From b7a0979daa6efc914c15d7626d791b2c965235ec Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 12 Sep 2019 13:48:21 +0200 Subject: [PATCH] added CV codes to regression --- .../Regression/html/._Regression-bs000.html | 52 +- .../Regression/html/._Regression-bs001.html | 50 +- .../Regression/html/._Regression-bs002.html | 50 +- .../Regression/html/._Regression-bs003.html | 50 +- .../Regression/html/._Regression-bs004.html | 50 +- .../Regression/html/._Regression-bs005.html | 50 +- .../Regression/html/._Regression-bs006.html | 50 +- .../Regression/html/._Regression-bs007.html | 50 +- .../Regression/html/._Regression-bs008.html | 50 +- .../Regression/html/._Regression-bs009.html | 50 +- .../Regression/html/._Regression-bs010.html | 50 +- .../Regression/html/._Regression-bs011.html | 50 +- .../Regression/html/._Regression-bs012.html | 50 +- .../Regression/html/._Regression-bs013.html | 50 +- .../Regression/html/._Regression-bs014.html | 50 +- .../Regression/html/._Regression-bs015.html | 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+-- doc/pub/Regression/html/Regression-bs.html | 52 +- .../Regression/html/Regression-reveal.html | 229 +- .../Regression/html/Regression-solarized.html | 253 +- doc/pub/Regression/html/Regression.html | 253 +- doc/pub/Regression/ipynb/Regression.ipynb | 268 +- .../ipynb/ipynb-Regression-src.tar.gz | Bin 212 -> 211 bytes doc/pub/Regression/pdf/Regression-minted.pdf | Bin 454773 -> 454910 bytes doc/src/Regression/Regression.do.txt | 210 + .../Results/FigureFiles/EoSfitting.png | Bin 0 -> 32264 bytes doc/src/Regression/datafiles/EoS.csv | 90 + doc/src/Regression/datafiles/MassEval2016.dat | 3475 +++++++++++++++++ doc/src/Regression/fit.py | 84 + doc/src/Regression/fit2.py | 68 + doc/src/Regression/test.py | 44 + doc/src/Regression/test2.py | 36 + 126 files changed, 9696 insertions(+), 3461 deletions(-) create mode 100644 doc/src/Regression/Results/FigureFiles/EoSfitting.png create mode 100644 doc/src/Regression/datafiles/EoS.csv create mode 100644 doc/src/Regression/datafiles/MassEval2016.dat create mode 100644 doc/src/Regression/fit.py create mode 100644 doc/src/Regression/fit2.py create mode 100644 doc/src/Regression/test.py create mode 100644 doc/src/Regression/test2.py diff --git a/doc/pub/Regression/html/._Regression-bs000.html b/doc/pub/Regression/html/._Regression-bs000.html index 9fe8c4983..6bf1b9e0f 100644 --- a/doc/pub/Regression/html/._Regression-bs000.html +++ b/doc/pub/Regression/html/._Regression-bs000.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -413,7 +425,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 10, 2019

