dimensional Ising model</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Ridge regression</a></li>
      <!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">LASSO regression</a></li>
@@ -402,7 +402,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>
-<center><h4>Aug 29, 2019</h4></center> <!-- date -->
+<center><h4>Aug 30, 2019</h4></center> <!-- date -->
 <br>
 <p>
 
diff --git a/doc/pub/Regression/html/Regression-reveal.html b/doc/pub/Regression/html/Regression-reveal.html
index 43534fae6..4f66cfa17 100644
--- a/doc/pub/Regression/html/Regression-reveal.html
+++ b/doc/pub/Regression/html/Regression-reveal.html
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>&nbsp;<br>
-<center><h4>Aug 29, 2019</h4></center> <!-- date -->
+<center><h4>Aug 30, 2019</h4></center> <!-- date -->
 <br>
 <p>
 
@@ -1490,8 +1490,7 @@ The examples we have looked at so far are cases where we normally can
 invert the matrix \( \boldsymbol{X}^T\boldsymbol{X} \). Using a polynomial expansion as we
 did both for the masses and the fitting of the equation of state,
 leads to row vectors of the design matrix which are essentially
-orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. 
-More material to come here.
+orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition.
 
 <p>
 This may
@@ -1509,276 +1508,11 @@ inversion algorithm. Thereafter we dive into the math of the SVD.
 
 
 </div>
-
-<p>
-<!-- todo: change model here. -->
 </section>
 
 
 <section>
-<h2 id="___sec30">The Ising model </h2>
-
-<p>
-The one-dimensional Ising model with nearest neighbor interaction, no
-external field and a constant coupling constant \( J \) is given by
-
-<p>&nbsp;<br>
-$$
-\begin{align}
-    H = -J \sum_{k}^L s_k s_{k + 1},
-\tag{1}
-\end{align}
-$$
-<p>&nbsp;<br>
-
-<p>
-where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
-in the system is determined by \( L \). For the one-dimensional system
-there is no phase transition.
-
-<p>
-We will look at a system of \( L = 40 \) spins with a coupling constant of
-\( J = 1 \). To get enough training data we will generate 10000 states
-with their respective energies.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
-<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">mpl_toolkits.axes_grid1</span> <span style="color: #8B008B; font-weight: bold">import</span> make_axes_locatable
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">seaborn</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">sns</span>
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">scipy.linalg</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">scl</span>
-<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">tqdm</span>
-sns.set(color_codes=<span style="color: #658b00">True</span>)
-cmap_args=<span style="color: #658b00">dict</span>(vmin=-<span style="color: #B452CD">1.</span>, vmax=<span style="color: #B452CD">1.</span>, cmap=<span style="color: #CD5555">&#39;seismic&#39;</span>)
-
-L = <span style="color: #B452CD">40</span>
-n = <span style="color: #658b00">int</span>(<span style="color: #B452CD">1e4</span>)
-
-spins = np.random.choice([-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>], size=(n, L))
-J = <span style="color: #B452CD">1.0</span>
-
-energies = np.zeros(n)
-
-<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
-    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], <span style="color: #B452CD">1</span>))
-</pre></div>
-<p>
-Here we use ordinary least squares
-regression to predict the energy for the nearest neighbor
-one-dimensional Ising model on a ring, i.e., the endpoints wrap
-around. We will use linear regression to fit a value for
-the coupling constant to achieve this.
-</section>
-
-
-<section>
-<h2 id="___sec31">Reformulating the problem to suit regression </h2>
-
-<p>
-A more general form for the one-dimensional Ising model is
-
-<p>&nbsp;<br>
-$$
-\begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\tag{2}
-\end{align}
-$$
-<p>&nbsp;<br>
-
-<p>
-Here we allow for interactions beyond the nearest neighbors and a state dependent
-coupling constant. This latter expression can be formulated as
-a matrix-product
-<p>&nbsp;<br>
-$$
-\begin{align}
-    \boldsymbol{H} = \boldsymbol{X} J,
-\tag{3}
-\end{align}
-$$
-<p>&nbsp;<br>
-
-<p>
-where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
-elements \( -J_{jk} \). This form of writing the energy fits perfectly
-with the form utilized in linear regression, that is
-
-<p>&nbsp;<br>
-$$
-\begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
-\tag{4}
-\end{align}
-$$
-<p>&nbsp;<br>
-
-<p>
-We split the data in training and test data as discussed in the previous example
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>X = np.zeros((n, L ** <span style="color: #B452CD">2</span>))
-<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
-    X[i] = np.outer(spins[i], spins[i]).ravel()
-y = energies
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=<span style="color: #B452CD">0.2</span>)
-</pre></div>
-</section>
-
-
-<section>
-<h2 id="___sec32">Linear regression </h2>
-
-<p>
-In the ordinary least squares method we choose the cost function
-
-<p>&nbsp;<br>
-$$
-\begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
-\tag{5}
-\end{align}
-$$
-<p>&nbsp;<br>
-
-<p>
-We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
-This yields the expression for \( \boldsymbol{\beta} \) to be
-
-<p>&nbsp;<br>
-$$
-    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
-$$
-<p>&nbsp;<br>
-
-<p>
-which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
-an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
-intercept, i.e., a constant term, we must make sure that the
-first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>X_train_own = np.concatenate(
-    (np.ones(<span style="color: #658b00">len</span>(X_train))[:, np.newaxis], X_train),
-    axis=<span style="color: #B452CD">1</span>
-)
-X_test_own = np.concatenate(
-    (np.ones(<span style="color: #658b00">len</span>(X_test))[:, np.newaxis], X_test),
-    axis=<span style="color: #B452CD">1</span>
-)
-</pre></div>
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_inv</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
-    <span style="color: #8B008B; font-weight: bold">return</span> scl.inv(x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> x) <span style="color: #a61717; background-color: #e3d2d2">@</span> (x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y)
-beta = ols_inv(X_train_own, y_train)
-</pre></div>
-</section>
-
-
-<section>
-<h2 id="___sec33">Singular Value decomposition </h2>
-
-<p>
-Doing the inversion directly turns out to be a bad idea since the matrix
-\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
-value decomposition</b>. Using the definition of the Moore-Penrose
-pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
-
-<p>&nbsp;<br>
-$$
-    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
-$$
-<p>&nbsp;<br>
-
-<p>
-where the pseudoinverse of \( \boldsymbol{X} \) is given by
-
-<p>&nbsp;<br>
-$$
-    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
-$$
-<p>&nbsp;<br>
-
-<p>
-Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
-where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
-where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
-\( \omega \) to
-<p>&nbsp;<br>
-$$
-\begin{align}
-    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
-\tag{6}
-\end{align}
-$$
-<p>&nbsp;<br>
-
-<p>
-Note that solving this equation by actually doing the pseudoinverse
-(which is what we will do) is not a good idea as this operation scales
-as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
-general matrix. Instead, doing \( QR \)-factorization and solving the
-linear system as an equation would reduce this down to
-\( \mathcal{O}(n^2) \) operations.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_svd</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
-    u, s, v = scl.svd(x)
-    <span style="color: #8B008B; font-weight: bold">return</span> v.T <span style="color: #a61717; background-color: #e3d2d2">@</span> scl.pinv(scl.diagsvd(s, u.shape[<span style="color: #B452CD">0</span>], v.shape[<span style="color: #B452CD">0</span>])) <span style="color: #a61717; background-color: #e3d2d2">@</span> u.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y
-</pre></div>
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>beta = ols_svd(X_train_own,y_train)
-</pre></div>
-<p>
-When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>J = beta[<span style="color: #B452CD">1</span>:].reshape(L, L)
-</pre></div>
-<p>
-A way of looking at the coefficients in \( J \) is to plot the matrices as images.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>fig = plt.figure(figsize=(<span style="color: #B452CD">20</span>, <span style="color: #B452CD">14</span>))
-im = plt.imshow(J, **cmap_args)
-plt.title(<span style="color: #CD5555">&quot;OLS&quot;</span>, fontsize=<span style="color: #B452CD">18</span>)
-plt.xticks(fontsize=<span style="color: #B452CD">18</span>)
-plt.yticks(fontsize=<span style="color: #B452CD">18</span>)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=<span style="color: #B452CD">18</span>)
-plt.show()
-</pre></div>
-<p>
-It is interesting to note that OLS
-considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
-valid matrix elements for \( J \).
-In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
-this problem can be removed, partly and only with Lasso regression.
-
-<p>
-In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
-</section>
-
-
-<section>
-<h2 id="___sec34">Linear Regression Problems </h2>
+<h2 id="___sec30">Linear Regression Problems </h2>
 
 <p>
 One of the typical problems we encounter with linear regression, in particular 
@@ -1813,7 +1547,7 @@ of linearly independent columns. In this particular case the matrix has rank 2.
 
 <p>
 Super-collinearity of an \( (n \times p) \)-dimensional design matrix \( \mathbf{X} \) implies
-that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{x} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
+that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
 <p>&nbsp;<br>
 $$
 \begin{align*}
@@ -1833,7 +1567,7 @@ This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least a
 
 
 <section>
-<h2 id="___sec35">Fixing the singularity </h2>
+<h2 id="___sec31">Fixing the singularity </h2>
 
 <p>
 If our design matrix \( \boldsymbol{X} \) which enters the linear regression problem
@@ -1841,7 +1575,7 @@ If our design matrix \( \boldsymbol{X} \) which enters the linear regression pro
 $$
 \begin{align}
 \boldsymbol{\beta} & =  (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y},
-\tag{7}
+\tag{1}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -1865,7 +1599,7 @@ where \( \boldsymbol{I} \) is the identity matrix.  When we discuss <b>Ridge</b>
 
 
 <section>
-<h2 id="___sec36">Basic math of the SVD </h2>
+<h2 id="___sec32">Basic math of the SVD </h2>
 
 <p>
 From standard linear algebra we know that a square matrix \( \boldsymbol{X} \) can be diagonalized if and only it is 
@@ -1912,7 +1646,7 @@ is not diagonalizable, it is a so-called <a href="https://en.wikipedia.org/wiki/
 
 
 <section>
-<h2 id="___sec37">The SVD, a Fantastic Algorithm </h2>
+<h2 id="___sec33">The SVD, a Fantastic Algorithm </h2>
 
 <p>
 However, and this is the strength of the SVD algorithm, any general
@@ -1947,7 +1681,7 @@ The SVD exits always!
 
 
 <section>
-<h2 id="___sec38">Another Example </h2>
+<h2 id="___sec34">Another Example </h2>
 
 <p>
 Consider the following matrix which can be SVD decomposed as
@@ -1974,7 +1708,7 @@ In the general case, where our design matrix \( \boldsymbol{X} \) has dimension
 \( n\times p \), the matrix is thus decomposed into an \( n\times n \)
 orthogonal matrix \( \boldsymbol{U} \), a \( p\times p \) orthogonal matrix \( \boldsymbol{V} \)
 and a diagonal matrix \( \boldsymbol{\Sigma} \) with \( r=\mathrm{min}(n,p) \)
-singular values \( \sigma_i\lg 0 \) on the main diagonal and zeros filling
+singular values \( \sigma_i\geq 0 \) on the main diagonal and zeros filling
 the rest of the matrix.  There are at most \( p \) singular values
 assuming that \( n > p \). In our regression examples for the nuclear
 masses and the equation of state this is indeed the case, while for
@@ -1987,7 +1721,7 @@ The columns of \( \boldsymbol{U} \) are called the left singular vectors while t
 
 
 <section>
-<h2 id="___sec39">Economy-size SVD </h2>
+<h2 id="___sec35">Economy-size SVD </h2>
 
 <p>
 If we assume that \( n > p \), then our matrix \( \boldsymbol{U} \) has dimension \( n
@@ -2012,7 +1746,7 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
 
 
 <section>
-<h2 id="___sec40">Mathematical Properties </h2>
+<h2 id="___sec36">Mathematical Properties </h2>
 
 <p>
 There are several interesting mathematical properties which will be
@@ -2076,7 +1810,7 @@ We will come back to this expression when we discuss Ridge regression.
 
 
 <section>
-<h2 id="___sec41">Ridge and LASSO Regression </h2>
+<h2 id="___sec37">Ridge and LASSO Regression </h2>
 
 <p>
 Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
@@ -2148,7 +1882,7 @@ $$
 
 
 <section>
-<h2 id="___sec42">More on Ridge Regression </h2>
+<h2 id="___sec38">More on Ridge Regression </h2>
 
 <p>
 Using the matrix-vector expression for Ridge regression,
@@ -2219,7 +1953,7 @@ with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \
 
 
 <section>
-<h2 id="___sec43">Interpreting the Ridge results </h2>
+<h2 id="___sec39">Interpreting the Ridge results </h2>
 
 <p>
 Since \( \lambda \geq 0 \), it means that compared to OLS, we have
@@ -2245,7 +1979,7 @@ With a parameter \( \lambda \) we can thus shrink the role of specific parameter
 
 
 <section>
-<h2 id="___sec44">More interpretations </h2>
+<h2 id="___sec40">More interpretations </h2>
 
 <p>
 For the sake of simplicity, let us assume that the design matrix is orthonormal, that is
@@ -2288,7 +2022,7 @@ Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et a
 
 
 <section>
-<h2 id="___sec45">Where are we going? </h2>
+<h2 id="___sec41">Where are we going? </h2>
 
 <p>
 Before we proceed, we need to rethink what we have been doing. In our
@@ -2306,7 +2040,7 @@ This will allow us to link the standard linear algebra methods we have discussed
 
 
 <section>
-<h2 id="___sec46">Resampling methods </h2>
+<h2 id="___sec42">Resampling methods </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2325,7 +2059,7 @@ once using the original training sample.
 
 
 <section>
-<h2 id="___sec47">Resampling approaches can be computationally expensive </h2>
+<h2 id="___sec43">Resampling approaches can be computationally expensive </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2351,7 +2085,7 @@ bootstrap is widely used.
 
 
 <section>
-<h2 id="___sec48">Why resampling methods ? </h2>
+<h2 id="___sec44">Why resampling methods ? </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b>Statistical analysis.</b>
 <ul>
@@ -2364,7 +2098,7 @@ bootstrap is widely used.
 
 
 <section>
-<h2 id="___sec49">Statistical analysis </h2>
+<h2 id="___sec45">Statistical analysis </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <ul>
@@ -2384,7 +2118,7 @@ bootstrap is widely used.
 
 
 <section>
-<h2 id="___sec50">Statistics  </h2>
+<h2 id="___sec46">Statistics  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2418,7 +2152,7 @@ selection of a large set of these numbers reproduces this PDF.
 
 
 <section>
-<h2 id="___sec51">Statistics, moments </h2>
+<h2 id="___sec47">Statistics, moments </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2444,7 +2178,7 @@ $$
 
 
 <section>
-<h2 id="___sec52">Statistics, central moments </h2>
+<h2 id="___sec48">Statistics, central moments </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2465,14 +2199,14 @@ $$
 \begin{align}
 \sigma^2_X\ \ =\ \ \mathrm{var}(X) & =  \langle (x-\langle x\rangle)^2\rangle =
 \int\! (x-\langle x\rangle)^2 p(x)\,dx
-\tag{8}\\
+\tag{2}\\
 & =  \int\! \left(x^2 - 2 x \langle x\rangle^{2} +
   \langle x\rangle^2\right)p(x)\,dx
-\tag{9}\\
+\tag{3}\\
 & =  \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2
-\tag{10}\\
+\tag{4}\\
 & =  \langle x^2\rangle - \langle x\rangle^2
-\tag{11}
+\tag{5}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -2485,7 +2219,7 @@ qualitatively as the <em>spread</em> of \( p \) around its mean.
 
 
 <section>
-<h2 id="___sec53">Statistics, covariance </h2>
+<h2 id="___sec49">Statistics, covariance </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2502,7 +2236,7 @@ $$
 &=
 \int\!\cdots\!\int\!(x_i-\langle x_i \rangle)(x_j-\langle x_j \rangle)\,
 P(x_1,\dots,x_n)\,dx_1\dots dx_n
-\tag{12}
+\tag{6}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -2519,7 +2253,7 @@ $$
 
 
 <section>
-<h2 id="___sec54">Statistics, more covariance  </h2>
+<h2 id="___sec50">Statistics, more covariance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2533,17 +2267,17 @@ the expectation value. Consider the stochastic variables \( X_i \) and
 $$
 \begin{align}
 \mathrm{cov}(X_i,\,X_j) &= \langle(x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
-\tag{13}\\
+\tag{7}\\
 &=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle 
-\tag{14}\\
+\tag{8}\\
 &=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j\rangle +
 \langle \langle x_i\rangle\langle x_j\rangle\rangle
-\tag{15}\\
+\tag{9}\\
 &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle +
 \langle x_i\rangle\langle x_j\rangle
-\tag{16}\\
+\tag{10}\\
 &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle
-\tag{17}
+\tag{11}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -2552,7 +2286,7 @@ $$
 
 
 <section>
-<h2 id="___sec55">Covariance example </h2>
+<h2 id="___sec51">Covariance example </h2>
 
 <p>
 Suppose we have defined three vectors \( \hat{x}, \hat{y}, \hat{z} \) with
@@ -2607,7 +2341,7 @@ function.
 
 
 <section>
-<h2 id="___sec56">Covariance in numpy </h2>
+<h2 id="___sec52">Covariance in numpy </h2>
 
 <p>
 
@@ -2647,7 +2381,7 @@ plt.show()
 
 
 <section>
-<h2 id="___sec57">Statistics, independent variables  </h2>
+<h2 id="___sec53">Statistics, independent variables  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2676,7 +2410,7 @@ $$
 
 
 <section>
-<h2 id="___sec58">Statistics, more variance  </h2>
+<h2 id="___sec54">Statistics, more variance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2686,7 +2420,7 @@ the variance of the linear combination \( U = \sum_i a_i X_i \):
 $$
 \begin{equation}
 \mathrm{var}(U) = \sum_{i,j}a_i a_j \mathrm{cov}(X_i, X_j)
-\tag{18}
+\tag{12}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -2713,7 +2447,7 @@ value of a set of measurements.
 
 
 <section>
-<h2 id="___sec59">Statistics and stochastic processes  </h2>
+<h2 id="___sec55">Statistics and stochastic processes  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2742,7 +2476,7 @@ interested in finding the few lowest moments, like the mean
 
 
 <section>
-<h2 id="___sec60">Statistics and sample variables  </h2>
+<h2 id="___sec56">Statistics and sample variables  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2773,12 +2507,12 @@ $$
 
 
 <section>
-<h2 id="___sec61">Statistics, sample variance and covariance </h2>
+<h2 id="___sec57">Statistics, sample variance and covariance </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
 Note that the sample variance is the sample covariance without the
-cross terms. In a similar manner as the covariance in Eq.&nbsp;<a href="#mjx-eqn-12">(12)</a> is a measure of the correlation between
+cross terms. In a similar manner as the covariance in Eq.&nbsp;<a href="#mjx-eqn-6">(6)</a> is a measure of the correlation between
 two stochastic variables, the above defined sample covariance is a
 measure of the sequential correlation between succeeding measurements
 of a sample.
@@ -2793,7 +2527,7 @@ and covariance \( \mathrm{cov}(X,Y) \).
 
 
 <section>
-<h2 id="___sec62">Statistics, law of large numbers </h2>
+<h2 id="___sec58">Statistics, law of large numbers </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2826,7 +2560,7 @@ true PDFs behind, which we usually do not have.
 
 
 <section>
-<h2 id="___sec63">Statistics, more on sample error </h2>
+<h2 id="___sec59">Statistics, more on sample error </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2849,7 +2583,7 @@ means.
 
 
 <section>
-<h2 id="___sec64">Statistics  </h2>
+<h2 id="___sec60">Statistics  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2871,7 +2605,7 @@ And in particular we are interested in its variance \( \mathrm{var}(\overline X_
 
 
 <section>
-<h2 id="___sec65">Statistics, central limit theorem </h2>
+<h2 id="___sec61">Statistics, central limit theorem </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2888,7 +2622,7 @@ $$
 \lim_{n\to\infty} p_{\overline X_n}(x) =
 \left(\frac{n}{2\pi\mathrm{var}(X)}\right)^{1/2}
 e^{-\frac{n(x-\bar x_n)^2}{2\mathrm{var}(X)}}
-\tag{19}
+\tag{13}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -2897,7 +2631,7 @@ $$
 
 
 <section>
-<h2 id="___sec66">Statistics, more technicalities  </h2>
+<h2 id="___sec62">Statistics, more technicalities  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2909,7 +2643,7 @@ $$
 \begin{equation}
 \mathrm{err}_X^2 = \mathrm{var}(\overline X_n) = \frac{1}{n^2}
 \sum_{ij} \mathrm{cov}(X_i, X_j)
-\tag{20}
+\tag{14}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -2927,7 +2661,7 @@ estimate of the PDF of each of the \( X_i \), estimating all properties of
 
 
 <section>
-<h2 id="___sec67">Statistics </h2>
+<h2 id="___sec63">Statistics </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2940,7 +2674,7 @@ $$
 <p>&nbsp;<br>
 
 Using \( \bar x \) in place of \( \mu_{X_i}^{\phantom X} \) we can give an
-<em>estimate</em> of the covariance in Eq.&nbsp;<a href="#mjx-eqn-20">(20)</a>
+<em>estimate</em> of the covariance in Eq.&nbsp;<a href="#mjx-eqn-14">(14)</a>
 <p>&nbsp;<br>
 $$
 \mathrm{cov}(X_i, X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
@@ -2959,7 +2693,7 @@ $$
 
 
 <section>
-<h2 id="___sec68">Statistics and sample variance  </h2>
+<h2 id="___sec64">Statistics and sample variance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2976,7 +2710,7 @@ which is approximated as
 $$
 \begin{equation}
 \mathrm{var}(X_i)\approx \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)=\mathrm{var}(x)
-\tag{21}
+\tag{15}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -2991,7 +2725,7 @@ $$
 &=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) \nonumber \\
 &\approx&\frac{1}{n^2}\sum_{ij}\frac{1}{n}\mathrm{cov}(x) =\frac{1}{n^2}n^2\frac{1}{n}\mathrm{cov}(x)\nonumber\\
 &=\frac{1}{n}\mathrm{cov}(x)
-\tag{22}
+\tag{16}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -3003,7 +2737,7 @@ measurements in the sample.
 
 
 <section>
-<h2 id="___sec69">Statistics, uncorrelated results </h2>
+<h2 id="___sec65">Statistics, uncorrelated results </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3023,12 +2757,12 @@ resulting in
 $$
 \begin{equation}
 \mathrm{err}_X^2\approx \frac{1}{n^2} \sum_i \mathrm{var}(x)= \frac{1}{n}\mathrm{var}(x)
-\tag{23}
+\tag{17}
 \end{equation}
 $$
 <p>&nbsp;<br>
 
-where in the second step we have used Eq.&nbsp;<a href="#mjx-eqn-21">(21)</a>.
+where in the second step we have used Eq.&nbsp;<a href="#mjx-eqn-15">(15)</a>.
 The error of the sample is then just its standard deviation divided by
 the square root of the number of measurements the sample contains.
 This is a very useful formula which is easy to compute. It acts as a
@@ -3039,12 +2773,12 @@ cannot overlook the always present correlations.
 
 
 <section>
-<h2 id="___sec70">Statistics, computations </h2>
+<h2 id="___sec66">Statistics, computations </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
 For computational purposes one usually splits up the estimate of
-\( \mathrm{err}_X^2 \), given by Eq.&nbsp;<a href="#mjx-eqn-22">(22)</a>, into two
+\( \mathrm{err}_X^2 \), given by Eq.&nbsp;<a href="#mjx-eqn-16">(16)</a>, into two
 parts
 <p>&nbsp;<br>
 $$
@@ -3057,13 +2791,13 @@ which equals
 $$
 \begin{equation}
 \frac{1}{n^2}\sum_{k=1}^n (x_k - \bar x_n)^2 +\frac{2}{n^2}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n)
-\tag{24}
+\tag{18}
 \end{equation}
 $$
 <p>&nbsp;<br>
 
 The first term is the same as the error in the uncorrelated case,
-Eq.&nbsp;<a href="#mjx-eqn-23">(23)</a>. This means that the second
+Eq.&nbsp;<a href="#mjx-eqn-17">(17)</a>. This means that the second
 term accounts for the error correction due to correlation between the
 measurements. For uncorrelated measurements this second term is zero.
 </div>
@@ -3071,7 +2805,7 @@ measurements. For uncorrelated measurements this second term is zero.
 
 
 <section>
-<h2 id="___sec71">Statistics, more on computations of errors </h2>
+<h2 id="___sec67">Statistics, more on computations of errors </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3094,7 +2828,7 @@ have to be stored throughout the experiment.
 
 
 <section>
-<h2 id="___sec72">Statistics, wrapping up 1 </h2>
+<h2 id="___sec68">Statistics, wrapping up 1 </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3132,11 +2866,11 @@ starting always at \( 1 \) for \( d=0 \).
 
 
 <section>
-<h2 id="___sec73">Statistics, final expression </h2>
+<h2 id="___sec69">Statistics, final expression </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
-The sample error (see eq.&nbsp;<a href="#mjx-eqn-24">(24)</a>) can now be
+The sample error (see eq.&nbsp;<a href="#mjx-eqn-18">(18)</a>) can now be
 written in terms of the autocorrelation function:
 <p>&nbsp;<br>
 $$
@@ -3146,7 +2880,7 @@ $$
 \frac{f_d}{\mathrm{var}(x)}\nonumber\\ &=&
 \left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\mathrm{var}(x)\nonumber\\
 &=\frac{\tau}{n}\cdot\mathrm{var}(x)
-\tag{25}
+\tag{19}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -3159,7 +2893,7 @@ correction factor the <em>autocorrelation time</em>:
 $$
 \begin{equation}
 \tau = 1+2\sum_{d=1}^{n-1}\kappa_d
-\tag{26}
+\tag{20}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -3168,13 +2902,13 @@ $$
 
 
 <section>
-<h2 id="___sec74">Statistics, effective number of correlations </h2>
+<h2 id="___sec70">Statistics, effective number of correlations </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
 For a correlation free experiment, \( \tau \)
 equals 1. From the point of view of
-eq.&nbsp;<a href="#mjx-eqn-25">(25)</a> we can interpret a sequential
+eq.&nbsp;<a href="#mjx-eqn-19">(19)</a> we can interpret a sequential
 correlation as an effective reduction of the number of measurements by
 a factor \( \tau \). The effective number of measurements becomes:
 <p>&nbsp;<br>
@@ -3194,7 +2928,7 @@ measurements is very large.
 
 
 <section>
-<h2 id="___sec75">Linking the regression analysis with a statistical interpretation </h2>
+<h2 id="___sec71">Linking the regression analysis with a statistical interpretation </h2>
 
 <p>
 Finally, we are going to discuss several statistical properties which can be obtained in terms of analytical expressions. 
@@ -3234,7 +2968,7 @@ row number \( i \) and perform a sum over all values \( p \).
 
 
 <section>
-<h2 id="___sec76">Assumptions made </h2>
+<h2 id="___sec72">Assumptions made </h2>
 
 <p>
 The assumption we have made here can be summarized as (and this is going to useful when we discuss the bias-variance trade off)
@@ -3258,7 +2992,7 @@ $$
 
 
 <section>
-<h2 id="___sec77">Expectation value and variance </h2>
+<h2 id="___sec73">Expectation value and variance </h2>
 
 <p>
 We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) 
@@ -3297,7 +3031,7 @@ mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (n
 
 
 <section>
-<h2 id="___sec78">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
+<h2 id="___sec74">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
 
 <p>
 With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
@@ -3392,7 +3126,7 @@ This means the variance we obtain with the standard OLS will always for \( \lamb
 
 
 <section>
-<h2 id="___sec79">Cross-validation </h2>
+<h2 id="___sec75">Cross-validation </h2>
 
 <p>
 Instead of choosing the penalty parameter to balance model fit with
@@ -3426,7 +3160,7 @@ some sense) is then selected.
 
 
 <section>
-<h2 id="___sec80">Computationally expensive </h2>
+<h2 id="___sec76">Computationally expensive </h2>
 
 <p>
 The validation set approach is conceptually simple and is easy to implement. But it has two potential drawbacks:
@@ -3439,7 +3173,7 @@ The validation set approach is conceptually simple and is easy to implement. But
 
 
 <section>
-<h2 id="___sec81">Various steps in cross-validation </h2>
+<h2 id="___sec77">Various steps in cross-validation </h2>
 
 <p>
 When the repetitive splitting of the data set is done randomly,
@@ -3460,7 +3194,7 @@ cross-validation (LOOCV).
 
 
 <section>
-<h2 id="___sec82">How to set up the cross-validation for Ridge and/or Lasso </h2>
+<h2 id="___sec78">How to set up the cross-validation for Ridge and/or Lasso </h2>
 
 <ul>
 <p><li> Define a range of interest for the penalty parameter.</li>
@@ -3499,7 +3233,7 @@ $$
 
 
 <section>
-<h2 id="___sec83">Resampling methods: Jackknife and Bootstrap </h2>
+<h2 id="___sec79">Resampling methods: Jackknife and Bootstrap </h2>
 
 <p>
 Two famous
@@ -3522,7 +3256,7 @@ need for bootstrapping.
 
 
 <section>
-<h2 id="___sec84">Resampling methods: Jackknife </h2>
+<h2 id="___sec80">Resampling methods: Jackknife </h2>
 
 <p>
 The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). 
@@ -3543,7 +3277,7 @@ number \( i \) is left out. Using this notation, define
 
 
 <section>
-<h2 id="___sec85">Jackknife code example </h2>
+<h2 id="___sec81">Jackknife code example </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -3580,7 +3314,7 @@ t = jackknife(x, stat)
 
 
 <section>
-<h2 id="___sec86">Resampling methods: Bootstrap </h2>
+<h2 id="___sec82">Resampling methods: Bootstrap </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3602,7 +3336,7 @@ advantages:
 
 
 <section>
-<h2 id="___sec87">Resampling methods: Bootstrap background </h2>
+<h2 id="___sec83">Resampling methods: Bootstrap background </h2>
 
 <p>
 Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
@@ -3618,7 +3352,7 @@ estimators.
 
 
 <section>
-<h2 id="___sec88">Resampling methods: More Bootstrap background </h2>
+<h2 id="___sec84">Resampling methods: More Bootstrap background </h2>
 
 <p>
 In the case that \( \widehat{\theta} \) has
@@ -3641,7 +3375,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
 
 
 <section>
-<h2 id="___sec89">Resampling methods: Bootstrap approach </h2>
+<h2 id="___sec85">Resampling methods: Bootstrap approach </h2>
 
 <p>
 But
@@ -3662,7 +3396,7 @@ frequency of the observation \( X_i \), just draw the values
 
 
 <section>
-<h2 id="___sec90">Resampling methods: Bootstrap steps </h2>
+<h2 id="___sec86">Resampling methods: Bootstrap steps </h2>
 
 <p>
 The independent bootstrap works like this: 
@@ -3688,7 +3422,7 @@ example, if you are interested in estimating the variance of \( \widehat
 
 
 <section>
-<h2 id="___sec91">Code example for the Bootstrap method </h2>
+<h2 id="___sec87">Code example for the Bootstrap method </h2>
 
 <p>
 The following code starts with a Gaussian distribution with mean value
@@ -3753,7 +3487,7 @@ plt.show()
 
 
 <section>
-<h2 id="___sec92">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
+<h2 id="___sec88">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
 
 <p>
 The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
@@ -3854,7 +3588,7 @@ plt.show()
 
 
 <section>
-<h2 id="___sec93">The bias-variance tradeoff </h2>
+<h2 id="___sec89">The bias-variance tradeoff </h2>
 
 <p>
 We will discuss the bias-variance tradeoff in the context of
@@ -3932,7 +3666,7 @@ that is the rewriting in terms of the so-called bias, the variance of the model
 
 
 <section>
-<h2 id="___sec94">Example code for Bias-Variance tradeoff </h2>
+<h2 id="___sec90">Example code for Bias-Variance tradeoff </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -3995,7 +3729,7 @@ plt.show()
 
 
 <section>
-<h2 id="___sec95">Understanding what happens </h2>
+<h2 id="___sec91">Understanding what happens </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -4050,7 +3784,7 @@ plt.show()
 
 
 <section>
-<h2 id="___sec96">Summing up </h2>
+<h2 id="___sec92">Summing up </h2>
 
 <p>
 The bias-variance tradeoff summarizes the fundamental tension in
@@ -4084,7 +3818,7 @@ flexible statistical methods have higher variance.
 