    +

    Sep 12, 2019


    @@ -437,7 +449,7 @@ MathJax.Hub.Config({

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    • diff --git a/doc/pub/Regression/html/._Regression-bs001.html b/doc/pub/Regression/html/._Regression-bs001.html index 2c575ea7a..3f91277c2 100644 --- a/doc/pub/Regression/html/._Regression-bs001.html +++ b/doc/pub/Regression/html/._Regression-bs001.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -431,7 +443,7 @@ Similarly, Mehta et al
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -435,7 +447,7 @@ A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\ver
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    • diff --git a/doc/pub/Regression/html/._Regression-bs003.html b/doc/pub/Regression/html/._Regression-bs003.html index fafe4bb92..4031edf18 100644 --- a/doc/pub/Regression/html/._Regression-bs003.html +++ b/doc/pub/Regression/html/._Regression-bs003.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -444,7 +456,7 @@ Linear regression gives us a set of analytical equations for the parameters \( \
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    • diff --git a/doc/pub/Regression/html/._Regression-bs004.html b/doc/pub/Regression/html/._Regression-bs004.html index 1b4ac2f54..94fe8f7e5 100644 --- a/doc/pub/Regression/html/._Regression-bs004.html +++ b/doc/pub/Regression/html/._Regression-bs004.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -441,7 +453,7 @@ so-called 13
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    • diff --git a/doc/pub/Regression/html/._Regression-bs005.html b/doc/pub/Regression/html/._Regression-bs005.html index de68a450f..4d086afb3 100644 --- a/doc/pub/Regression/html/._Regression-bs005.html +++ b/doc/pub/Regression/html/._Regression-bs005.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +446,7 @@ where \( \epsilon_i \) is the error in our approximation.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs006.html b/doc/pub/Regression/html/._Regression-bs006.html index 3678baaec..930d11d42 100644 --- a/doc/pub/Regression/html/._Regression-bs006.html +++ b/doc/pub/Regression/html/._Regression-bs006.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +446,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs007.html b/doc/pub/Regression/html/._Regression-bs007.html index ee0df9c40..5ecbc229b 100644 --- a/doc/pub/Regression/html/._Regression-bs007.html +++ b/doc/pub/Regression/html/._Regression-bs007.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -458,7 +470,7 @@ The above design matrix is called a 16
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    • diff --git a/doc/pub/Regression/html/._Regression-bs008.html b/doc/pub/Regression/html/._Regression-bs008.html index d00966166..1eb7a8ca4 100644 --- a/doc/pub/Regression/html/._Regression-bs008.html +++ b/doc/pub/Regression/html/._Regression-bs008.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -448,7 +460,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs009.html b/doc/pub/Regression/html/._Regression-bs009.html index db983e26f..4ab7f31ca 100644 --- a/doc/pub/Regression/html/._Regression-bs009.html +++ b/doc/pub/Regression/html/._Regression-bs009.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -445,7 +457,7 @@ The left-hand side of this equation is kwown. Our error vector \( \boldsymbol{\e
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    • diff --git a/doc/pub/Regression/html/._Regression-bs010.html b/doc/pub/Regression/html/._Regression-bs010.html index ac5dfd3d8..374d73b70 100644 --- a/doc/pub/Regression/html/._Regression-bs010.html +++ b/doc/pub/Regression/html/._Regression-bs010.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -447,7 +459,7 @@ our matrix as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predict
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    • diff --git a/doc/pub/Regression/html/._Regression-bs011.html b/doc/pub/Regression/html/._Regression-bs011.html index f386f292c..e63f3d6f2 100644 --- a/doc/pub/Regression/html/._Regression-bs011.html +++ b/doc/pub/Regression/html/._Regression-bs011.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -508,7 +520,7 @@ throughout these lectures.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs012.html b/doc/pub/Regression/html/._Regression-bs012.html index 37e71376d..eb797f9fa 100644 --- a/doc/pub/Regression/html/._Regression-bs012.html +++ b/doc/pub/Regression/html/._Regression-bs012.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -454,7 +466,7 @@ since when taking the first derivative with respect to the unknown parameters \(
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    • diff --git a/doc/pub/Regression/html/._Regression-bs013.html b/doc/pub/Regression/html/._Regression-bs013.html index 75993b0d0..b36988c5c 100644 --- a/doc/pub/Regression/html/._Regression-bs013.html +++ b/doc/pub/Regression/html/._Regression-bs013.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -474,7 +486,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs014.html b/doc/pub/Regression/html/._Regression-bs014.html index 7b0aa46d0..f869eebf5 100644 --- a/doc/pub/Regression/html/._Regression-bs014.html +++ b/doc/pub/Regression/html/._Regression-bs014.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -463,7 +475,7 @@ allow for the usage of direct linear algebra methods such as LU decomposi
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -441,7 +453,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs016.html b/doc/pub/Regression/html/._Regression-bs016.html index 7140e9f7e..a952e9302 100644 --- a/doc/pub/Regression/html/._Regression-bs016.html +++ b/doc/pub/Regression/html/._Regression-bs016.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -449,7 +461,7 @@ Let us now return to our nuclear binding energies and simply code the above equa
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    • diff --git a/doc/pub/Regression/html/._Regression-bs017.html b/doc/pub/Regression/html/._Regression-bs017.html index c9132fa3b..50cca8770 100644 --- a/doc/pub/Regression/html/._Regression-bs017.html +++ b/doc/pub/Regression/html/._Regression-bs017.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -459,7 +471,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs018.html b/doc/pub/Regression/html/._Regression-bs018.html index 4d4decc1c..96b9210ef 100644 --- a/doc/pub/Regression/html/._Regression-bs018.html +++ b/doc/pub/Regression/html/._Regression-bs018.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -458,7 +470,7 @@ and finally the relative error as
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    • diff --git a/doc/pub/Regression/html/._Regression-bs019.html b/doc/pub/Regression/html/._Regression-bs019.html index 88e8340d9..158dc1eac 100644 --- a/doc/pub/Regression/html/._Regression-bs019.html +++ b/doc/pub/Regression/html/._Regression-bs019.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -450,7 +462,7 @@ where the matrix \( \boldsymbol{\Sigma} \) is a diagonal matrix with \( \sigma_i
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    • diff --git a/doc/pub/Regression/html/._Regression-bs020.html b/doc/pub/Regression/html/._Regression-bs020.html index bf5817807..72b6a7cb6 100644 --- a/doc/pub/Regression/html/._Regression-bs020.html +++ b/doc/pub/Regression/html/._Regression-bs020.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -446,7 +458,7 @@ where we have defined the matrix \( \boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\
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    • diff --git a/doc/pub/Regression/html/._Regression-bs021.html b/doc/pub/Regression/html/._Regression-bs021.html index f7c538ed0..7928b81f7 100644 --- a/doc/pub/Regression/html/._Regression-bs021.html +++ b/doc/pub/Regression/html/._Regression-bs021.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -444,7 +456,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs022.html b/doc/pub/Regression/html/._Regression-bs022.html index 68ea39495..6733641a9 100644 --- a/doc/pub/Regression/html/._Regression-bs022.html +++ b/doc/pub/Regression/html/._Regression-bs022.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -449,7 +461,7 @@ $$
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -442,7 +454,7 @@ $$
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -476,7 +488,7 @@ Lasso and Ridge regression. See below.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs025.html b/doc/pub/Regression/html/._Regression-bs025.html index d9ba7796f..45a904e7e 100644 --- a/doc/pub/Regression/html/._Regression-bs025.html +++ b/doc/pub/Regression/html/._Regression-bs025.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -442,7 +454,7 @@ hyperparameter \( \lambda \), also to be explained below.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs026.html b/doc/pub/Regression/html/._Regression-bs026.html index 0e9c7cc4c..4119ed779 100644 --- a/doc/pub/Regression/html/._Regression-bs026.html +++ b/doc/pub/Regression/html/._Regression-bs026.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -520,7 +532,7 @@ below.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs027.html b/doc/pub/Regression/html/._Regression-bs027.html index 7ccfeb294..50a6d9610 100644 --- a/doc/pub/Regression/html/._Regression-bs027.html +++ b/doc/pub/Regression/html/._Regression-bs027.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -502,7 +514,7 @@ ypredict = X_test @ beta
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    • diff --git a/doc/pub/Regression/html/._Regression-bs028.html b/doc/pub/Regression/html/._Regression-bs028.html index 6eb87d03f..ead62cfa8 100644 --- a/doc/pub/Regression/html/._Regression-bs028.html +++ b/doc/pub/Regression/html/._Regression-bs028.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -448,7 +460,7 @@ The features/predictors are
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    • diff --git a/doc/pub/Regression/html/._Regression-bs029.html b/doc/pub/Regression/html/._Regression-bs029.html index 6f39fdd78..2466827d1 100644 --- a/doc/pub/Regression/html/._Regression-bs029.html +++ b/doc/pub/Regression/html/._Regression-bs029.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -574,7 +586,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs030.html b/doc/pub/Regression/html/._Regression-bs030.html index 751e0fd61..2f2661a7a 100644 --- a/doc/pub/Regression/html/._Regression-bs030.html +++ b/doc/pub/Regression/html/._Regression-bs030.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -453,7 +465,7 @@ inversion algorithm. Thereafter we dive into the math of the SVD.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs031.html b/doc/pub/Regression/html/._Regression-bs031.html index 6f83f9e48..303ff48de 100644 --- a/doc/pub/Regression/html/._Regression-bs031.html +++ b/doc/pub/Regression/html/._Regression-bs031.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -468,7 +480,7 @@ This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least a
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    • diff --git a/doc/pub/Regression/html/._Regression-bs032.html b/doc/pub/Regression/html/._Regression-bs032.html index c5aa1000e..b12f75c5d 100644 --- a/doc/pub/Regression/html/._Regression-bs032.html +++ b/doc/pub/Regression/html/._Regression-bs032.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -445,7 +457,7 @@ where \( \boldsymbol{I} \) is the identity matrix. When we discuss Ridge
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -456,7 +468,7 @@ is not diagonalizable, it is a so-called 42
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    • diff --git a/doc/pub/Regression/html/._Regression-bs034.html b/doc/pub/Regression/html/._Regression-bs034.html index 25177fa44..951dc89a5 100644 --- a/doc/pub/Regression/html/._Regression-bs034.html +++ b/doc/pub/Regression/html/._Regression-bs034.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -448,7 +460,7 @@ The SVD exits always!
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    • diff --git a/doc/pub/Regression/html/._Regression-bs035.html b/doc/pub/Regression/html/._Regression-bs035.html index a66cf23e5..1a2008c66 100644 --- a/doc/pub/Regression/html/._Regression-bs035.html +++ b/doc/pub/Regression/html/._Regression-bs035.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -455,7 +467,7 @@ The columns of \( \boldsymbol{U} \) are called the left singular vectors while t
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    • diff --git a/doc/pub/Regression/html/._Regression-bs036.html b/doc/pub/Regression/html/._Regression-bs036.html index 2bbbda526..0c25985ff 100644 --- a/doc/pub/Regression/html/._Regression-bs036.html +++ b/doc/pub/Regression/html/._Regression-bs036.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -442,7 +454,7 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
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    • diff --git a/doc/pub/Regression/html/._Regression-bs037.html b/doc/pub/Regression/html/._Regression-bs037.html index 5b2c71dc3..238ff8127 100644 --- a/doc/pub/Regression/html/._Regression-bs037.html +++ b/doc/pub/Regression/html/._Regression-bs037.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -469,7 +481,7 @@ We will come back to this expression when we discuss Ridge regression.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs038.html b/doc/pub/Regression/html/._Regression-bs038.html index 5dc54dd5e..ec616a156 100644 --- a/doc/pub/Regression/html/._Regression-bs038.html +++ b/doc/pub/Regression/html/._Regression-bs038.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -475,7 +487,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs039.html b/doc/pub/Regression/html/._Regression-bs039.html index 9ffa52557..9c8b3ed84 100644 --- a/doc/pub/Regression/html/._Regression-bs039.html +++ b/doc/pub/Regression/html/._Regression-bs039.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -476,7 +488,7 @@ with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -441,7 +453,7 @@ With a parameter \( \lambda \) we can thus shrink the role of specific parameter
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -454,7 +466,7 @@ Similarly, Mehta et al