 
 <section>
-<h2 id="___sec97">Another Example rom Scikit-Learn's Repository </h2>
+<h2 id="___sec93">Another Example rom Scikit-Learn's Repository </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -4163,6 +3897,268 @@ plt.show()
 </section>
 
 
+<section>
+<h2 id="___sec94">The Ising model </h2>
+
+<p>
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant \( J \) is given by
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{21}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
+in the system is determined by \( L \). For the one-dimensional system
+there is no phase transition.
+
+<p>
+We will look at a system of \( L = 40 \) spins with a coupling constant of
+\( J = 1 \). To get enough training data we will generate 10000 states
+with their respective energies.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">mpl_toolkits.axes_grid1</span> <span style="color: #8B008B; font-weight: bold">import</span> make_axes_locatable
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">seaborn</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">sns</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">scipy.linalg</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">scl</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">tqdm</span>
+sns.set(color_codes=<span style="color: #658b00">True</span>)
+cmap_args=<span style="color: #658b00">dict</span>(vmin=-<span style="color: #B452CD">1.</span>, vmax=<span style="color: #B452CD">1.</span>, cmap=<span style="color: #CD5555">&#39;seismic&#39;</span>)
+
+L = <span style="color: #B452CD">40</span>
+n = <span style="color: #658b00">int</span>(<span style="color: #B452CD">1e4</span>)
+
+spins = np.random.choice([-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>], size=(n, L))
+J = <span style="color: #B452CD">1.0</span>
+
+energies = np.zeros(n)
+
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], <span style="color: #B452CD">1</span>))
+</pre></div>
+<p>
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+</section>
+
+
+<section>
+<h2 id="___sec95">Reformulating the problem to suit regression </h2>
+
+<p>
+A more general form for the one-dimensional Ising model is
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\tag{22}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+<p>&nbsp;<br>
+$$
+\begin{align}
+    \boldsymbol{H} = \boldsymbol{X} J,
+\tag{23}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>
+where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\tag{24}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>
+We split the data in training and test data as discussed in the previous example
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>X = np.zeros((n, L ** <span style="color: #B452CD">2</span>))
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=<span style="color: #B452CD">0.2</span>)
+</pre></div>
+</section>
+
+
+<section>
+<h2 id="___sec96">Linear regression </h2>
+
+<p>
+In the ordinary least squares method we choose the cost function
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\tag{25}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>
+We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
+This yields the expression for \( \boldsymbol{\beta} \) to be
+
+<p>&nbsp;<br>
+$$
+    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+$$
+<p>&nbsp;<br>
+
+<p>
+which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
+an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>X_train_own = np.concatenate(
+    (np.ones(<span style="color: #658b00">len</span>(X_train))[:, np.newaxis], X_train),
+    axis=<span style="color: #B452CD">1</span>
+)
+X_test_own = np.concatenate(
+    (np.ones(<span style="color: #658b00">len</span>(X_test))[:, np.newaxis], X_test),
+    axis=<span style="color: #B452CD">1</span>
+)
+</pre></div>
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_inv</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
+    <span style="color: #8B008B; font-weight: bold">return</span> scl.inv(x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> x) <span style="color: #a61717; background-color: #e3d2d2">@</span> (x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y)
+beta = ols_inv(X_train_own, y_train)
+</pre></div>
+</section>
+
+
+<section>
+<h2 id="___sec97">Singular Value decomposition </h2>
+
+<p>
+Doing the inversion directly turns out to be a bad idea since the matrix
+\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
+value decomposition</b>. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+
+<p>&nbsp;<br>
+$$
+    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+<p>&nbsp;<br>
+
+<p>
+where the pseudoinverse of \( \boldsymbol{X} \) is given by
+
+<p>&nbsp;<br>
+$$
+    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+<p>&nbsp;<br>
+
+<p>
+Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
+where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
+where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
+\( \omega \) to
+<p>&nbsp;<br>
+$$
+\begin{align}
+    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\tag{26}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
+general matrix. Instead, doing \( QR \)-factorization and solving the
+linear system as an equation would reduce this down to
+\( \mathcal{O}(n^2) \) operations.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_svd</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
+    u, s, v = scl.svd(x)
+    <span style="color: #8B008B; font-weight: bold">return</span> v.T <span style="color: #a61717; background-color: #e3d2d2">@</span> scl.pinv(scl.diagsvd(s, u.shape[<span style="color: #B452CD">0</span>], v.shape[<span style="color: #B452CD">0</span>])) <span style="color: #a61717; background-color: #e3d2d2">@</span> u.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y
+</pre></div>
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>beta = ols_svd(X_train_own,y_train)
+</pre></div>
+<p>
+When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>J = beta[<span style="color: #B452CD">1</span>:].reshape(L, L)
+</pre></div>
+<p>
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>fig = plt.figure(figsize=(<span style="color: #B452CD">20</span>, <span style="color: #B452CD">14</span>))
+im = plt.imshow(J, **cmap_args)
+plt.title(<span style="color: #CD5555">&quot;OLS&quot;</span>, fontsize=<span style="color: #B452CD">18</span>)
+plt.xticks(fontsize=<span style="color: #B452CD">18</span>)
+plt.yticks(fontsize=<span style="color: #B452CD">18</span>)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=<span style="color: #B452CD">18</span>)
+plt.show()
+</pre></div>
+<p>
+It is interesting to note that OLS
+considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
+valid matrix elements for \( J \).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+<p>
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+</section>
+
+
 <section>
 <h2 id="___sec98">The one-dimensional Ising model </h2>
 
diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html
index 29262f9ae..0fd8fbde6 100644
--- a/doc/pub/Regression/html/Regression-solarized.html
+++ b/doc/pub/Regression/html/Regression-solarized.html
@@ -131,112 +131,112 @@ div { text-align: justify; text-justify: inter-word; }
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -289,7 +289,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>
-<center><h4>Aug 29, 2019</h4></center> <!-- date -->
+<center><h4>Aug 30, 2019</h4></center> <!-- date -->
 <br>
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1554,8 +1554,7 @@ The examples we have looked at so far are cases where we normally can
 invert the matrix \( \boldsymbol{X}^T\boldsymbol{X} \). Using a polynomial expansion as we
 did both for the masses and the fitting of the equation of state,
 leads to row vectors of the design matrix which are essentially
-orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. 
-More material to come here.
+orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition.
 
 <p>
 This may
@@ -1575,255 +1574,10 @@ inversion algorithm. Thereafter we dive into the math of the SVD.
 </div>
 
 
-<p>
-<!-- todo: change model here. -->
-
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec30">The Ising model </h2>
-
-<p>
-The one-dimensional Ising model with nearest neighbor interaction, no
-external field and a constant coupling constant \( J \) is given by
-
-$$
-\begin{align}
-    H = -J \sum_{k}^L s_k s_{k + 1},
-\label{_auto1}
-\end{align}
-$$
-
-<p>
-where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
-in the system is determined by \( L \). For the one-dimensional system
-there is no phase transition.
-
-<p>
-We will look at a system of \( L = 40 \) spins with a coupling constant of
-\( J = 1 \). To get enough training data we will generate 10000 states
-with their respective energies.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
-<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">mpl_toolkits.axes_grid1</span> <span style="color: #8B008B; font-weight: bold">import</span> make_axes_locatable
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">seaborn</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">sns</span>
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">scipy.linalg</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">scl</span>
-<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
-<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">tqdm</span>
-sns.set(color_codes=<span style="color: #658b00">True</span>)
-cmap_args=<span style="color: #658b00">dict</span>(vmin=-<span style="color: #B452CD">1.</span>, vmax=<span style="color: #B452CD">1.</span>, cmap=<span style="color: #CD5555">&#39;seismic&#39;</span>)
-
-L = <span style="color: #B452CD">40</span>
-n = <span style="color: #658b00">int</span>(<span style="color: #B452CD">1e4</span>)
-
-spins = np.random.choice([-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>], size=(n, L))
-J = <span style="color: #B452CD">1.0</span>
-
-energies = np.zeros(n)
-
-<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
-    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], <span style="color: #B452CD">1</span>))
-</pre></div>
-<p>
-Here we use ordinary least squares
-regression to predict the energy for the nearest neighbor
-one-dimensional Ising model on a ring, i.e., the endpoints wrap
-around. We will use linear regression to fit a value for
-the coupling constant to achieve this.
-
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec31">Reformulating the problem to suit regression </h2>
-
-<p>
-A more general form for the one-dimensional Ising model is
-
-$$
-\begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\label{_auto2}
-\end{align}
-$$
-
-<p>
-Here we allow for interactions beyond the nearest neighbors and a state dependent
-coupling constant. This latter expression can be formulated as
-a matrix-product
-$$
-\begin{align}
-    \boldsymbol{H} = \boldsymbol{X} J,
-\label{_auto3}
-\end{align}
-$$
-
-<p>
-where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
-elements \( -J_{jk} \). This form of writing the energy fits perfectly
-with the form utilized in linear regression, that is
-
-$$
-\begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
-\label{_auto4}
-\end{align}
-$$
-
-<p>
-We split the data in training and test data as discussed in the previous example
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>X = np.zeros((n, L ** <span style="color: #B452CD">2</span>))
-<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
-    X[i] = np.outer(spins[i], spins[i]).ravel()
-y = energies
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=<span style="color: #B452CD">0.2</span>)
-</pre></div>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec32">Linear regression </h2>
-
-<p>
-In the ordinary least squares method we choose the cost function
-
-$$
-\begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
-\label{_auto5}
-\end{align}
-$$
-
-<p>
-We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
-This yields the expression for \( \boldsymbol{\beta} \) to be
-
-$$
-    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
-$$
-
-<p>
-which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
-an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
-intercept, i.e., a constant term, we must make sure that the
-first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>X_train_own = np.concatenate(
-    (np.ones(<span style="color: #658b00">len</span>(X_train))[:, np.newaxis], X_train),
-    axis=<span style="color: #B452CD">1</span>
-)
-X_test_own = np.concatenate(
-    (np.ones(<span style="color: #658b00">len</span>(X_test))[:, np.newaxis], X_test),
-    axis=<span style="color: #B452CD">1</span>
-)
-</pre></div>
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_inv</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
-    <span style="color: #8B008B; font-weight: bold">return</span> scl.inv(x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> x) <span style="color: #a61717; background-color: #e3d2d2">@</span> (x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y)
-beta = ols_inv(X_train_own, y_train)
-</pre></div>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec33">Singular Value decomposition </h2>
-
-<p>
-Doing the inversion directly turns out to be a bad idea since the matrix
-\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
-value decomposition</b>. Using the definition of the Moore-Penrose
-pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
-
-$$
-    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
-$$
-
-<p>
-where the pseudoinverse of \( \boldsymbol{X} \) is given by
-
-$$
-    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
-$$
-
-<p>
-Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
-where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
-where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
-\( \omega \) to
-$$
-\begin{align}
-    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
-\label{_auto6}
-\end{align}
-$$
-
-<p>
-Note that solving this equation by actually doing the pseudoinverse
-(which is what we will do) is not a good idea as this operation scales
-as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
-general matrix. Instead, doing \( QR \)-factorization and solving the
-linear system as an equation would reduce this down to
-\( \mathcal{O}(n^2) \) operations.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_svd</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
-    u, s, v = scl.svd(x)
-    <span style="color: #8B008B; font-weight: bold">return</span> v.T <span style="color: #a61717; background-color: #e3d2d2">@</span> scl.pinv(scl.diagsvd(s, u.shape[<span style="color: #B452CD">0</span>], v.shape[<span style="color: #B452CD">0</span>])) <span style="color: #a61717; background-color: #e3d2d2">@</span> u.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y
-</pre></div>
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>beta = ols_svd(X_train_own,y_train)
-</pre></div>
-<p>
-When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>J = beta[<span style="color: #B452CD">1</span>:].reshape(L, L)
-</pre></div>
-<p>
-A way of looking at the coefficients in \( J \) is to plot the matrices as images.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
-<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>fig = plt.figure(figsize=(<span style="color: #B452CD">20</span>, <span style="color: #B452CD">14</span>))
-im = plt.imshow(J, **cmap_args)
-plt.title(<span style="color: #CD5555">&quot;OLS&quot;</span>, fontsize=<span style="color: #B452CD">18</span>)
-plt.xticks(fontsize=<span style="color: #B452CD">18</span>)
-plt.yticks(fontsize=<span style="color: #B452CD">18</span>)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=<span style="color: #B452CD">18</span>)
-plt.show()
-</pre></div>
-<p>
-It is interesting to note that OLS
-considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
-valid matrix elements for \( J \).
-In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
-this problem can be removed, partly and only with Lasso regression.
-
-<p>
-In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
-
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec34">Linear Regression Problems </h2>
+<h2 id="___sec30">Linear Regression Problems </h2>
 
 <p>
 One of the typical problems we encounter with linear regression, in particular 
@@ -1856,7 +1610,7 @@ of linearly independent columns. In this particular case the matrix has rank 2.
 
 <p>
 Super-collinearity of an \( (n \times p) \)-dimensional design matrix \( \mathbf{X} \) implies
-that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{x} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
+that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
 $$
 \begin{align*}
 \boldsymbol{X} & =  \left[
@@ -1874,14 +1628,14 @@ This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least a
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec35">Fixing the singularity </h2>
+<h2 id="___sec31">Fixing the singularity </h2>
 
 <p>
 If our design matrix \( \boldsymbol{X} \) which enters the linear regression problem
 $$
 \begin{align}
 \boldsymbol{\beta} & =  (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y},
-\label{_auto7}
+\label{_auto1}
 \end{align}
 $$
 
@@ -1902,7 +1656,7 @@ where \( \boldsymbol{I} \) is the identity matrix.  When we discuss <b>Ridge</b>
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec36">Basic math of the SVD </h2>
+<h2 id="___sec32">Basic math of the SVD </h2>
 
 <p>
 From standard linear algebra we know that a square matrix \( \boldsymbol{X} \) can be diagonalized if and only it is 
@@ -1941,7 +1695,7 @@ is not diagonalizable, it is a so-called <a href="https://en.wikipedia.org/wiki/
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec37">The SVD, a Fantastic Algorithm </h2>
+<h2 id="___sec33">The SVD, a Fantastic Algorithm </h2>
 
 <p>
 However, and this is the strength of the SVD algorithm, any general
@@ -1972,7 +1726,7 @@ The SVD exits always!
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec38">Another Example </h2>
+<h2 id="___sec34">Another Example </h2>
 
 <p>
 Consider the following matrix which can be SVD decomposed as
@@ -1997,7 +1751,7 @@ In the general case, where our design matrix \( \boldsymbol{X} \) has dimension
 \( n\times p \), the matrix is thus decomposed into an \( n\times n \)
 orthogonal matrix \( \boldsymbol{U} \), a \( p\times p \) orthogonal matrix \( \boldsymbol{V} \)
 and a diagonal matrix \( \boldsymbol{\Sigma} \) with \( r=\mathrm{min}(n,p) \)
-singular values \( \sigma_i\lg 0 \) on the main diagonal and zeros filling
+singular values \( \sigma_i\geq 0 \) on the main diagonal and zeros filling
 the rest of the matrix.  There are at most \( p \) singular values
 assuming that \( n > p \). In our regression examples for the nuclear
 masses and the equation of state this is indeed the case, while for
@@ -2010,7 +1764,7 @@ The columns of \( \boldsymbol{U} \) are called the left singular vectors while t
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec39">Economy-size SVD </h2>
+<h2 id="___sec35">Economy-size SVD </h2>
 
 <p>
 If we assume that \( n > p \), then our matrix \( \boldsymbol{U} \) has dimension \( n
@@ -2035,7 +1789,7 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec40">Mathematical Properties </h2>
+<h2 id="___sec36">Mathematical Properties </h2>
 
 <p>
 There are several interesting mathematical properties which will be
@@ -2087,7 +1841,7 @@ We will come back to this expression when we discuss Ridge regression.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec41">Ridge and LASSO Regression </h2>
+<h2 id="___sec37">Ridge and LASSO Regression </h2>
 
 <p>
 Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
@@ -2145,7 +1899,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec42">More on Ridge Regression </h2>
+<h2 id="___sec38">More on Ridge Regression </h2>
 
 <p>
 Using the matrix-vector expression for Ridge regression,
@@ -2204,7 +1958,7 @@ with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec43">Interpreting the Ridge results </h2>
+<h2 id="___sec39">Interpreting the Ridge results </h2>
 
 <p>
 Since \( \lambda \geq 0 \), it means that compared to OLS, we have 
@@ -2228,7 +1982,7 @@ With a parameter \( \lambda \) we can thus shrink the role of specific parameter
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec44">More interpretations </h2>
+<h2 id="___sec40">More interpretations </h2>
 
 <p>
 For the sake of simplicity, let us assume that the design matrix is orthonormal, that is 
@@ -2265,7 +2019,7 @@ Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et a
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec45">Where are we going? </h2>
+<h2 id="___sec41">Where are we going? </h2>
 
 <p>
 Before we proceed, we need to rethink what we have been doing. In our
@@ -2282,7 +2036,7 @@ This will allow us to link the standard linear algebra methods we have discussed
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec46">Resampling methods </h2>
+<h2 id="___sec42">Resampling methods </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2302,7 +2056,7 @@ once using the original training sample.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec47">Resampling approaches can be computationally expensive </h2>
+<h2 id="___sec43">Resampling approaches can be computationally expensive </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2331,7 +2085,7 @@ bootstrap is widely used.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec48">Why resampling methods ? </h2>
+<h2 id="___sec44">Why resampling methods ? </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b>Statistical analysis.</b>
 <p>
@@ -2347,7 +2101,7 @@ bootstrap is widely used.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec49">Statistical analysis </h2>
+<h2 id="___sec45">Statistical analysis </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2369,7 +2123,7 @@ bootstrap is widely used.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec50">Statistics  </h2>
+<h2 id="___sec46">Statistics  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2400,7 +2154,7 @@ selection of a large set of these numbers reproduces this PDF.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec51">Statistics, moments </h2>
+<h2 id="___sec47">Statistics, moments </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2423,7 +2177,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec52">Statistics, central moments </h2>
+<h2 id="___sec48">Statistics, central moments </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2441,14 +2195,14 @@ $$
 \begin{align}
 \sigma^2_X\ \ =\ \ \mathrm{var}(X) & =  \langle (x-\langle x\rangle)^2\rangle =
 \int\! (x-\langle x\rangle)^2 p(x)\,dx
-\label{_auto8}\\
+\label{_auto2}\\
 & =  \int\! \left(x^2 - 2 x \langle x\rangle^{2} +
   \langle x\rangle^2\right)p(x)\,dx
-\label{_auto9}\\
+\label{_auto3}\\
 & =  \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2
-\label{_auto10}\\
+\label{_auto4}\\
 & =  \langle x^2\rangle - \langle x\rangle^2
-\label{_auto11}
+\label{_auto5}
 \end{align}
 $$
 