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -468,7 +480,7 @@ in the program terminating due to a singular matrix.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs043.html b/doc/pub/Regression/html/._Regression-bs043.html index 66a0ba257..7cf2f531c 100644 --- a/doc/pub/Regression/html/._Regression-bs043.html +++ b/doc/pub/Regression/html/._Regression-bs043.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -436,7 +448,7 @@ affected by changing the parameter \( \lambda \).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs044.html b/doc/pub/Regression/html/._Regression-bs044.html index 1c7d64d63..94d7157dc 100644 --- a/doc/pub/Regression/html/._Regression-bs044.html +++ b/doc/pub/Regression/html/._Regression-bs044.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -437,7 +449,7 @@ The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonorm
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    • diff --git a/doc/pub/Regression/html/._Regression-bs045.html b/doc/pub/Regression/html/._Regression-bs045.html index 885220862..0e7821802 100644 --- a/doc/pub/Regression/html/._Regression-bs045.html +++ b/doc/pub/Regression/html/._Regression-bs045.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +437,7 @@ More material to be added here
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    • diff --git a/doc/pub/Regression/html/._Regression-bs046.html b/doc/pub/Regression/html/._Regression-bs046.html index 0bff44ccd..0b5c1184c 100644 --- a/doc/pub/Regression/html/._Regression-bs046.html +++ b/doc/pub/Regression/html/._Regression-bs046.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +446,7 @@ This will allow us to link the standard linear algebra methods we have discussed
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    • diff --git a/doc/pub/Regression/html/._Regression-bs047.html b/doc/pub/Regression/html/._Regression-bs047.html index d35e081c7..5d8a03d38 100644 --- a/doc/pub/Regression/html/._Regression-bs047.html +++ b/doc/pub/Regression/html/._Regression-bs047.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -451,7 +463,7 @@ cross-validation and the bootstrap method.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs048.html b/doc/pub/Regression/html/._Regression-bs048.html index 4ee13b750..a60bd6372 100644 --- a/doc/pub/Regression/html/._Regression-bs048.html +++ b/doc/pub/Regression/html/._Regression-bs048.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -447,7 +459,7 @@ bootstrap is widely used.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs049.html b/doc/pub/Regression/html/._Regression-bs049.html index 65d300b72..d0bce87ee 100644 --- a/doc/pub/Regression/html/._Regression-bs049.html +++ b/doc/pub/Regression/html/._Regression-bs049.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +446,7 @@ MathJax.Hub.Config({
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -440,7 +452,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/Regression/html/._Regression-bs051.html b/doc/pub/Regression/html/._Regression-bs051.html index 06ad547b9..714c82d94 100644 --- a/doc/pub/Regression/html/._Regression-bs051.html +++ b/doc/pub/Regression/html/._Regression-bs051.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -449,7 +461,7 @@ selection of a large set of these numbers reproduces this PDF.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs052.html b/doc/pub/Regression/html/._Regression-bs052.html index e43260763..a71ccb73d 100644 --- a/doc/pub/Regression/html/._Regression-bs052.html +++ b/doc/pub/Regression/html/._Regression-bs052.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -441,7 +453,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs053.html b/doc/pub/Regression/html/._Regression-bs053.html index cb7eb7f42..619de3250 100644 --- a/doc/pub/Regression/html/._Regression-bs053.html +++ b/doc/pub/Regression/html/._Regression-bs053.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -456,7 +468,7 @@ qualitatively as the spread of \( p \) around its mean.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs054.html b/doc/pub/Regression/html/._Regression-bs054.html index fa460268b..5d9059392 100644 --- a/doc/pub/Regression/html/._Regression-bs054.html +++ b/doc/pub/Regression/html/._Regression-bs054.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -449,7 +461,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs055.html b/doc/pub/Regression/html/._Regression-bs055.html index d21e51e28..41c9b66ca 100644 --- a/doc/pub/Regression/html/._Regression-bs055.html +++ b/doc/pub/Regression/html/._Regression-bs055.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -450,7 +462,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs056.html b/doc/pub/Regression/html/._Regression-bs056.html index d9350a266..ad52bf9a3 100644 --- a/doc/pub/Regression/html/._Regression-bs056.html +++ b/doc/pub/Regression/html/._Regression-bs056.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -466,7 +478,7 @@ function.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs057.html b/doc/pub/Regression/html/._Regression-bs057.html index 98b5b209a..c22a65952 100644 --- a/doc/pub/Regression/html/._Regression-bs057.html +++ b/doc/pub/Regression/html/._Regression-bs057.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -456,7 +468,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs058.html b/doc/pub/Regression/html/._Regression-bs058.html index cef636af6..0b17dc972 100644 --- a/doc/pub/Regression/html/._Regression-bs058.html +++ b/doc/pub/Regression/html/._Regression-bs058.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -444,7 +456,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs059.html b/doc/pub/Regression/html/._Regression-bs059.html index 0c9ab9257..c331c4a4a 100644 --- a/doc/pub/Regression/html/._Regression-bs059.html +++ b/doc/pub/Regression/html/._Regression-bs059.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -450,7 +462,7 @@ value of a set of measurements.
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -446,7 +458,7 @@ interested in finding the few lowest moments, like the mean
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    • diff --git a/doc/pub/Regression/html/._Regression-bs061.html b/doc/pub/Regression/html/._Regression-bs061.html index ecade430f..beed2c3fb 100644 --- a/doc/pub/Regression/html/._Regression-bs061.html +++ b/doc/pub/Regression/html/._Regression-bs061.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -444,7 +456,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs062.html b/doc/pub/Regression/html/._Regression-bs062.html index de8443db1..c8cd497ca 100644 --- a/doc/pub/Regression/html/._Regression-bs062.html +++ b/doc/pub/Regression/html/._Regression-bs062.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -439,7 +451,7 @@ and covariance \( \mathrm{cov}(X,Y) \).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs063.html b/doc/pub/Regression/html/._Regression-bs063.html index 81293886d..f1bb232d0 100644 --- a/doc/pub/Regression/html/._Regression-bs063.html +++ b/doc/pub/Regression/html/._Regression-bs063.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -450,7 +462,7 @@ true PDFs behind, which we usually do not have.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs064.html b/doc/pub/Regression/html/._Regression-bs064.html index 7005f6ffa..6ef16e921 100644 --- a/doc/pub/Regression/html/._Regression-bs064.html +++ b/doc/pub/Regression/html/._Regression-bs064.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -440,7 +452,7 @@ means.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs065.html b/doc/pub/Regression/html/._Regression-bs065.html index a660be363..b54fe490e 100644 --- a/doc/pub/Regression/html/._Regression-bs065.html +++ b/doc/pub/Regression/html/._Regression-bs065.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -439,7 +451,7 @@ And in particular we are interested in its variance \( \mathrm{var}(\overline X_