@@ -2461,7 +2215,7 @@ qualitatively as the <em>spread</em> of \( p \) around its mean.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec53">Statistics, covariance </h2>
+<h2 id="___sec49">Statistics, covariance </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2492,7 +2246,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec54">Statistics, more covariance  </h2>
+<h2 id="___sec50">Statistics, more covariance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2505,17 +2259,17 @@ the expectation value. Consider the stochastic variables \( X_i \) and
 $$
 \begin{align}
 \mathrm{cov}(X_i,\,X_j) &= \langle(x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
-\label{_auto12}\\
+\label{_auto6}\\
 &=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle 
-\label{_auto13}\\
+\label{_auto7}\\
 &=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j\rangle +
 \langle \langle x_i\rangle\langle x_j\rangle\rangle
-\label{_auto14}\\
+\label{_auto8}\\
 &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle +
 \langle x_i\rangle\langle x_j\rangle
-\label{_auto15}\\
+\label{_auto9}\\
 &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle
-\label{_auto16}
+\label{_auto10}
 \end{align}
 $$
 </div>
@@ -2524,7 +2278,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec55">Covariance example </h2>
+<h2 id="___sec51">Covariance example </h2>
 
 <p>
 Suppose we have defined three vectors \( \hat{x}, \hat{y}, \hat{z} \) with
@@ -2573,7 +2327,7 @@ function.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec56">Covariance in numpy </h2>
+<h2 id="___sec52">Covariance in numpy </h2>
 
 <p>
 
@@ -2612,7 +2366,7 @@ plt.show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec57">Statistics, independent variables  </h2>
+<h2 id="___sec53">Statistics, independent variables  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2638,7 +2392,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec58">Statistics, more variance  </h2>
+<h2 id="___sec54">Statistics, more variance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2670,7 +2424,7 @@ value of a set of measurements.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec59">Statistics and stochastic processes  </h2>
+<h2 id="___sec55">Statistics and stochastic processes  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2698,7 +2452,7 @@ interested in finding the few lowest moments, like the mean
 <p>
 <!-- !split  -->
 
-<h2 id="___sec60">Statistics and sample variables  </h2>
+<h2 id="___sec56">Statistics and sample variables  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2724,7 +2478,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec61">Statistics, sample variance and covariance </h2>
+<h2 id="___sec57">Statistics, sample variance and covariance </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2745,7 +2499,7 @@ and covariance \( \mathrm{cov}(X,Y) \).
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec62">Statistics, law of large numbers </h2>
+<h2 id="___sec58">Statistics, law of large numbers </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2777,7 +2531,7 @@ true PDFs behind, which we usually do not have.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec63">Statistics, more on sample error </h2>
+<h2 id="___sec59">Statistics, more on sample error </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2799,7 +2553,7 @@ means.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec64">Statistics  </h2>
+<h2 id="___sec60">Statistics  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2820,7 +2574,7 @@ And in particular we are interested in its variance \( \mathrm{var}(\overline X_
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec65">Statistics, central limit theorem </h2>
+<h2 id="___sec61">Statistics, central limit theorem </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2845,7 +2599,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec66">Statistics, more technicalities  </h2>
+<h2 id="___sec62">Statistics, more technicalities  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2874,7 +2628,7 @@ estimate of the PDF of each of the \( X_i \), estimating all properties of
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec67">Statistics </h2>
+<h2 id="___sec63">Statistics </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2901,7 +2655,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec68">Statistics and sample variance  </h2>
+<h2 id="___sec64">Statistics and sample variance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2940,7 +2694,7 @@ measurements in the sample.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec69">Statistics, uncorrelated results </h2>
+<h2 id="___sec65">Statistics, uncorrelated results </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2975,7 +2729,7 @@ cannot overlook the always present correlations.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec70">Statistics, computations </h2>
+<h2 id="___sec66">Statistics, computations </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3004,7 +2758,7 @@ measurements. For uncorrelated measurements this second term is zero.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec71">Statistics, more on computations of errors </h2>
+<h2 id="___sec67">Statistics, more on computations of errors </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3026,7 +2780,7 @@ have to be stored throughout the experiment.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec72">Statistics, wrapping up 1 </h2>
+<h2 id="___sec68">Statistics, wrapping up 1 </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3059,7 +2813,7 @@ starting always at \( 1 \) for \( d=0 \).
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec73">Statistics, final expression </h2>
+<h2 id="___sec69">Statistics, final expression </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3092,7 +2846,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec74">Statistics, effective number of correlations </h2>
+<h2 id="___sec70">Statistics, effective number of correlations </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3117,7 +2871,7 @@ measurements is very large.
 <p>
 <!-- !split  -->
 
-<h2 id="___sec75">Linking the regression analysis with a statistical interpretation </h2>
+<h2 id="___sec71">Linking the regression analysis with a statistical interpretation </h2>
 
 <p>
 Finally, we are going to discuss several statistical properties which can be obtained in terms of analytical expressions. 
@@ -3155,7 +2909,7 @@ row number \( i \) and perform a sum over all values \( p \).
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec76">Assumptions made </h2>
+<h2 id="___sec72">Assumptions made </h2>
 
 <p>
 The assumption we have made here can be summarized as (and this is going to useful when we discuss the bias-variance trade off)
@@ -3175,7 +2929,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec77">Expectation value and variance </h2>
+<h2 id="___sec73">Expectation value and variance </h2>
 
 <p>
 We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) 
@@ -3210,7 +2964,7 @@ mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (n
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec78">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
+<h2 id="___sec74">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
 
 <p>
 With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
@@ -3295,7 +3049,7 @@ This means the variance we obtain with the standard OLS will always for \( \lamb
 <p>
 <!-- !split  -->
 
-<h2 id="___sec79">Cross-validation </h2>
+<h2 id="___sec75">Cross-validation </h2>
 
 <p>
 Instead of choosing the penalty parameter to balance model fit with
@@ -3329,7 +3083,7 @@ some sense) is then selected.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec80">Computationally expensive </h2>
+<h2 id="___sec76">Computationally expensive </h2>
 
 <p>
 The validation set approach is conceptually simple and is easy to implement. But it has two potential drawbacks:
@@ -3341,7 +3095,7 @@ The validation set approach is conceptually simple and is easy to implement. But
 
 <!-- !split  -->
 
-<h2 id="___sec81">Various steps in cross-validation </h2>
+<h2 id="___sec77">Various steps in cross-validation </h2>
 
 <p>
 When the repetitive splitting of the data set is done randomly,
@@ -3362,7 +3116,7 @@ cross-validation (LOOCV).
 <p>
 <!-- !split  -->
 
-<h2 id="___sec82">How to set up the cross-validation for Ridge and/or Lasso </h2>
+<h2 id="___sec78">How to set up the cross-validation for Ridge and/or Lasso </h2>
 
 <ul>
 <li> Define a range of interest for the penalty parameter.</li>
@@ -3398,7 +3152,7 @@ $$
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec83">Resampling methods: Jackknife and Bootstrap </h2>
+<h2 id="___sec79">Resampling methods: Jackknife and Bootstrap </h2>
 
 <p>
 Two famous
@@ -3421,7 +3175,7 @@ need for bootstrapping.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec84">Resampling methods: Jackknife </h2>
+<h2 id="___sec80">Resampling methods: Jackknife </h2>
 
 <p>
 The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). 
@@ -3440,7 +3194,7 @@ number \( i \) is left out. Using this notation, define
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec85">Jackknife code example </h2>
+<h2 id="___sec81">Jackknife code example </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -3476,7 +3230,7 @@ t = jackknife(x, stat)
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec86">Resampling methods: Bootstrap </h2>
+<h2 id="___sec82">Resampling methods: Bootstrap </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3497,7 +3251,7 @@ advantages:
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec87">Resampling methods: Bootstrap background </h2>
+<h2 id="___sec83">Resampling methods: Bootstrap background </h2>
 
 <p>
 Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
@@ -3513,7 +3267,7 @@ estimators.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec88">Resampling methods: More Bootstrap background </h2>
+<h2 id="___sec84">Resampling methods: More Bootstrap background </h2>
 
 <p>
 In the case that \( \widehat{\theta} \) has
@@ -3535,7 +3289,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec89">Resampling methods: Bootstrap approach </h2>
+<h2 id="___sec85">Resampling methods: Bootstrap approach </h2>
 
 <p>
 But
@@ -3556,7 +3310,7 @@ frequency of the observation \( X_i \), just draw the values
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec90">Resampling methods: Bootstrap steps </h2>
+<h2 id="___sec86">Resampling methods: Bootstrap steps </h2>
 
 <p>
 The independent bootstrap works like this: 
@@ -3581,7 +3335,7 @@ example, if you are interested in estimating the variance of \( \widehat
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec91">Code example for the Bootstrap method </h2>
+<h2 id="___sec87">Code example for the Bootstrap method </h2>
 
 <p>
 The following code starts with a Gaussian distribution with mean value
@@ -3645,7 +3399,7 @@ plt.show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec92">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
+<h2 id="___sec88">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
 
 <p>
 The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
@@ -3745,7 +3499,7 @@ plt.show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec93">The bias-variance tradeoff </h2>
+<h2 id="___sec89">The bias-variance tradeoff </h2>
 
 <p>
 We will discuss the bias-variance tradeoff in the context of
@@ -3811,7 +3565,7 @@ that is the rewriting in terms of the so-called bias, the variance of the model
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec94">Example code for Bias-Variance tradeoff </h2>
+<h2 id="___sec90">Example code for Bias-Variance tradeoff </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -3873,7 +3627,7 @@ plt.show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec95">Understanding what happens </h2>
+<h2 id="___sec91">Understanding what happens </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -3927,7 +3681,7 @@ plt.show()
 <p>
 <!-- !split  -->
 
-<h2 id="___sec96">Summing up </h2>
+<h2 id="___sec92">Summing up </h2>
 
 <p>
 The bias-variance tradeoff summarizes the fundamental tension in
@@ -3961,7 +3715,7 @@ flexible statistical methods have higher variance.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec97">Another Example rom Scikit-Learn's Repository </h2>
+<h2 id="___sec93">Another Example rom Scikit-Learn's Repository </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
@@ -4040,6 +3794,248 @@ plt.show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
+<h2 id="___sec94">The Ising model </h2>
+
+<p>
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant \( J \) is given by
+
+$$
+\begin{align}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\label{_auto11}
+\end{align}
+$$
+
+<p>
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
+in the system is determined by \( L \). For the one-dimensional system
+there is no phase transition.
+
+<p>
+We will look at a system of \( L = 40 \) spins with a coupling constant of
+\( J = 1 \). To get enough training data we will generate 10000 states
+with their respective energies.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">mpl_toolkits.axes_grid1</span> <span style="color: #8B008B; font-weight: bold">import</span> make_axes_locatable
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">seaborn</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">sns</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">scipy.linalg</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">scl</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">tqdm</span>
+sns.set(color_codes=<span style="color: #658b00">True</span>)
+cmap_args=<span style="color: #658b00">dict</span>(vmin=-<span style="color: #B452CD">1.</span>, vmax=<span style="color: #B452CD">1.</span>, cmap=<span style="color: #CD5555">&#39;seismic&#39;</span>)
+
+L = <span style="color: #B452CD">40</span>
+n = <span style="color: #658b00">int</span>(<span style="color: #B452CD">1e4</span>)
+
+spins = np.random.choice([-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>], size=(n, L))
+J = <span style="color: #B452CD">1.0</span>
+
+energies = np.zeros(n)
+
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], <span style="color: #B452CD">1</span>))
+</pre></div>
+<p>
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
+<h2 id="___sec95">Reformulating the problem to suit regression </h2>
+
+<p>
+A more general form for the one-dimensional Ising model is
+
+$$
+\begin{align}
+    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\label{_auto12}
+\end{align}
+$$
+
+<p>
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+$$
+\begin{align}
+    \boldsymbol{H} = \boldsymbol{X} J,
+\label{_auto13}
+\end{align}
+$$
+
+<p>
+where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+$$
+\begin{align}
+    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\label{_auto14}
+\end{align}
+$$
+
+<p>
+We split the data in training and test data as discussed in the previous example
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>X = np.zeros((n, L ** <span style="color: #B452CD">2</span>))
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=<span style="color: #B452CD">0.2</span>)
+</pre></div>
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
+<h2 id="___sec96">Linear regression </h2>
+
+<p>
+In the ordinary least squares method we choose the cost function
+
+$$
+\begin{align}
+    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\label{_auto15}
+\end{align}
+$$
+
+<p>
+We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
+This yields the expression for \( \boldsymbol{\beta} \) to be
+
+$$
+    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+$$
+
+<p>
+which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
+an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>X_train_own = np.concatenate(
+    (np.ones(<span style="color: #658b00">len</span>(X_train))[:, np.newaxis], X_train),
+    axis=<span style="color: #B452CD">1</span>
+)
+X_test_own = np.concatenate(
+    (np.ones(<span style="color: #658b00">len</span>(X_test))[:, np.newaxis], X_test),
+    axis=<span style="color: #B452CD">1</span>
+)
+</pre></div>
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_inv</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
+    <span style="color: #8B008B; font-weight: bold">return</span> scl.inv(x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> x) <span style="color: #a61717; background-color: #e3d2d2">@</span> (x.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y)
+beta = ols_inv(X_train_own, y_train)
+</pre></div>
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
+<h2 id="___sec97">Singular Value decomposition </h2>
+
+<p>
+Doing the inversion directly turns out to be a bad idea since the matrix
+\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
+value decomposition</b>. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+
+$$
+    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+
+<p>
+where the pseudoinverse of \( \boldsymbol{X} \) is given by
+
+$$
+    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+
+<p>
+Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
+where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
+where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
+\( \omega \) to
+$$
+\begin{align}
+    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\label{_auto16}
+\end{align}
+$$
+
+<p>
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
+general matrix. Instead, doing \( QR \)-factorization and solving the
+linear system as an equation would reduce this down to
+\( \mathcal{O}(n^2) \) operations.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">ols_svd</span>(x: np.ndarray, y: np.ndarray) -&gt; np.ndarray:
+    u, s, v = scl.svd(x)
+    <span style="color: #8B008B; font-weight: bold">return</span> v.T <span style="color: #a61717; background-color: #e3d2d2">@</span> scl.pinv(scl.diagsvd(s, u.shape[<span style="color: #B452CD">0</span>], v.shape[<span style="color: #B452CD">0</span>])) <span style="color: #a61717; background-color: #e3d2d2">@</span> u.T <span style="color: #a61717; background-color: #e3d2d2">@</span> y
+</pre></div>
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>beta = ols_svd(X_train_own,y_train)
+</pre></div>
+<p>
+When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>J = beta[<span style="color: #B452CD">1</span>:].reshape(L, L)
+</pre></div>
+<p>
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>fig = plt.figure(figsize=(<span style="color: #B452CD">20</span>, <span style="color: #B452CD">14</span>))
+im = plt.imshow(J, **cmap_args)
+plt.title(<span style="color: #CD5555">&quot;OLS&quot;</span>, fontsize=<span style="color: #B452CD">18</span>)
+plt.xticks(fontsize=<span style="color: #B452CD">18</span>)
+plt.yticks(fontsize=<span style="color: #B452CD">18</span>)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=<span style="color: #B452CD">18</span>)
+plt.show()
+</pre></div>
+<p>
+It is interesting to note that OLS
+considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
+valid matrix elements for \( J \).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+<p>
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
 <h2 id="___sec98">The one-dimensional Ising model </h2>
 