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    • diff --git a/doc/pub/Regression/html/._Regression-bs066.html b/doc/pub/Regression/html/._Regression-bs066.html index b16c98085..9a338f111 100644 --- a/doc/pub/Regression/html/._Regression-bs066.html +++ b/doc/pub/Regression/html/._Regression-bs066.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -443,7 +455,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs067.html b/doc/pub/Regression/html/._Regression-bs067.html index c7d0d9bbb..4849fc411 100644 --- a/doc/pub/Regression/html/._Regression-bs067.html +++ b/doc/pub/Regression/html/._Regression-bs067.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -447,7 +459,7 @@ estimate of the PDF of each of the \( X_i \), estimating all properties of
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    • diff --git a/doc/pub/Regression/html/._Regression-bs068.html b/doc/pub/Regression/html/._Regression-bs068.html index 50773c9f6..199e82495 100644 --- a/doc/pub/Regression/html/._Regression-bs068.html +++ b/doc/pub/Regression/html/._Regression-bs068.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -445,7 +457,7 @@ $$
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -457,7 +469,7 @@ measurements in the sample.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs070.html b/doc/pub/Regression/html/._Regression-bs070.html index afeb62b88..e87088bbc 100644 --- a/doc/pub/Regression/html/._Regression-bs070.html +++ b/doc/pub/Regression/html/._Regression-bs070.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -453,7 +465,7 @@ cannot overlook the always present correlations.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs071.html b/doc/pub/Regression/html/._Regression-bs071.html index 96a3e366a..2ec47498d 100644 --- a/doc/pub/Regression/html/._Regression-bs071.html +++ b/doc/pub/Regression/html/._Regression-bs071.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -447,7 +459,7 @@ measurements. For uncorrelated measurements this second term is zero.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs072.html b/doc/pub/Regression/html/._Regression-bs072.html index 436e07701..f7e2db3bf 100644 --- a/doc/pub/Regression/html/._Regression-bs072.html +++ b/doc/pub/Regression/html/._Regression-bs072.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -440,7 +452,7 @@ have to be stored throughout the experiment.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs073.html b/doc/pub/Regression/html/._Regression-bs073.html index d00dc8dea..b331217f7 100644 --- a/doc/pub/Regression/html/._Regression-bs073.html +++ b/doc/pub/Regression/html/._Regression-bs073.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -451,7 +463,7 @@ starting always at \( 1 \) for \( d=0 \).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs074.html b/doc/pub/Regression/html/._Regression-bs074.html index f0c17524f..36a9f2393 100644 --- a/doc/pub/Regression/html/._Regression-bs074.html +++ b/doc/pub/Regression/html/._Regression-bs074.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -451,7 +463,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs075.html b/doc/pub/Regression/html/._Regression-bs075.html index 15731ed46..8fe169408 100644 --- a/doc/pub/Regression/html/._Regression-bs075.html +++ b/doc/pub/Regression/html/._Regression-bs075.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -443,7 +455,7 @@ measurements is very large.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs076.html b/doc/pub/Regression/html/._Regression-bs076.html index fece5949b..6ac324e55 100644 --- a/doc/pub/Regression/html/._Regression-bs076.html +++ b/doc/pub/Regression/html/._Regression-bs076.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -455,7 +467,7 @@ row number \( i \) and perform a sum over all values \( p \).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs077.html b/doc/pub/Regression/html/._Regression-bs077.html index cf22c9dea..57d7c5898 100644 --- a/doc/pub/Regression/html/._Regression-bs077.html +++ b/doc/pub/Regression/html/._Regression-bs077.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -437,7 +449,7 @@ $$
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -452,7 +464,7 @@ mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (n
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  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -502,7 +514,7 @@ This means the variance we obtain with the standard OLS will always for \( \lamb
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    • diff --git a/doc/pub/Regression/html/._Regression-bs080.html b/doc/pub/Regression/html/._Regression-bs080.html index 8d58a45c2..12a1b485a 100644 --- a/doc/pub/Regression/html/._Regression-bs080.html +++ b/doc/pub/Regression/html/._Regression-bs080.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -446,7 +458,7 @@ training error reaches a saturation.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs081.html b/doc/pub/Regression/html/._Regression-bs081.html index 624b31fbe..d22009600 100644 --- a/doc/pub/Regression/html/._Regression-bs081.html +++ b/doc/pub/Regression/html/._Regression-bs081.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -440,7 +452,7 @@ need for bootstrapping.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs082.html b/doc/pub/Regression/html/._Regression-bs082.html index f8bac47ca..04a273b2a 100644 --- a/doc/pub/Regression/html/._Regression-bs082.html +++ b/doc/pub/Regression/html/._Regression-bs082.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -436,7 +448,7 @@ number \( i \) is left out. Using this notation, define
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    • diff --git a/doc/pub/Regression/html/._Regression-bs083.html b/doc/pub/Regression/html/._Regression-bs083.html index 3782b716d..63f85eb49 100644 --- a/doc/pub/Regression/html/._Regression-bs083.html +++ b/doc/pub/Regression/html/._Regression-bs083.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -453,7 +465,7 @@ t = jackknife(x, stat)
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    • diff --git a/doc/pub/Regression/html/._Regression-bs084.html b/doc/pub/Regression/html/._Regression-bs084.html index 13cea1646..caa7b866f 100644 --- a/doc/pub/Regression/html/._Regression-bs084.html +++ b/doc/pub/Regression/html/._Regression-bs084.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -439,7 +451,7 @@ advantages:
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    • diff --git a/doc/pub/Regression/html/._Regression-bs085.html b/doc/pub/Regression/html/._Regression-bs085.html index 8d5308837..64f2c0fd8 100644 --- a/doc/pub/Regression/html/._Regression-bs085.html +++ b/doc/pub/Regression/html/._Regression-bs085.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -433,7 +445,7 @@ estimators.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs086.html b/doc/pub/Regression/html/._Regression-bs086.html index e6cad178b..1516ce771 100644 --- a/doc/pub/Regression/html/._Regression-bs086.html +++ b/doc/pub/Regression/html/._Regression-bs086.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -439,7 +451,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
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    • diff --git a/doc/pub/Regression/html/._Regression-bs087.html b/doc/pub/Regression/html/._Regression-bs087.html index 15a8f15c6..114da782f 100644 --- a/doc/pub/Regression/html/._Regression-bs087.html +++ b/doc/pub/Regression/html/._Regression-bs087.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -438,7 +450,7 @@ frequency of the observation \( X_i \), just draw the values
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    • diff --git a/doc/pub/Regression/html/._Regression-bs088.html b/doc/pub/Regression/html/._Regression-bs088.html index 7bdb8afce..beb27c8a2 100644 --- a/doc/pub/Regression/html/._Regression-bs088.html +++ b/doc/pub/Regression/html/._Regression-bs088.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -442,7 +454,7 @@ example, if you are interested in estimating the variance of \( \widehat
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    • diff --git a/doc/pub/Regression/html/._Regression-bs089.html b/doc/pub/Regression/html/._Regression-bs089.html index d0dc58ecd..ba7de4d0d 100644 --- a/doc/pub/Regression/html/._Regression-bs089.html +++ b/doc/pub/Regression/html/._Regression-bs089.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -481,7 +493,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs090.html b/doc/pub/Regression/html/._Regression-bs090.html index abd310b3e..2767eb454 100644 --- a/doc/pub/Regression/html/._Regression-bs090.html +++ b/doc/pub/Regression/html/._Regression-bs090.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -422,7 +434,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/Regression/html/._Regression-bs091.html b/doc/pub/Regression/html/._Regression-bs091.html index b64a70b41..82ac178e7 100644 --- a/doc/pub/Regression/html/._Regression-bs091.html +++ b/doc/pub/Regression/html/._Regression-bs091.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -438,7 +450,7 @@ cross-validation (LOOCV).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs092.html b/doc/pub/Regression/html/._Regression-bs092.html index beb16e141..a1037f491 100644 --- a/doc/pub/Regression/html/._Regression-bs092.html +++ b/doc/pub/Regression/html/._Regression-bs092.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -453,7 +465,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs093.html b/doc/pub/Regression/html/._Regression-bs093.html index 0fcb249b8..e5ba761c9 100644 --- a/doc/pub/Regression/html/._Regression-bs093.html +++ b/doc/pub/Regression/html/._Regression-bs093.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -517,7 +529,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs094.html b/doc/pub/Regression/html/._Regression-bs094.html index 17c98089b..df7064149 100644 --- a/doc/pub/Regression/html/._Regression-bs094.html +++ b/doc/pub/Regression/html/._Regression-bs094.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -483,7 +495,7 @@ that is the rewriting in terms of the so-called bias, the variance of the model
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    • diff --git a/doc/pub/Regression/html/._Regression-bs095.html b/doc/pub/Regression/html/._Regression-bs095.html index e4a6beb31..f4861caa8 100644 --- a/doc/pub/Regression/html/._Regression-bs095.html +++ b/doc/pub/Regression/html/._Regression-bs095.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -479,7 +491,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs096.html b/doc/pub/Regression/html/._Regression-bs096.html index ccd443435..86441f877 100644 --- a/doc/pub/Regression/html/._Regression-bs096.html +++ b/doc/pub/Regression/html/._Regression-bs096.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -471,7 +483,7 @@ plt.show()
  • 105
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    • -
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs097.html b/doc/pub/Regression/html/._Regression-bs097.html index 69b393ee8..74ddbcbd2 100644 --- a/doc/pub/Regression/html/._Regression-bs097.html +++ b/doc/pub/Regression/html/._Regression-bs097.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -451,7 +463,7 @@ flexible statistical methods have higher variance.
  • 106
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    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs098.html b/doc/pub/Regression/html/._Regression-bs098.html index b2116367c..7ada610c0 100644 --- a/doc/pub/Regression/html/._Regression-bs098.html +++ b/doc/pub/Regression/html/._Regression-bs098.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -495,6 +507,8 @@ plt.show()
  • 106
  • 107
  • 108
    • +
  • ...
    • +
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs099.html b/doc/pub/Regression/html/._Regression-bs099.html index e84c51328..3f639eada 100644 --- a/doc/pub/Regression/html/._Regression-bs099.html +++ b/doc/pub/Regression/html/._Regression-bs099.html @@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec97'), - ('The Ising model', 2, None, '___sec98'), - ('Reformulating the problem to suit regression', + ('More examples on bootstrap and cross-validation and errors', + 2, + None, + '___sec98'), + ('The same example but now with cross-validation', 2, None, '___sec99'), - ('Linear regression', 2, None, '___sec100'), - ('Singular Value decomposition', 2, None, '___sec101'), - ('The one-dimensional Ising model', 2, None, '___sec102'), - ('Ridge regression', 2, None, '___sec103'), - ('LASSO regression', 2, None, '___sec104'), + ('Cross-validation with Ridge', 2, None, '___sec100'), + ('The Ising model', 2, None, '___sec101'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec102'), + ('Linear regression', 2, None, '___sec103'), + ('Singular Value decomposition', 2, None, '___sec104'), + ('The one-dimensional Ising model', 2, None, '___sec105'), + ('Ridge regression', 2, None, '___sec106'), + ('LASSO regression', 2, None, '___sec107'), ('Performance as function of the regularization parameter', 2, None, - '___sec105'), + '___sec108'), ('Finding the optimal value of $\\lambda$', 2, None, - '___sec106')]} + '___sec109')]} end of tocinfo --> @@ -370,15 +379,18 @@ MathJax.Hub.Config({
  • Understanding what happens
  • Summing up
  • Another Example from Scikit-Learn's Repository
    • -
  • The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • +
  • Cross-validation with Ridge
    • +
  • The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -394,60 +406,90 @@ MathJax.Hub.Config({ -