 <p>
diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html
index 5a5c74677..b03683500 100644
--- a/doc/pub/Regression/html/Regression.html
+++ b/doc/pub/Regression/html/Regression.html
@@ -136,112 +136,112 @@ div { text-align: justify; text-justify: inter-word; }
               ('The Boston housing data example', 2, None, '___sec27'),
               ('Housing data, the code', 2, None, '___sec28'),
               ('The singular value decomposition', 2, None, '___sec29'),
-              ('The Ising model', 2, None, '___sec30'),
-              ('Reformulating the problem to suit regression',
-               2,
-               None,
-               '___sec31'),
-              ('Linear regression', 2, None, '___sec32'),
-              ('Singular Value decomposition', 2, None, '___sec33'),
-              ('Linear Regression Problems', 2, None, '___sec34'),
-              ('Fixing the singularity', 2, None, '___sec35'),
-              ('Basic math of the SVD', 2, None, '___sec36'),
-              ('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
-              ('Another Example', 2, None, '___sec38'),
-              ('Economy-size SVD', 2, None, '___sec39'),
-              ('Mathematical Properties', 2, None, '___sec40'),
-              ('Ridge and LASSO Regression', 2, None, '___sec41'),
-              ('More on Ridge Regression', 2, None, '___sec42'),
-              ('Interpreting the Ridge results', 2, None, '___sec43'),
-              ('More interpretations', 2, None, '___sec44'),
-              ('Where are we going?', 2, None, '___sec45'),
-              ('Resampling methods', 2, None, '___sec46'),
+              ('Linear Regression Problems', 2, None, '___sec30'),
+              ('Fixing the singularity', 2, None, '___sec31'),
+              ('Basic math of the SVD', 2, None, '___sec32'),
+              ('The SVD, a Fantastic Algorithm', 2, None, '___sec33'),
+              ('Another Example', 2, None, '___sec34'),
+              ('Economy-size SVD', 2, None, '___sec35'),
+              ('Mathematical Properties', 2, None, '___sec36'),
+              ('Ridge and LASSO Regression', 2, None, '___sec37'),
+              ('More on Ridge Regression', 2, None, '___sec38'),
+              ('Interpreting the Ridge results', 2, None, '___sec39'),
+              ('More interpretations', 2, None, '___sec40'),
+              ('Where are we going?', 2, None, '___sec41'),
+              ('Resampling methods', 2, None, '___sec42'),
               ('Resampling approaches can be computationally expensive',
                2,
                None,
-               '___sec47'),
-              ('Why resampling methods ?', 2, None, '___sec48'),
-              ('Statistical analysis', 2, None, '___sec49'),
-              ('Statistics', 2, None, '___sec50'),
-              ('Statistics, moments', 2, None, '___sec51'),
-              ('Statistics, central moments', 2, None, '___sec52'),
-              ('Statistics, covariance', 2, None, '___sec53'),
-              ('Statistics, more covariance', 2, None, '___sec54'),
-              ('Covariance example', 2, None, '___sec55'),
-              ('Covariance in numpy', 2, None, '___sec56'),
-              ('Statistics, independent variables', 2, None, '___sec57'),
-              ('Statistics, more variance', 2, None, '___sec58'),
-              ('Statistics and stochastic processes', 2, None, '___sec59'),
-              ('Statistics and sample variables', 2, None, '___sec60'),
+               '___sec43'),
+              ('Why resampling methods ?', 2, None, '___sec44'),
+              ('Statistical analysis', 2, None, '___sec45'),
+              ('Statistics', 2, None, '___sec46'),
+              ('Statistics, moments', 2, None, '___sec47'),
+              ('Statistics, central moments', 2, None, '___sec48'),
+              ('Statistics, covariance', 2, None, '___sec49'),
+              ('Statistics, more covariance', 2, None, '___sec50'),
+              ('Covariance example', 2, None, '___sec51'),
+              ('Covariance in numpy', 2, None, '___sec52'),
+              ('Statistics, independent variables', 2, None, '___sec53'),
+              ('Statistics, more variance', 2, None, '___sec54'),
+              ('Statistics and stochastic processes', 2, None, '___sec55'),
+              ('Statistics and sample variables', 2, None, '___sec56'),
               ('Statistics, sample variance and covariance',
                2,
                None,
-               '___sec61'),
-              ('Statistics, law of large numbers', 2, None, '___sec62'),
-              ('Statistics, more on sample error', 2, None, '___sec63'),
-              ('Statistics', 2, None, '___sec64'),
-              ('Statistics, central limit theorem', 2, None, '___sec65'),
-              ('Statistics, more technicalities', 2, None, '___sec66'),
-              ('Statistics', 2, None, '___sec67'),
-              ('Statistics and sample variance', 2, None, '___sec68'),
-              ('Statistics, uncorrelated results', 2, None, '___sec69'),
-              ('Statistics, computations', 2, None, '___sec70'),
+               '___sec57'),
+              ('Statistics, law of large numbers', 2, None, '___sec58'),
+              ('Statistics, more on sample error', 2, None, '___sec59'),
+              ('Statistics', 2, None, '___sec60'),
+              ('Statistics, central limit theorem', 2, None, '___sec61'),
+              ('Statistics, more technicalities', 2, None, '___sec62'),
+              ('Statistics', 2, None, '___sec63'),
+              ('Statistics and sample variance', 2, None, '___sec64'),
+              ('Statistics, uncorrelated results', 2, None, '___sec65'),
+              ('Statistics, computations', 2, None, '___sec66'),
               ('Statistics, more on computations of errors',
                2,
                None,
-               '___sec71'),
-              ('Statistics, wrapping up 1', 2, None, '___sec72'),
-              ('Statistics, final expression', 2, None, '___sec73'),
+               '___sec67'),
+              ('Statistics, wrapping up 1', 2, None, '___sec68'),
+              ('Statistics, final expression', 2, None, '___sec69'),
               ('Statistics, effective number of correlations',
                2,
                None,
-               '___sec74'),
+               '___sec70'),
               ('Linking the regression analysis with a statistical '
                'interpretation',
                2,
                None,
-               '___sec75'),
-              ('Assumptions made', 2, None, '___sec76'),
-              ('Expectation value and variance', 2, None, '___sec77'),
+               '___sec71'),
+              ('Assumptions made', 2, None, '___sec72'),
+              ('Expectation value and variance', 2, None, '___sec73'),
               ('Expectation value and variance for $\\boldsymbol{\\beta}$',
                2,
                None,
-               '___sec78'),
-              ('Cross-validation', 2, None, '___sec79'),
-              ('Computationally expensive', 2, None, '___sec80'),
-              ('Various steps in cross-validation', 2, None, '___sec81'),
+               '___sec74'),
+              ('Cross-validation', 2, None, '___sec75'),
+              ('Computationally expensive', 2, None, '___sec76'),
+              ('Various steps in cross-validation', 2, None, '___sec77'),
               ('How to set up the cross-validation for Ridge and/or Lasso',
                2,
                None,
-               '___sec82'),
+               '___sec78'),
               ('Resampling methods: Jackknife and Bootstrap',
                2,
                None,
-               '___sec83'),
-              ('Resampling methods: Jackknife', 2, None, '___sec84'),
-              ('Jackknife code example', 2, None, '___sec85'),
-              ('Resampling methods: Bootstrap', 2, None, '___sec86'),
-              ('Resampling methods: Bootstrap background', 2, None, '___sec87'),
+               '___sec79'),
+              ('Resampling methods: Jackknife', 2, None, '___sec80'),
+              ('Jackknife code example', 2, None, '___sec81'),
+              ('Resampling methods: Bootstrap', 2, None, '___sec82'),
+              ('Resampling methods: Bootstrap background', 2, None, '___sec83'),
               ('Resampling methods: More Bootstrap background',
                2,
                None,
-               '___sec88'),
-              ('Resampling methods: Bootstrap approach', 2, None, '___sec89'),
-              ('Resampling methods: Bootstrap steps', 2, None, '___sec90'),
-              ('Code example for the Bootstrap method', 2, None, '___sec91'),
+               '___sec84'),
+              ('Resampling methods: Bootstrap approach', 2, None, '___sec85'),
+              ('Resampling methods: Bootstrap steps', 2, None, '___sec86'),
+              ('Code example for the Bootstrap method', 2, None, '___sec87'),
               ('Code Example for Cross-validation and $k$-fold '
                'Cross-validation',
                2,
                None,
-               '___sec92'),
-              ('The bias-variance tradeoff', 2, None, '___sec93'),
-              ('Example code for Bias-Variance tradeoff', 2, None, '___sec94'),
-              ('Understanding what happens', 2, None, '___sec95'),
-              ('Summing up', 2, None, '___sec96'),
+               '___sec88'),
+              ('The bias-variance tradeoff', 2, None, '___sec89'),
+              ('Example code for Bias-Variance tradeoff', 2, None, '___sec90'),
+              ('Understanding what happens', 2, None, '___sec91'),
+              ('Summing up', 2, None, '___sec92'),
               ("Another Example rom Scikit-Learn's Repository",
                2,
                None,
-               '___sec97'),
+               '___sec93'),
+              ('The Ising model', 2, None, '___sec94'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec95'),
+              ('Linear regression', 2, None, '___sec96'),
+              ('Singular Value decomposition', 2, None, '___sec97'),
               ('The one-dimensional Ising model', 2, None, '___sec98'),
               ('Ridge regression', 2, None, '___sec99'),
               ('LASSO regression', 2, None, '___sec100'),
@@ -294,7 +294,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>
-<center><h4>Aug 29, 2019</h4></center> <!-- date -->
+<center><h4>Aug 30, 2019</h4></center> <!-- date -->
 <br>
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1559,8 +1559,7 @@ The examples we have looked at so far are cases where we normally can
 invert the matrix \( \boldsymbol{X}^T\boldsymbol{X} \). Using a polynomial expansion as we
 did both for the masses and the fitting of the equation of state,
 leads to row vectors of the design matrix which are essentially
-orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. 
-More material to come here.
+orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition.
 
 <p>
 This may
@@ -1580,255 +1579,10 @@ inversion algorithm. Thereafter we dive into the math of the SVD.
 </div>
 
 
-<p>
-<!-- todo: change model here. -->
-
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec30">The Ising model </h2>
-
-<p>
-The one-dimensional Ising model with nearest neighbor interaction, no
-external field and a constant coupling constant \( J \) is given by
-
-$$
-\begin{align}
-    H = -J \sum_{k}^L s_k s_{k + 1},
-\label{_auto1}
-\end{align}
-$$
-
-<p>
-where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
-in the system is determined by \( L \). For the one-dimensional system
-there is no phase transition.
-
-<p>
-We will look at a system of \( L = 40 \) spins with a coupling constant of
-\( J = 1 \). To get enough training data we will generate 10000 states
-with their respective energies.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
-sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000">True</span>)
-cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&#39;seismic&#39;</span>)
-
-L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
-n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
-
-spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
-J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
-
-energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
-
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
-    energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
-</pre></div>
-<p>
-Here we use ordinary least squares
-regression to predict the energy for the nearest neighbor
-one-dimensional Ising model on a ring, i.e., the endpoints wrap
-around. We will use linear regression to fit a value for
-the coupling constant to achieve this.
-
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec31">Reformulating the problem to suit regression </h2>
-
-<p>
-A more general form for the one-dimensional Ising model is
-
-$$
-\begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\label{_auto2}
-\end{align}
-$$
-
-<p>
-Here we allow for interactions beyond the nearest neighbors and a state dependent
-coupling constant. This latter expression can be formulated as
-a matrix-product
-$$
-\begin{align}
-    \boldsymbol{H} = \boldsymbol{X} J,
-\label{_auto3}
-\end{align}
-$$
-
-<p>
-where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
-elements \( -J_{jk} \). This form of writing the energy fits perfectly
-with the form utilized in linear regression, that is
-
-$$
-\begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
-\label{_auto4}
-\end{align}
-$$
-
-<p>
-We split the data in training and test data as discussed in the previous example
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
-    X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
-y <span style="color: #666666">=</span> energies
-X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
-</pre></div>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec32">Linear regression </h2>
-
-<p>
-In the ordinary least squares method we choose the cost function
-
-$$
-\begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
-\label{_auto5}
-\end{align}
-$$
-
-<p>
-We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
-This yields the expression for \( \boldsymbol{\beta} \) to be
-
-$$
-    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
-$$
-
-<p>
-which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
-an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
-intercept, i.e., a constant term, we must make sure that the
-first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
-    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
-    axis<span style="color: #666666">=1</span>
-)
-X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
-    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
-    axis<span style="color: #666666">=1</span>
-)
-</pre></div>
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_inv</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
-    <span style="color: #008000; font-weight: bold">return</span> scl<span style="color: #666666">.</span>inv(x<span style="color: #666666">.</span>T @ x) @ (x<span style="color: #666666">.</span>T @ y)
-beta <span style="color: #666666">=</span> ols_inv(X_train_own, y_train)
-</pre></div>
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec33">Singular Value decomposition </h2>
-
-<p>
-Doing the inversion directly turns out to be a bad idea since the matrix
-\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
-value decomposition</b>. Using the definition of the Moore-Penrose
-pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
-
-$$
-    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
-$$
-
-<p>
-where the pseudoinverse of \( \boldsymbol{X} \) is given by
-
-$$
-    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
-$$
-
-<p>
-Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
-where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
-where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
-\( \omega \) to
-$$
-\begin{align}
-    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
-\label{_auto6}
-\end{align}
-$$
-
-<p>
-Note that solving this equation by actually doing the pseudoinverse
-(which is what we will do) is not a good idea as this operation scales
-as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
-general matrix. Instead, doing \( QR \)-factorization and solving the
-linear system as an equation would reduce this down to
-\( \mathcal{O}(n^2) \) operations.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_svd</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
-    u, s, v <span style="color: #666666">=</span> scl<span style="color: #666666">.</span>svd(x)
-    <span style="color: #008000; font-weight: bold">return</span> v<span style="color: #666666">.</span>T @ scl<span style="color: #666666">.</span>pinv(scl<span style="color: #666666">.</span>diagsvd(s, u<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], v<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])) @ u<span style="color: #666666">.</span>T @ y
-</pre></div>
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>beta <span style="color: #666666">=</span> ols_svd(X_train_own,y_train)
-</pre></div>
-<p>
-When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J <span style="color: #666666">=</span> beta[<span style="color: #666666">1</span>:]<span style="color: #666666">.</span>reshape(L, L)
-</pre></div>
-<p>
-A way of looking at the coefficients in \( J \) is to plot the matrices as images.
-
-<p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
-im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J, <span style="color: #666666">**</span>cmap_args)
-plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;OLS&quot;</span>, fontsize<span style="color: #666666">=18</span>)
-plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
-plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
-cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
-cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
-plt<span style="color: #666666">.</span>show()
-</pre></div>
-<p>
-It is interesting to note that OLS
-considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
-valid matrix elements for \( J \).
-In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
-this problem can be removed, partly and only with Lasso regression.
-
-<p>
-In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
-
-<p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-
-<h2 id="___sec34">Linear Regression Problems </h2>
+<h2 id="___sec30">Linear Regression Problems </h2>
 
 <p>
 One of the typical problems we encounter with linear regression, in particular 
@@ -1861,7 +1615,7 @@ of linearly independent columns. In this particular case the matrix has rank 2.
 
 <p>
 Super-collinearity of an \( (n \times p) \)-dimensional design matrix \( \mathbf{X} \) implies
-that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{x} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
+that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
 $$
 \begin{align*}
 \boldsymbol{X} & =  \left[
@@ -1879,14 +1633,14 @@ This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least a
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec35">Fixing the singularity </h2>
+<h2 id="___sec31">Fixing the singularity </h2>
 
 <p>
 If our design matrix \( \boldsymbol{X} \) which enters the linear regression problem
 $$
 \begin{align}
 \boldsymbol{\beta} & =  (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y},
-\label{_auto7}
+\label{_auto1}
 \end{align}
 $$
 
@@ -1907,7 +1661,7 @@ where \( \boldsymbol{I} \) is the identity matrix.  When we discuss <b>Ridge</b>
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec36">Basic math of the SVD </h2>
+<h2 id="___sec32">Basic math of the SVD </h2>
 
 <p>
 From standard linear algebra we know that a square matrix \( \boldsymbol{X} \) can be diagonalized if and only it is 
@@ -1946,7 +1700,7 @@ is not diagonalizable, it is a so-called <a href="https://en.wikipedia.org/wiki/
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec37">The SVD, a Fantastic Algorithm </h2>
+<h2 id="___sec33">The SVD, a Fantastic Algorithm </h2>
 
 <p>
 However, and this is the strength of the SVD algorithm, any general
@@ -1977,7 +1731,7 @@ The SVD exits always!
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec38">Another Example </h2>
+<h2 id="___sec34">Another Example </h2>
 
 <p>
 Consider the following matrix which can be SVD decomposed as
@@ -2002,7 +1756,7 @@ In the general case, where our design matrix \( \boldsymbol{X} \) has dimension
 \( n\times p \), the matrix is thus decomposed into an \( n\times n \)
 orthogonal matrix \( \boldsymbol{U} \), a \( p\times p \) orthogonal matrix \( \boldsymbol{V} \)
 and a diagonal matrix \( \boldsymbol{\Sigma} \) with \( r=\mathrm{min}(n,p) \)
-singular values \( \sigma_i\lg 0 \) on the main diagonal and zeros filling
+singular values \( \sigma_i\geq 0 \) on the main diagonal and zeros filling
 the rest of the matrix.  There are at most \( p \) singular values
 assuming that \( n > p \). In our regression examples for the nuclear
 masses and the equation of state this is indeed the case, while for
@@ -2015,7 +1769,7 @@ The columns of \( \boldsymbol{U} \) are called the left singular vectors while t
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec39">Economy-size SVD </h2>
+<h2 id="___sec35">Economy-size SVD </h2>
 
 <p>
 If we assume that \( n > p \), then our matrix \( \boldsymbol{U} \) has dimension \( n
@@ -2040,7 +1794,7 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec40">Mathematical Properties </h2>
+<h2 id="___sec36">Mathematical Properties </h2>
 
 <p>
 There are several interesting mathematical properties which will be
@@ -2092,7 +1846,7 @@ We will come back to this expression when we discuss Ridge regression.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec41">Ridge and LASSO Regression </h2>
+<h2 id="___sec37">Ridge and LASSO Regression </h2>
 
 <p>
 Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
@@ -2150,7 +1904,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec42">More on Ridge Regression </h2>
+<h2 id="___sec38">More on Ridge Regression </h2>
 
 <p>
 Using the matrix-vector expression for Ridge regression,
@@ -2209,7 +1963,7 @@ with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec43">Interpreting the Ridge results </h2>
+<h2 id="___sec39">Interpreting the Ridge results </h2>
 
 <p>
 Since \( \lambda \geq 0 \), it means that compared to OLS, we have 
@@ -2233,7 +1987,7 @@ With a parameter \( \lambda \) we can thus shrink the role of specific parameter
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec44">More interpretations </h2>
+<h2 id="___sec40">More interpretations </h2>
 
 <p>
 For the sake of simplicity, let us assume that the design matrix is orthonormal, that is 
@@ -2270,7 +2024,7 @@ Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et a
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec45">Where are we going? </h2>
+<h2 id="___sec41">Where are we going? </h2>
 