    The Ising model

    - -

    -The one-dimensional Ising model with nearest neighbor interaction, no -external field and a constant coupling constant \( J \) is given by - -$$ -\begin{align} - H = -J \sum_{k}^L s_k s_{k + 1}, -\tag{21} -\end{align} -$$ - -

    -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. +

    More examples on bootstrap and cross-validation and errors

    -

    import numpy as np
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
 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.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.model_selection import train_test_split
-import tqdm
-sns.set(color_codes=True)
-cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
 
-L = 40
-n = int(1e4)
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
 
-spins = np.random.choice([-1, 1], size=(n, L))
-J = 1.0
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
 
-energies = np.zeros(n)
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
 
-for i in range(n):
-    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+testerror = np.zeros(Maxpolydegree)
+trainingerror = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+
+trials = 100
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+
+# loop over trials in order to estimate the expectation value of the MSE
+    testerror[polydegree] = 0.0
+    trainingerror[polydegree] = 0.0
+    for samples in range(trials):
+        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
+        ypred = model.predict(x_train)
+        ytilde = model.predict(x_test)
+        testerror[polydegree] += mean_squared_error(y_test, ytilde)
+        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
+
+    testerror[polydegree] /= trials
+    trainingerror[polydegree] /= trials
+    print("Degree of polynomial: %3d"% polynomial[polydegree])
+    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
+    print("Mean squared error on test data: %.8f" % testerror[polydegree])
+
+plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
+plt.plot(polynomial, np.log10(testerror), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
 
    
-
    
-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.
-
 
    
 
    
 
@@ -472,6 +514,9 @@ the coupling constant to achieve this.
   