 <p>
 Before we proceed, we need to rethink what we have been doing. In our
@@ -2287,7 +2041,7 @@ This will allow us to link the standard linear algebra methods we have discussed
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec46">Resampling methods </h2>
+<h2 id="___sec42">Resampling methods </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2307,7 +2061,7 @@ once using the original training sample.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec47">Resampling approaches can be computationally expensive </h2>
+<h2 id="___sec43">Resampling approaches can be computationally expensive </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2336,7 +2090,7 @@ bootstrap is widely used.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec48">Why resampling methods ? </h2>
+<h2 id="___sec44">Why resampling methods ? </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b>Statistical analysis.</b>
 <p>
@@ -2352,7 +2106,7 @@ bootstrap is widely used.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec49">Statistical analysis </h2>
+<h2 id="___sec45">Statistical analysis </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2374,7 +2128,7 @@ bootstrap is widely used.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec50">Statistics  </h2>
+<h2 id="___sec46">Statistics  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2405,7 +2159,7 @@ selection of a large set of these numbers reproduces this PDF.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec51">Statistics, moments </h2>
+<h2 id="___sec47">Statistics, moments </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2428,7 +2182,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec52">Statistics, central moments </h2>
+<h2 id="___sec48">Statistics, central moments </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2446,14 +2200,14 @@ $$
 \begin{align}
 \sigma^2_X\ \ =\ \ \mathrm{var}(X) & =  \langle (x-\langle x\rangle)^2\rangle =
 \int\! (x-\langle x\rangle)^2 p(x)\,dx
-\label{_auto8}\\
+\label{_auto2}\\
 & =  \int\! \left(x^2 - 2 x \langle x\rangle^{2} +
   \langle x\rangle^2\right)p(x)\,dx
-\label{_auto9}\\
+\label{_auto3}\\
 & =  \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2
-\label{_auto10}\\
+\label{_auto4}\\
 & =  \langle x^2\rangle - \langle x\rangle^2
-\label{_auto11}
+\label{_auto5}
 \end{align}
 $$
 
@@ -2466,7 +2220,7 @@ qualitatively as the <em>spread</em> of \( p \) around its mean.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec53">Statistics, covariance </h2>
+<h2 id="___sec49">Statistics, covariance </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2497,7 +2251,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec54">Statistics, more covariance  </h2>
+<h2 id="___sec50">Statistics, more covariance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2510,17 +2264,17 @@ the expectation value. Consider the stochastic variables \( X_i \) and
 $$
 \begin{align}
 \mathrm{cov}(X_i,\,X_j) &= \langle(x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle
-\label{_auto12}\\
+\label{_auto6}\\
 &=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle 
-\label{_auto13}\\
+\label{_auto7}\\
 &=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j\rangle +
 \langle \langle x_i\rangle\langle x_j\rangle\rangle
-\label{_auto14}\\
+\label{_auto8}\\
 &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle +
 \langle x_i\rangle\langle x_j\rangle
-\label{_auto15}\\
+\label{_auto9}\\
 &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle
-\label{_auto16}
+\label{_auto10}
 \end{align}
 $$
 </div>
@@ -2529,7 +2283,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec55">Covariance example </h2>
+<h2 id="___sec51">Covariance example </h2>
 
 <p>
 Suppose we have defined three vectors \( \hat{x}, \hat{y}, \hat{z} \) with
@@ -2578,7 +2332,7 @@ function.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec56">Covariance in numpy </h2>
+<h2 id="___sec52">Covariance in numpy </h2>
 
 <p>
 
@@ -2617,7 +2371,7 @@ plt<span style="color: #666666">.</span>show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec57">Statistics, independent variables  </h2>
+<h2 id="___sec53">Statistics, independent variables  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2643,7 +2397,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec58">Statistics, more variance  </h2>
+<h2 id="___sec54">Statistics, more variance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2675,7 +2429,7 @@ value of a set of measurements.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec59">Statistics and stochastic processes  </h2>
+<h2 id="___sec55">Statistics and stochastic processes  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2703,7 +2457,7 @@ interested in finding the few lowest moments, like the mean
 <p>
 <!-- !split  -->
 
-<h2 id="___sec60">Statistics and sample variables  </h2>
+<h2 id="___sec56">Statistics and sample variables  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2729,7 +2483,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec61">Statistics, sample variance and covariance </h2>
+<h2 id="___sec57">Statistics, sample variance and covariance </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2750,7 +2504,7 @@ and covariance \( \mathrm{cov}(X,Y) \).
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec62">Statistics, law of large numbers </h2>
+<h2 id="___sec58">Statistics, law of large numbers </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2782,7 +2536,7 @@ true PDFs behind, which we usually do not have.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec63">Statistics, more on sample error </h2>
+<h2 id="___sec59">Statistics, more on sample error </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2804,7 +2558,7 @@ means.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec64">Statistics  </h2>
+<h2 id="___sec60">Statistics  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2825,7 +2579,7 @@ And in particular we are interested in its variance \( \mathrm{var}(\overline X_
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec65">Statistics, central limit theorem </h2>
+<h2 id="___sec61">Statistics, central limit theorem </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2850,7 +2604,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec66">Statistics, more technicalities  </h2>
+<h2 id="___sec62">Statistics, more technicalities  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2879,7 +2633,7 @@ estimate of the PDF of each of the \( X_i \), estimating all properties of
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec67">Statistics </h2>
+<h2 id="___sec63">Statistics </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2906,7 +2660,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec68">Statistics and sample variance  </h2>
+<h2 id="___sec64">Statistics and sample variance  </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2945,7 +2699,7 @@ measurements in the sample.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec69">Statistics, uncorrelated results </h2>
+<h2 id="___sec65">Statistics, uncorrelated results </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -2980,7 +2734,7 @@ cannot overlook the always present correlations.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec70">Statistics, computations </h2>
+<h2 id="___sec66">Statistics, computations </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3009,7 +2763,7 @@ measurements. For uncorrelated measurements this second term is zero.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec71">Statistics, more on computations of errors </h2>
+<h2 id="___sec67">Statistics, more on computations of errors </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3031,7 +2785,7 @@ have to be stored throughout the experiment.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec72">Statistics, wrapping up 1 </h2>
+<h2 id="___sec68">Statistics, wrapping up 1 </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3064,7 +2818,7 @@ starting always at \( 1 \) for \( d=0 \).
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec73">Statistics, final expression </h2>
+<h2 id="___sec69">Statistics, final expression </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3097,7 +2851,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec74">Statistics, effective number of correlations </h2>
+<h2 id="___sec70">Statistics, effective number of correlations </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3122,7 +2876,7 @@ measurements is very large.
 <p>
 <!-- !split  -->
 
-<h2 id="___sec75">Linking the regression analysis with a statistical interpretation </h2>
+<h2 id="___sec71">Linking the regression analysis with a statistical interpretation </h2>
 
 <p>
 Finally, we are going to discuss several statistical properties which can be obtained in terms of analytical expressions. 
@@ -3160,7 +2914,7 @@ row number \( i \) and perform a sum over all values \( p \).
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec76">Assumptions made </h2>
+<h2 id="___sec72">Assumptions made </h2>
 
 <p>
 The assumption we have made here can be summarized as (and this is going to useful when we discuss the bias-variance trade off)
@@ -3180,7 +2934,7 @@ $$
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec77">Expectation value and variance </h2>
+<h2 id="___sec73">Expectation value and variance </h2>
 
 <p>
 We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) 
@@ -3215,7 +2969,7 @@ mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (n
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec78">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
+<h2 id="___sec74">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
 
 <p>
 With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
@@ -3300,7 +3054,7 @@ This means the variance we obtain with the standard OLS will always for \( \lamb
 <p>
 <!-- !split  -->
 
-<h2 id="___sec79">Cross-validation </h2>
+<h2 id="___sec75">Cross-validation </h2>
 
 <p>
 Instead of choosing the penalty parameter to balance model fit with
@@ -3334,7 +3088,7 @@ some sense) is then selected.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec80">Computationally expensive </h2>
+<h2 id="___sec76">Computationally expensive </h2>
 
 <p>
 The validation set approach is conceptually simple and is easy to implement. But it has two potential drawbacks:
@@ -3346,7 +3100,7 @@ The validation set approach is conceptually simple and is easy to implement. But
 
 <!-- !split  -->
 
-<h2 id="___sec81">Various steps in cross-validation </h2>
+<h2 id="___sec77">Various steps in cross-validation </h2>
 
 <p>
 When the repetitive splitting of the data set is done randomly,
@@ -3367,7 +3121,7 @@ cross-validation (LOOCV).
 <p>
 <!-- !split  -->
 
-<h2 id="___sec82">How to set up the cross-validation for Ridge and/or Lasso </h2>
+<h2 id="___sec78">How to set up the cross-validation for Ridge and/or Lasso </h2>
 
 <ul>
 <li> Define a range of interest for the penalty parameter.</li>
@@ -3403,7 +3157,7 @@ $$
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec83">Resampling methods: Jackknife and Bootstrap </h2>
+<h2 id="___sec79">Resampling methods: Jackknife and Bootstrap </h2>
 
 <p>
 Two famous
@@ -3426,7 +3180,7 @@ need for bootstrapping.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec84">Resampling methods: Jackknife </h2>
+<h2 id="___sec80">Resampling methods: Jackknife </h2>
 
 <p>
 The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). 
@@ -3445,7 +3199,7 @@ number \( i \) is left out. Using this notation, define
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec85">Jackknife code example </h2>
+<h2 id="___sec81">Jackknife code example </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -3481,7 +3235,7 @@ t <span style="color: #666666">=</span> jackknife(x, stat)
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec86">Resampling methods: Bootstrap </h2>
+<h2 id="___sec82">Resampling methods: Bootstrap </h2>
 <div class="alert alert-block alert-block alert-text-normal">
 <b></b>
 <p>
@@ -3502,7 +3256,7 @@ advantages:
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec87">Resampling methods: Bootstrap background </h2>
+<h2 id="___sec83">Resampling methods: Bootstrap background </h2>
 
 <p>
 Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
@@ -3518,7 +3272,7 @@ estimators.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec88">Resampling methods: More Bootstrap background </h2>
+<h2 id="___sec84">Resampling methods: More Bootstrap background </h2>
 
 <p>
 In the case that \( \widehat{\theta} \) has
@@ -3540,7 +3294,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec89">Resampling methods: Bootstrap approach </h2>
+<h2 id="___sec85">Resampling methods: Bootstrap approach </h2>
 
 <p>
 But
@@ -3561,7 +3315,7 @@ frequency of the observation \( X_i \), just draw the values
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec90">Resampling methods: Bootstrap steps </h2>
+<h2 id="___sec86">Resampling methods: Bootstrap steps </h2>
 
 <p>
 The independent bootstrap works like this: 
@@ -3586,7 +3340,7 @@ example, if you are interested in estimating the variance of \( \widehat
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec91">Code example for the Bootstrap method </h2>
+<h2 id="___sec87">Code example for the Bootstrap method </h2>
 
 <p>
 The following code starts with a Gaussian distribution with mean value
@@ -3650,7 +3404,7 @@ plt<span style="color: #666666">.</span>show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec92">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
+<h2 id="___sec88">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
 
 <p>
 The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
@@ -3750,7 +3504,7 @@ plt<span style="color: #666666">.</span>show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec93">The bias-variance tradeoff </h2>
+<h2 id="___sec89">The bias-variance tradeoff </h2>
 
 <p>
 We will discuss the bias-variance tradeoff in the context of
@@ -3816,7 +3570,7 @@ that is the rewriting in terms of the so-called bias, the variance of the model
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec94">Example code for Bias-Variance tradeoff </h2>
+<h2 id="___sec90">Example code for Bias-Variance tradeoff </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -3878,7 +3632,7 @@ plt<span style="color: #666666">.</span>show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec95">Understanding what happens </h2>
+<h2 id="___sec91">Understanding what happens </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -3932,7 +3686,7 @@ plt<span style="color: #666666">.</span>show()
 <p>
 <!-- !split  -->
 
-<h2 id="___sec96">Summing up </h2>
+<h2 id="___sec92">Summing up </h2>
 
 <p>
 The bias-variance tradeoff summarizes the fundamental tension in
@@ -3966,7 +3720,7 @@ flexible statistical methods have higher variance.
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec97">Another Example rom Scikit-Learn's Repository </h2>
+<h2 id="___sec93">Another Example rom Scikit-Learn's Repository </h2>
 <p>
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -4045,6 +3799,248 @@ plt<span style="color: #666666">.</span>show()
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
+<h2 id="___sec94">The Ising model </h2>
+
+<p>
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant \( J \) is given by
+
+$$
+\begin{align}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\label{_auto11}
+\end{align}
+$$
+
+<p>
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
+in the system is determined by \( L \). For the one-dimensional system
+there is no phase transition.
+
+<p>
+We will look at a system of \( L = 40 \) spins with a coupling constant of
+\( J = 1 \). To get enough training data we will generate 10000 states
+with their respective energies.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
+sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000">True</span>)
+cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&#39;seismic&#39;</span>)
+
+L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
+n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
+
+spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
+J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+
+energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
+
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
+    energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
+</pre></div>
+<p>
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
+<h2 id="___sec95">Reformulating the problem to suit regression </h2>
+
+<p>
+A more general form for the one-dimensional Ising model is
+
+$$
+\begin{align}
+    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\label{_auto12}
+\end{align}
+$$
+
+<p>
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+$$
+\begin{align}
+    \boldsymbol{H} = \boldsymbol{X} J,
+\label{_auto13}
+\end{align}
+$$
+
+<p>
+where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+$$
+\begin{align}
+    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\label{_auto14}
+\end{align}
+$$
+
+<p>
+We split the data in training and test data as discussed in the previous example
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
+    X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
+y <span style="color: #666666">=</span> energies
+X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
+</pre></div>
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
+<h2 id="___sec96">Linear regression </h2>
+
+<p>
+In the ordinary least squares method we choose the cost function
+
+$$
+\begin{align}
+    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\label{_auto15}
+\end{align}
+$$
+
+<p>
+We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
+This yields the expression for \( \boldsymbol{\beta} \) to be
+
+$$
+    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+$$
+
+<p>
+which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
+an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
+    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
+    axis<span style="color: #666666">=1</span>
+)
+X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
+    (np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
+    axis<span style="color: #666666">=1</span>
+)
+</pre></div>
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_inv</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
+    <span style="color: #008000; font-weight: bold">return</span> scl<span style="color: #666666">.</span>inv(x<span style="color: #666666">.</span>T @ x) @ (x<span style="color: #666666">.</span>T @ y)
+beta <span style="color: #666666">=</span> ols_inv(X_train_own, y_train)
+</pre></div>
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
+<h2 id="___sec97">Singular Value decomposition </h2>
+
+<p>
+Doing the inversion directly turns out to be a bad idea since the matrix
+\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
+value decomposition</b>. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+
+$$
+    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+
+<p>
+where the pseudoinverse of \( \boldsymbol{X} \) is given by
+
+$$
+    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+
+<p>
+Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
+where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
+where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
+\( \omega \) to
+$$
+\begin{align}
+    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\label{_auto16}
+\end{align}
+$$
+
+<p>
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
+general matrix. Instead, doing \( QR \)-factorization and solving the
+linear system as an equation would reduce this down to
+\( \mathcal{O}(n^2) \) operations.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_svd</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
+    u, s, v <span style="color: #666666">=</span> scl<span style="color: #666666">.</span>svd(x)
+    <span style="color: #008000; font-weight: bold">return</span> v<span style="color: #666666">.</span>T @ scl<span style="color: #666666">.</span>pinv(scl<span style="color: #666666">.</span>diagsvd(s, u<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], v<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])) @ u<span style="color: #666666">.</span>T @ y
+</pre></div>
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>beta <span style="color: #666666">=</span> ols_svd(X_train_own,y_train)
+</pre></div>
+<p>
+When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J <span style="color: #666666">=</span> beta[<span style="color: #666666">1</span>:]<span style="color: #666666">.</span>reshape(L, L)
+</pre></div>
+<p>
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
+
+<p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
+im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J, <span style="color: #666666">**</span>cmap_args)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;OLS&quot;</span>, fontsize<span style="color: #666666">=18</span>)
+plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
+plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
+cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
+cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
+plt<span style="color: #666666">.</span>show()
+</pre></div>
+<p>
+It is interesting to note that OLS
+considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
+valid matrix elements for \( J \).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+<p>
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
+<p>
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+
 <h2 id="___sec98">The one-dimensional Ising model </h2>
 
 <p>
diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb
index 8f1b1cbe8..38d684b11 100644
--- a/doc/pub/Regression/ipynb/Regression.ipynb
+++ b/doc/pub/Regression/ipynb/Regression.ipynb
@@ -10,7 +10,7 @@
     "<!-- Author: -->  \n",
     "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
     "\n",
-    "Date: **Aug 29, 2019**\n",
+    "Date: **Aug 30, 2019**\n",
     "\n",
     "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
     "\n",
@@ -1905,7 +1905,7 @@
     "did both for the masses and the fitting of the equation of state,\n",
     "leads to row vectors of the design matrix which are essentially\n",
     "orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. \n",
-    "More material to come here. \n",
+    "\n",
     "\n",
     "\n",
     "This may\n",
@@ -1922,407 +1922,6 @@
     "\n",
     "\n",
     "\n",
-    "<!-- todo: change model here. -->\n",
-    "\n",
-    "## The Ising model\n",
-    "\n",
-    "The one-dimensional Ising model with nearest neighbor interaction, no\n",
-    "external field and a constant coupling constant $J$ is given by"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto1\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "    H = -J \\sum_{k}^L s_k s_{k + 1},\n",
-    "\\label{_auto1} \\tag{1}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n",
-    "in the system is determined by $L$. For the one-dimensional system\n",
-    "there is no phase transition.\n",
-    "\n",
-    "We will look at a system of $L = 40$ spins with a coupling constant of\n",
-    "$J = 1$. To get enough training data we will generate 10000 states\n",
-    "with their respective energies."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 22,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "import numpy as np\n",
-    "import matplotlib.pyplot as plt\n",
-    "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
-    "import seaborn as sns\n",
-    "import scipy.linalg as scl\n",
-    "from sklearn.model_selection import train_test_split\n",
-    "import tqdm\n",
-    "sns.set(color_codes=True)\n",
-    "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
-    "\n",
-    "L = 40\n",
-    "n = int(1e4)\n",
-    "\n",
-    "spins = np.random.choice([-1, 1], size=(n, L))\n",
-    "J = 1.0\n",
-    "\n",
-    "energies = np.zeros(n)\n",
-    "\n",
-    "for i in range(n):\n",
-    "    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Here we use ordinary least squares\n",
-    "regression to predict the energy for the nearest neighbor\n",
-    "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n",
-    "around. We will use linear regression to fit a value for\n",
-    "the coupling constant to achieve this.\n",
-    "\n",
-    "## Reformulating the problem to suit regression\n",
-    "\n",
-    "A more general form for the one-dimensional Ising model is"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto2\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "    H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
-    "\\label{_auto2} \\tag{2}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Here we allow for interactions beyond the nearest neighbors and a state dependent\n",
-    "coupling constant. This latter expression can be formulated as\n",
-    "a matrix-product"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto3\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "    \\boldsymbol{H} = \\boldsymbol{X} J,\n",
-    "\\label{_auto3} \\tag{3}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n",
-    "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
-    "with the form utilized in linear regression, that is"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto4\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "    \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon},\n",
-    "\\label{_auto4} \\tag{4}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We split the data in training and test data as discussed in the previous example"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 23,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "X = np.zeros((n, L ** 2))\n",
-    "for i in range(n):\n",
-    "    X[i] = np.outer(spins[i], spins[i]).ravel()\n",
-    "y = energies\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Linear regression\n",
-    "\n",
-    "In the ordinary least squares method we choose the cost function"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto5\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "    C(\\boldsymbol{X}, \\boldsymbol{\\beta})= \\frac{1}{n}\\left\\{(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})\\right\\}.\n",
-    "\\label{_auto5} \\tag{5}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n",
-    "This yields the expression for $\\boldsymbol{\\beta}$ to be"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n",
-    "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n",
-    "intercept, i.e., a constant term, we must make sure that the\n",
-    "first column of $\\boldsymbol{X}$ consists of $1$. We do this here"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 24,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "X_train_own = np.concatenate(\n",
-    "    (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
-    "    axis=1\n",
-    ")\n",
-    "X_test_own = np.concatenate(\n",
-    "    (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
-    "    axis=1\n",
-    ")"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 25,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
-    "    return scl.inv(x.T @ x) @ (x.T @ y)\n",
-    "beta = ols_inv(X_train_own, y_train)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Singular Value decomposition\n",
-    "\n",
-    "Doing the inversion directly turns out to be a bad idea since the matrix\n",
-    "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n",
-    "value decomposition**. Using the definition of the Moore-Penrose\n",
-    "pseudoinverse we can write the equation for $\\boldsymbol{\\beta}$ as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "where the pseudoinverse of $\\boldsymbol{X}$ is given by"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Using singular value decomposition we can decompose the matrix  $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n",
-    "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n",
-    "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n",
-    "$\\omega$ to"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto6\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "    \\boldsymbol{\\beta} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^{+} \\boldsymbol{U}^T \\boldsymbol{y}.\n",
-    "\\label{_auto6} \\tag{6}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Note that solving this equation by actually doing the pseudoinverse\n",
-    "(which is what we will do) is not a good idea as this operation scales\n",
-    "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n",
-    "general matrix. Instead, doing $QR$-factorization and solving the\n",
-    "linear system as an equation would reduce this down to\n",
-    "$\\mathcal{O}(n^2)$ operations."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 26,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
-    "    u, s, v = scl.svd(x)\n",
-    "    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 27,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "beta = ols_svd(X_train_own,y_train)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "When extracting the $J$-matrix  we need to make sure that we remove the intercept, as is done here"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 28,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "J = beta[1:].reshape(L, L)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "A way of looking at the coefficients in $J$ is to plot the matrices as images."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 29,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [],
-   "source": [
-    "fig = plt.figure(figsize=(20, 14))\n",
-    "im = plt.imshow(J, **cmap_args)\n",
-    "plt.title(\"OLS\", fontsize=18)\n",
-    "plt.xticks(fontsize=18)\n",
-    "plt.yticks(fontsize=18)\n",
-    "cb = fig.colorbar(im)\n",
-    "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "It is interesting to note that OLS\n",
-    "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n",
-    "valid matrix elements for $J$.\n",
-    "In our discussion below on hyperparameters and Ridge and Lasso regression we will see that\n",
-    "this problem can be removed, partly and only with Lasso regression. \n",
-    "\n",
-    "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n",
-    "\n",
     "\n",
     "## Linear Regression Problems\n",
     "\n",
@@ -2365,7 +1964,7 @@
     "of linearly independent columns. In this particular case the matrix has rank 2.\n",
     "\n",
     "Super-collinearity of an $(n \\times p)$-dimensional design matrix $\\mathbf{X}$ implies\n",
-    "that the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{x}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this"
+    "that the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this"
    ]
   },
   {
@@ -2402,12 +2001,12 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto7\"></div>\n",
+    "<div id=\"_auto1\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
     "\\boldsymbol{\\beta}  =  (\\boldsymbol{X}^{T} \\boldsymbol{X})^{-1} \\boldsymbol{X}^{T} \\boldsymbol{y},\n",
-    "\\label{_auto7} \\tag{7}\n",
+    "\\label{_auto1} \\tag{1}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -2599,7 +2198,7 @@
     "$n\\times p$, the matrix is thus decomposed into an $n\\times n$\n",
     "orthogonal matrix $\\boldsymbol{U}$, a $p\\times p$ orthogonal matrix $\\boldsymbol{V}$\n",
     "and a diagonal matrix $\\boldsymbol{\\Sigma}$ with $r=\\mathrm{min}(n,p)$\n",
-    "singular values $\\sigma_i\\lg 0$ on the main diagonal and zeros filling\n",
+    "singular values $\\sigma_i\\geq 0$ on the main diagonal and zeros filling\n",
     "the rest of the matrix.  There are at most $p$ singular values\n",
     "assuming that $n > p$. In our regression examples for the nuclear\n",
     "masses and the equation of state this is indeed the case, while for\n",
@@ -3248,13 +2847,13 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto8\"></div>\n",
+    "<div id=\"_auto2\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
     "\\sigma^2_X\\ \\ =\\ \\ \\mathrm{var}(X)  =  \\langle (x-\\langle x\\rangle)^2\\rangle =\n",
     "\\int\\! (x-\\langle x\\rangle)^2 p(x)\\,dx\n",
-    "\\label{_auto8} \\tag{8}\n",
+    "\\label{_auto2} \\tag{2}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3264,13 +2863,13 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto9\"></div>\n",
+    "<div id=\"_auto3\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     " =  \\int\\! \\left(x^2 - 2 x \\langle x\\rangle^{2} +\n",
     "  \\langle x\\rangle^2\\right)p(x)\\,dx\n",
-    "\\label{_auto9} \\tag{9}\n",
+    "\\label{_auto3} \\tag{3}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3280,12 +2879,12 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto10\"></div>\n",
+    "<div id=\"_auto4\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     " =  \\langle x^2\\rangle - 2 \\langle x\\rangle\\langle x\\rangle + \\langle x\\rangle^2\n",
-    "\\label{_auto10} \\tag{10}\n",
+    "\\label{_auto4} \\tag{4}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3295,12 +2894,12 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto11\"></div>\n",
+    "<div id=\"_auto5\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     " =  \\langle x^2\\rangle - \\langle x\\rangle^2\n",
-    "\\label{_auto11} \\tag{11}\n",
+    "\\label{_auto5} \\tag{5}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3345,7 +2944,7 @@
     "=\n",
     "\\int\\!\\cdots\\!\\int\\!(x_i-\\langle x_i \\rangle)(x_j-\\langle x_j \\rangle)\\,\n",
     "P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n\n",
-    "\\label{eq:def_covariance} \\tag{12}\n",
+    "\\label{eq:def_covariance} \\tag{6}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3385,12 +2984,12 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto12\"></div>\n",
+    "<div id=\"_auto6\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
     "\\mathrm{cov}(X_i,\\,X_j) = \\langle(x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle\n",
-    "\\label{_auto12} \\tag{13}\n",
+    "\\label{_auto6} \\tag{7}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3400,12 +2999,12 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto13\"></div>\n",
+    "<div id=\"_auto7\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     "=\\langle x_i x_j - x_i\\langle x_j\\rangle - \\langle x_i\\rangle x_j + \\langle x_i\\rangle\\langle x_j\\rangle\\rangle \n",
-    "\\label{_auto13} \\tag{14}\n",
+    "\\label{_auto7} \\tag{8}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3415,13 +3014,13 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto14\"></div>\n",
+    "<div id=\"_auto8\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     "=\\langle x_i x_j\\rangle - \\langle x_i\\langle x_j\\rangle\\rangle - \\langle \\langle x_i\\rangle x_j\\rangle +\n",
     "\\langle \\langle x_i\\rangle\\langle x_j\\rangle\\rangle\n",
-    "\\label{_auto14} \\tag{15}\n",
+    "\\label{_auto8} \\tag{9}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3431,13 +3030,13 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto15\"></div>\n",
+    "<div id=\"_auto9\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     "=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle +\n",
     "\\langle x_i\\rangle\\langle x_j\\rangle\n",
-    "\\label{_auto15} \\tag{16}\n",
+    "\\label{_auto9} \\tag{10}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3447,12 +3046,12 @@
    "metadata": {},
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto16\"></div>\n",
+    "<div id=\"_auto10\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     "=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle\n",
-    "\\label{_auto16} \\tag{17}\n",
+    "\\label{_auto10} \\tag{11}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3540,7 +3139,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 22,
    "metadata": {
     "collapsed": false
    },
@@ -3565,7 +3164,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 23,
    "metadata": {
     "collapsed": false
    },
@@ -3642,7 +3241,7 @@
     "$$\n",
     "\\begin{equation}\n",
     "\\mathrm{var}(U) = \\sum_{i,j}a_i a_j \\mathrm{cov}(X_i, X_j)\n",
-    "\\label{eq:variance_linear_combination} \\tag{18}\n",
+    "\\label{eq:variance_linear_combination} \\tag{12}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3661,8 +3260,8 @@
    "metadata": {},
    "source": [
     "1\n",
-    "2\n",
-    "0\n",
+    "1\n",
+    "1\n",
     " \n",
     "<\n",
     "<\n",
@@ -3787,7 +3386,7 @@
    "source": [
     "## Statistics, sample variance and covariance\n",
     "Note that the sample variance is the sample covariance without the\n",
-    "cross terms. In a similar manner as the covariance in Eq. ([12](#eq:def_covariance)) is a measure of the correlation between\n",
+    "cross terms. In a similar manner as the covariance in Eq. ([6](#eq:def_covariance)) is a measure of the correlation between\n",
     "two stochastic variables, the above defined sample covariance is a\n",
     "measure of the sequential correlation between succeeding measurements\n",
     "of a sample.\n",
@@ -3913,7 +3512,7 @@
     "\\lim_{n\\to\\infty} p_{\\overline X_n}(x) =\n",
     "\\left(\\frac{n}{2\\pi\\mathrm{var}(X)}\\right)^{1/2}\n",
     "e^{-\\frac{n(x-\\bar x_n)^2}{2\\mathrm{var}(X)}}\n",
-    "\\label{eq:central_limit_gaussian} \\tag{19}\n",
+    "\\label{eq:central_limit_gaussian} \\tag{13}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3939,7 +3538,7 @@
     "\\begin{equation}\n",
     "\\mathrm{err}_X^2 = \\mathrm{var}(\\overline X_n) = \\frac{1}{n^2}\n",
     "\\sum_{ij} \\mathrm{cov}(X_i, X_j)\n",
-    "\\label{eq:error_exact} \\tag{20}\n",
+    "\\label{eq:error_exact} \\tag{14}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -3979,7 +3578,7 @@
    "metadata": {},
    "source": [
     "Using $\\bar x$ in place of $\\mu_{X_i}^{\\phantom X}$ we can give an\n",
-    "*estimate* of the covariance in Eq. ([20](#eq:error_exact))"
+    "*estimate* of the covariance in Eq. ([14](#eq:error_exact))"
    ]
   },
   {
@@ -4043,7 +3642,7 @@
     "$$\n",
     "\\begin{equation}\n",
     "\\mathrm{var}(X_i)\\approx \\frac{1}{n}\\sum_{k=1}^n (x_k - \\bar x_n)=\\mathrm{var}(x)\n",
-    "\\label{eq:var_estimate_i_think} \\tag{21}\n",
+    "\\label{eq:var_estimate_i_think} \\tag{15}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -4085,7 +3684,7 @@
     "$$\n",
     "\\begin{equation} \n",
     "=\\frac{1}{n}\\mathrm{cov}(x)\n",
-    "\\label{eq:error_estimate} \\tag{22}\n",
+    "\\label{eq:error_estimate} \\tag{16}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -4134,7 +3733,7 @@
     "$$\n",
     "\\begin{equation}\n",
     "\\mathrm{err}_X^2\\approx \\frac{1}{n^2} \\sum_i \\mathrm{var}(x)= \\frac{1}{n}\\mathrm{var}(x)\n",
-    "\\label{eq:error_estimate_uncorrel} \\tag{23}\n",
+    "\\label{eq:error_estimate_uncorrel} \\tag{17}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -4143,7 +3742,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "where in the second step we have used Eq. ([21](#eq:var_estimate_i_think)).\n",
+    "where in the second step we have used Eq. ([15](#eq:var_estimate_i_think)).\n",
     "The error of the sample is then just its standard deviation divided by\n",
     "the square root of the number of measurements the sample contains.\n",
     "This is a very useful formula which is easy to compute. It acts as a\n",
@@ -4154,7 +3753,7 @@
     "\n",
     "## Statistics, computations\n",
     "For computational purposes one usually splits up the estimate of\n",
-    "$\\mathrm{err}_X^2$, given by Eq. ([22](#eq:error_estimate)), into two\n",
+    "$\\mathrm{err}_X^2$, given by Eq. ([16](#eq:error_estimate)), into two\n",
     "parts"
    ]
   },
@@ -4184,7 +3783,7 @@
     "$$\n",
     "\\begin{equation}\n",
     "\\frac{1}{n^2}\\sum_{k=1}^n (x_k - \\bar x_n)^2 +\\frac{2}{n^2}\\sum_{k<l} (x_k - \\bar x_n)(x_l - \\bar x_n)\n",
-    "\\label{eq:error_estimate_split_up} \\tag{24}\n",
+    "\\label{eq:error_estimate_split_up} \\tag{18}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -4194,7 +3793,7 @@
    "metadata": {},
    "source": [
     "The first term is the same as the error in the uncorrelated case,\n",
-    "Eq. ([23](#eq:error_estimate_uncorrel)). This means that the second\n",
+    "Eq. ([17](#eq:error_estimate_uncorrel)). This means that the second\n",
     "term accounts for the error correction due to correlation between the\n",
     "measurements. For uncorrelated measurements this second term is zero.\n",
     "\n",
@@ -4292,7 +3891,7 @@
     "\n",
     "\n",
     "## Statistics, final expression\n",
-    "The sample error (see eq. ([24](#eq:error_estimate_split_up))) can now be\n",
+    "The sample error (see eq. ([18](#eq:error_estimate_split_up))) can now be\n",
     "written in terms of the autocorrelation function:"
    ]
   },
@@ -4327,7 +3926,7 @@
     "$$\n",
     "\\begin{equation} \n",
     "=\\frac{\\tau}{n}\\cdot\\mathrm{var}(x)\n",
-    "\\label{eq:error_estimate_corr_time} \\tag{25}\n",
+    "\\label{eq:error_estimate_corr_time} \\tag{19}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -4352,7 +3951,7 @@
     "$$\n",
     "\\begin{equation}\n",
     "\\tau = 1+2\\sum_{d=1}^{n-1}\\kappa_d\n",
-    "\\label{eq:autocorrelation_time} \\tag{26}\n",
+    "\\label{eq:autocorrelation_time} \\tag{20}\n",
     "\\end{equation}\n",
     "$$"
    ]
@@ -4364,7 +3963,7 @@
     "## Statistics, effective number of correlations\n",
     "For a correlation free experiment, $\\tau$\n",
     "equals 1. From the point of view of\n",
-    "eq. ([25](#eq:error_estimate_corr_time)) we can interpret a sequential\n",
+    "eq. ([19](#eq:error_estimate_corr_time)) we can interpret a sequential\n",
     "correlation as an effective reduction of the number of measurements by\n",
     "a factor $\\tau$. The effective number of measurements becomes:"
    ]
@@ -4813,7 +4412,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 24,
    "metadata": {
     "collapsed": false
    },
@@ -4956,7 +4555,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 25,
    "metadata": {
     "collapsed": false
    },
@@ -5016,7 +4615,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 26,
    "metadata": {
     "collapsed": false
    },
@@ -5247,7 +4846,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 27,
    "metadata": {
     "collapsed": false
    },
@@ -5318,7 +4917,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 28,
    "metadata": {
     "collapsed": false
    },
@@ -5417,7 +5016,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 29,
    "metadata": {
     "collapsed": false
    },
@@ -5500,6 +5099,409 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "## The Ising model\n",
+    "\n",
+    "The one-dimensional Ising model with nearest neighbor interaction, no\n",
+    "external field and a constant coupling constant $J$ is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto11\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    H = -J \\sum_{k}^L s_k s_{k + 1},\n",
+    "\\label{_auto11} \\tag{21}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n",
+    "in the system is determined by $L$. For the one-dimensional system\n",
+    "there is no phase transition.\n",
+    "\n",
+    "We will look at a system of $L = 40$ spins with a coupling constant of\n",
+    "$J = 1$. To get enough training data we will generate 10000 states\n",
+    "with their respective energies."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
+    "import seaborn as sns\n",
+    "import scipy.linalg as scl\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "import tqdm\n",
+    "sns.set(color_codes=True)\n",
+    "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
+    "\n",
+    "L = 40\n",
+    "n = int(1e4)\n",
+    "\n",
+    "spins = np.random.choice([-1, 1], size=(n, L))\n",
+    "J = 1.0\n",
+    "\n",
+    "energies = np.zeros(n)\n",
+    "\n",
+    "for i in range(n):\n",
+    "    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here we use ordinary least squares\n",
+    "regression to predict the energy for the nearest neighbor\n",
+    "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n",
+    "around. We will use linear regression to fit a value for\n",
+    "the coupling constant to achieve this.\n",
+    "\n",
+    "## Reformulating the problem to suit regression\n",
+    "\n",
+    "A more general form for the one-dimensional Ising model is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto12\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
+    "\\label{_auto12} \\tag{22}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here we allow for interactions beyond the nearest neighbors and a state dependent\n",
+    "coupling constant. This latter expression can be formulated as\n",
+    "a matrix-product"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto13\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    \\boldsymbol{H} = \\boldsymbol{X} J,\n",
+    "\\label{_auto13} \\tag{23}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n",
+    "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
+    "with the form utilized in linear regression, that is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto14\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon},\n",
+    "\\label{_auto14} \\tag{24}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We split the data in training and test data as discussed in the previous example"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "X = np.zeros((n, L ** 2))\n",
+    "for i in range(n):\n",
+    "    X[i] = np.outer(spins[i], spins[i]).ravel()\n",
+    "y = energies\n",
+    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Linear regression\n",
+    "\n",
+    "In the ordinary least squares method we choose the cost function"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto15\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    C(\\boldsymbol{X}, \\boldsymbol{\\beta})= \\frac{1}{n}\\left\\{(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})\\right\\}.\n",
+    "\\label{_auto15} \\tag{25}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n",
+    "This yields the expression for $\\boldsymbol{\\beta}$ to be"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n",
+    "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n",
+    "intercept, i.e., a constant term, we must make sure that the\n",
+    "first column of $\\boldsymbol{X}$ consists of $1$. We do this here"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "X_train_own = np.concatenate(\n",
+    "    (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
+    "    axis=1\n",
+    ")\n",
+    "X_test_own = np.concatenate(\n",
+    "    (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
+    "    axis=1\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
+    "    return scl.inv(x.T @ x) @ (x.T @ y)\n",
+    "beta = ols_inv(X_train_own, y_train)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Singular Value decomposition\n",
+    "\n",
+    "Doing the inversion directly turns out to be a bad idea since the matrix\n",
+    "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n",
+    "value decomposition**. Using the definition of the Moore-Penrose\n",
+    "pseudoinverse we can write the equation for $\\boldsymbol{\\beta}$ as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "where the pseudoinverse of $\\boldsymbol{X}$ is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Using singular value decomposition we can decompose the matrix  $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n",
+    "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n",
+    "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n",
+    "$\\omega$ to"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto16\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    \\boldsymbol{\\beta} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^{+} \\boldsymbol{U}^T \\boldsymbol{y}.\n",
+    "\\label{_auto16} \\tag{26}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note that solving this equation by actually doing the pseudoinverse\n",
+    "(which is what we will do) is not a good idea as this operation scales\n",
+    "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n",
+    "general matrix. Instead, doing $QR$-factorization and solving the\n",
+    "linear system as an equation would reduce this down to\n",
+    "$\\mathcal{O}(n^2)$ operations."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
+    "    u, s, v = scl.svd(x)\n",
+    "    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "beta = ols_svd(X_train_own,y_train)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "When extracting the $J$-matrix  we need to make sure that we remove the intercept, as is done here"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "J = beta[1:].reshape(L, L)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "A way of looking at the coefficients in $J$ is to plot the matrices as images."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "fig = plt.figure(figsize=(20, 14))\n",
+    "im = plt.imshow(J, **cmap_args)\n",
+    "plt.title(\"OLS\", fontsize=18)\n",
+    "plt.xticks(fontsize=18)\n",
+    "plt.yticks(fontsize=18)\n",
+    "cb = fig.colorbar(im)\n",
+    "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "It is interesting to note that OLS\n",
+    "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n",
+    "valid matrix elements for $J$.\n",
+    "In our discussion below on hyperparameters and Ridge and Lasso regression we will see that\n",
+    "this problem can be removed, partly and only with Lasso regression. \n",
+    "\n",
+    "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
     "## The one-dimensional Ising model\n",
     "\n",
     "Let us bring back the Ising model again, but now with an additional\n",
diff --git a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz
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diff --git a/doc/src/Regression/Regression.do.txt b/doc/src/Regression/Regression.do.txt
index 889e62561..1b261cd06 100644
--- a/doc/src/Regression/Regression.do.txt
+++ b/doc/src/Regression/Regression.do.txt
@@ -1130,7 +1130,7 @@ invert the matrix $\bm{X}^T\bm{X}$. Using a polynomial expansion as we
 did both for the masses and the fitting of the equation of state,
 leads to row vectors of the design matrix which are essentially
 orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. 
-More material to come here. 
+
 
 
 This may
@@ -1147,218 +1147,6 @@ inversion algorithm. Thereafter we dive into the math of the SVD.
 !eblock
 
 
-# todo: change model here.
-
-!split
-=====  The Ising model =====
-
-The one-dimensional Ising model with nearest neighbor interaction, no
-external field and a constant coupling constant $J$ is given by
-
-!bt
-\begin{align}
-    H = -J \sum_{k}^L s_k s_{k + 1},
-\end{align}
-!et
-
-where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins
-in the system is determined by $L$. For the one-dimensional system
-there is no phase transition.
-
-We will look at a system of $L = 40$ spins with a coupling constant of
-$J = 1$. To get enough training data we will generate 10000 states
-with their respective energies.
-
-
-!bc pycod
-import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.axes_grid1 import make_axes_locatable
-import seaborn as sns
-import scipy.linalg as scl
-from sklearn.model_selection import train_test_split
-import tqdm
-sns.set(color_codes=True)
-cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
-
-L = 40
-n = int(1e4)
-
-spins = np.random.choice([-1, 1], size=(n, L))
-J = 1.0
-
-energies = np.zeros(n)
-
-for i in range(n):
-    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
-!ec
-
-Here we use ordinary least squares
-regression to predict the energy for the nearest neighbor
-one-dimensional Ising model on a ring, i.e., the endpoints wrap
-around. We will use linear regression to fit a value for
-the coupling constant to achieve this.
-
-!split
-===== Reformulating the problem to suit regression =====
-
-A more general form for the one-dimensional Ising model is
-
-!bt
-\begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\end{align}
-!et
-
-Here we allow for interactions beyond the nearest neighbors and a state dependent
-coupling constant. This latter expression can be formulated as
-a matrix-product
-!bt
-\begin{align}
-    \bm{H} = \bm{X} J,
-\end{align}
-!et
-
-where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the
-elements $-J_{jk}$. This form of writing the energy fits perfectly
-with the form utilized in linear regression, that is
-
-!bt
-\begin{align}
-    \bm{y} = \bm{X}\bm{\beta} + \bm{\epsilon},
-\end{align}
-!et
-
-We split the data in training and test data as discussed in the previous example
-
-!bc pycod
-X = np.zeros((n, L ** 2))
-for i in range(n):
-    X[i] = np.outer(spins[i], spins[i]).ravel()
-y = energies
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-!ec
-
-!split
-===== Linear regression =====
-
-In the ordinary least squares method we choose the cost function
-
-!bt
-\begin{align}
-    C(\bm{X}, \bm{\beta})= \frac{1}{n}\left\{(\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y})\right\}.
-\end{align}
-!et
-
-We then find the extremal point of $C$ by taking the derivative with respect to $\bm{\beta}$ as discussed above.
-This yields the expression for $\bm{\beta}$ to be
-
-!bt
-\[
-    \bm{\beta} = \frac{\bm{X}^T \bm{y}}{\bm{X}^T \bm{X}},
-\]
-!et
-
-which immediately imposes some requirements on $\bm{X}$ as there must exist
-an inverse of $\bm{X}^T \bm{X}$. If the expression we are modeling contains an
-intercept, i.e., a constant term, we must make sure that the
-first column of $\bm{X}$ consists of $1$. We do this here
-
-!bc pycod
-X_train_own = np.concatenate(
-    (np.ones(len(X_train))[:, np.newaxis], X_train),
-    axis=1
-)
-X_test_own = np.concatenate(
-    (np.ones(len(X_test))[:, np.newaxis], X_test),
-    axis=1
-)
-!ec
-
-!bc pycod
-def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
-    return scl.inv(x.T @ x) @ (x.T @ y)
-beta = ols_inv(X_train_own, y_train)
-!ec
-
-
-!split
-===== Singular Value decomposition =====
-
-Doing the inversion directly turns out to be a bad idea since the matrix
-$\bm{X}^T\bm{X}$ is singular. An alternative approach is to use the _singular
-value decomposition_. Using the definition of the Moore-Penrose
-pseudoinverse we can write the equation for $\bm{\beta}$ as
-
-!bt
-\[
-    \bm{\beta} = \bm{X}^{+}\bm{y},
-\]
-!et
-
-where the pseudoinverse of $\bm{X}$ is given by
-
-!bt
-\[
-    \bm{X}^{+} = \frac{\bm{X}^T}{\bm{X}^T\bm{X}}.
-\]
-!et
-
-Using singular value decomposition we can decompose the matrix  $\bm{X} = \bm{U}\bm{\Sigma} \bm{V}^T$,
-where $\bm{U}$ and $\bm{V}$ are orthogonal(unitary) matrices and $\bm{\Sigma}$ contains the singular values (more details below).
-where $X^{+} = V\Sigma^{+} U^T$. This reduces the equation for
-$\omega$ to
-!bt
-\begin{align}
-    \bm{\beta} = \bm{V}\bm{\Sigma}^{+} \bm{U}^T \bm{y}.
-\end{align}
-!et
-
-Note that solving this equation by actually doing the pseudoinverse
-(which is what we will do) is not a good idea as this operation scales
-as $\mathcal{O}(n^3)$, where $n$ is the number of elements in a
-general matrix. Instead, doing $QR$-factorization and solving the
-linear system as an equation would reduce this down to
-$\mathcal{O}(n^2)$ operations.
-
-
-!bc pycod
-def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
-    u, s, v = scl.svd(x)
-    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
-!ec
-
-!bc pycod
-beta = ols_svd(X_train_own,y_train)
-!ec
-
-When extracting the $J$-matrix  we need to make sure that we remove the intercept, as is done here
-
-!bc pycod
-J = beta[1:].reshape(L, L)
-!ec
-
-A way of looking at the coefficients in $J$ is to plot the matrices as images.
-
-
-!bc pycod
-fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J, **cmap_args)
-plt.title("OLS", fontsize=18)
-plt.xticks(fontsize=18)
-plt.yticks(fontsize=18)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
-plt.show()
-!ec
-It is interesting to note that OLS
-considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as
-valid matrix elements for $J$.
-In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
-this problem can be removed, partly and only with Lasso regression. 
-
-In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
-
 
 !split
 ===== Linear Regression Problems =====
@@ -1391,7 +1179,7 @@ column vectors. Hence, the rank of $\mathbf{X}$ is equal to the number
 of linearly independent columns. In this particular case the matrix has rank 2.
 
 Super-collinearity of an $(n \times p)$-dimensional design matrix $\mathbf{X}$ implies
-that the inverse of the matrix $\bm{X}^T\bm{x}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
+that the inverse of the matrix $\bm{X}^T\bm{X}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
 !bt
 \begin{align*}
 \bm{X} & =  \left[
@@ -1527,7 +1315,7 @@ In the general case, where our design matrix $\bm{X}$ has dimension
 $n\times p$, the matrix is thus decomposed into an $n\times n$
 orthogonal matrix $\bm{U}$, a $p\times p$ orthogonal matrix $\bm{V}$
 and a diagonal matrix $\bm{\Sigma}$ with $r=\mathrm{min}(n,p)$
-singular values $\sigma_i\lg 0$ on the main diagonal and zeros filling
+singular values $\sigma_i\geq 0$ on the main diagonal and zeros filling
 the rest of the matrix.  There are at most $p$ singular values
 assuming that $n > p$. In our regression examples for the nuclear
 masses and the equation of state this is indeed the case, while for
@@ -3388,6 +3176,220 @@ plt.show()
 
 
 
+!split
+=====  The Ising model =====
+
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant $J$ is given by
+
+!bt
+\begin{align}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\end{align}
+!et
+
+where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins
+in the system is determined by $L$. For the one-dimensional system
+there is no phase transition.
+
+We will look at a system of $L = 40$ spins with a coupling constant of
+$J = 1$. To get enough training data we will generate 10000 states
+with their respective energies.
+
+
+!bc pycod
+import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+!ec
+
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
+!split
+===== Reformulating the problem to suit regression =====
+
+A more general form for the one-dimensional Ising model is
+
+!bt
+\begin{align}
+    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\end{align}
+!et
+
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+!bt
+\begin{align}
+    \bm{H} = \bm{X} J,
+\end{align}
+!et
+
+where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the
+elements $-J_{jk}$. This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+!bt
+\begin{align}
+    \bm{y} = \bm{X}\bm{\beta} + \bm{\epsilon},
+\end{align}
+!et
+
+We split the data in training and test data as discussed in the previous example
+
+!bc pycod
+X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+!ec
+
+!split
+===== Linear regression =====
+
+In the ordinary least squares method we choose the cost function
+
+!bt
+\begin{align}
+    C(\bm{X}, \bm{\beta})= \frac{1}{n}\left\{(\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y})\right\}.
+\end{align}
+!et
+
+We then find the extremal point of $C$ by taking the derivative with respect to $\bm{\beta}$ as discussed above.
+This yields the expression for $\bm{\beta}$ to be
+
+!bt
+\[
+    \bm{\beta} = \frac{\bm{X}^T \bm{y}}{\bm{X}^T \bm{X}},
+\]
+!et
+
+which immediately imposes some requirements on $\bm{X}$ as there must exist
+an inverse of $\bm{X}^T \bm{X}$. If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of $\bm{X}$ consists of $1$. We do this here
+
+!bc pycod
+X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+!ec
+
+!bc pycod
+def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    return scl.inv(x.T @ x) @ (x.T @ y)
+beta = ols_inv(X_train_own, y_train)
+!ec
+
+
+!split
+===== Singular Value decomposition =====
+
+Doing the inversion directly turns out to be a bad idea since the matrix
+$\bm{X}^T\bm{X}$ is singular. An alternative approach is to use the _singular
+value decomposition_. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for $\bm{\beta}$ as
+
+!bt
+\[
+    \bm{\beta} = \bm{X}^{+}\bm{y},
+\]
+!et
+
+where the pseudoinverse of $\bm{X}$ is given by
+
+!bt
+\[
+    \bm{X}^{+} = \frac{\bm{X}^T}{\bm{X}^T\bm{X}}.
+\]
+!et
+
+Using singular value decomposition we can decompose the matrix  $\bm{X} = \bm{U}\bm{\Sigma} \bm{V}^T$,
+where $\bm{U}$ and $\bm{V}$ are orthogonal(unitary) matrices and $\bm{\Sigma}$ contains the singular values (more details below).
+where $X^{+} = V\Sigma^{+} U^T$. This reduces the equation for
+$\omega$ to
+!bt
+\begin{align}
+    \bm{\beta} = \bm{V}\bm{\Sigma}^{+} \bm{U}^T \bm{y}.
+\end{align}
+!et
+
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as $\mathcal{O}(n^3)$, where $n$ is the number of elements in a
+general matrix. Instead, doing $QR$-factorization and solving the
+linear system as an equation would reduce this down to
+$\mathcal{O}(n^2)$ operations.
+
+
+!bc pycod
+def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    u, s, v = scl.svd(x)
+    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+!ec
+
+!bc pycod
+beta = ols_svd(X_train_own,y_train)
+!ec
+
+When extracting the $J$-matrix  we need to make sure that we remove the intercept, as is done here
+
+!bc pycod
+J = beta[1:].reshape(L, L)
+!ec
+
+A way of looking at the coefficients in $J$ is to plot the matrices as images.
+
+
+!bc pycod
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J, **cmap_args)
+plt.title("OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+!ec
+It is interesting to note that OLS
+considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as
+valid matrix elements for $J$.
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression. 
+
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
+
+
+
+
 !split
 ===== The one-dimensional Ising model =====