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diff --git a/doc/pub/Regression/html/._Regression-bs100.html b/doc/pub/Regression/html/._Regression-bs100.html
index d1c3929b7..5522d119d 100644
--- a/doc/pub/Regression/html/._Regression-bs100.html
+++ b/doc/pub/Regression/html/._Regression-bs100.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -392,54 +404,80 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Reformulating the problem to suit regression 
    
-
-
    
-A more general form for the one-dimensional Ising model is
-
-$$
-\begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\tag{22}
-\end{align}
-$$
-
-
    
-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
-$$
-\begin{align}
-    \boldsymbol{H} = \boldsymbol{X} J,
-\tag{23}
-\end{align}
-$$
-
-
    
-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
-
-$$
-\begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
-\tag{24}
-\end{align}
-$$
-
-
    
-We split the data in training and test data as discussed in the previous example
+
    The same example but now with cross-validation 
    
 
 
    
 
 
-
    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)
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import KFold
+from sklearn.model_selection import cross_val_score
+
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+estimated_mse_sklearn = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+k =5
+kfold = KFold(n_splits = k)
+
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+        OLS = LinearRegression()
+# loop over trials in order to estimate the expectation value of the MSE
+    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
+#[:, np.newaxis]
+    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
+
+plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
 
    
 
    
 
    
@@ -464,6 +502,10 @@ X_train, X_test, y_train, y_test = train_tes
   
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diff --git a/doc/pub/Regression/html/._Regression-bs101.html b/doc/pub/Regression/html/._Regression-bs101.html
index 4d1f5638f..a95a1a320 100644
--- a/doc/pub/Regression/html/._Regression-bs101.html
+++ b/doc/pub/Regression/html/._Regression-bs101.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -394,50 +406,45 @@ MathJax.Hub.Config({
 
 
 
-
    Linear regression 
    
-
-
    
-In the ordinary least squares method we choose the cost function
-
-$$
-\begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
-\tag{25}
-\end{align}
-$$
-
-
    
-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
-
-$$
-    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
-$$
-
-
    
-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
-
+
    Cross-validation with Ridge 
    
 
    
 
 
-
    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
-)
-
    
-
    
+
    import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
 
-
-
    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)
+# A seed just to ensure that the random numbers are the same for every run.
+np.random.seed(3155)
+# Generate the data.
+n = 100
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 10)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+    i += 1
+plt.figure()
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
 
    
 
    
 
    
@@ -461,6 +468,9 @@ beta = ols_inv(X_train_own, y_train)
   
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    • 
 
 
diff --git a/doc/pub/Regression/html/._Regression-bs102.html b/doc/pub/Regression/html/._Regression-bs102.html
index 640471e63..e866d223d 100644
--- a/doc/pub/Regression/html/._Regression-bs102.html
+++ b/doc/pub/Regression/html/._Regression-bs102.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -394,89 +406,59 @@ MathJax.Hub.Config({
 
 
 
-
    Singular Value decomposition 
    
+
    The Ising model 
    
 
 
    
-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 singular
-value decomposition. Using the definition of the Moore-Penrose
-pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant \( J \) is given by
 
-$$
-    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
-$$
-
-
    
-where the pseudoinverse of \( \boldsymbol{X} \) is given by
-
-$$
-    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
-$$
-
-
    
-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
 $$
 \begin{align}
-    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
-\tag{26}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{21}
 \end{align}
 $$
 
 
    
-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.
+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.
 
 
    
 
 
-
    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
+
    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))
 
    
 
    
-
-
-
    beta = ols_svd(X_train_own,y_train)
-
    
-
    
-When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
-
-
    
-
-
-
    J = beta[1:].reshape(L, L)
-
    
-
    
-A way of looking at the coefficients in \( J \) is to plot the matrices as images.
-
-
    
-
-
-
    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()
-
    
-
    
-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?
+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.
 
 
    
 
    
@@ -499,6 +481,9 @@ In this case our matrix inversion was actually possible. The obvious question no
   
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diff --git a/doc/pub/Regression/html/._Regression-bs103.html b/doc/pub/Regression/html/._Regression-bs103.html
index e2c22d358..298b6bf22 100644
--- a/doc/pub/Regression/html/._Regression-bs103.html
+++ b/doc/pub/Regression/html/._Regression-bs103.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -394,86 +406,44 @@ MathJax.Hub.Config({
 
 
 
-
    The one-dimensional Ising model 
    
+
    Reformulating the problem to suit regression 
    
 
-
    
-Let us bring back the Ising model again, but now with an additional
-focus on Ridge and Lasso regression as well. We repeat some of the
-basic parts of the Ising model and the setup of the training and test
-data.  The one-dimensional Ising model with nearest neighbor
-interaction, no external field and a constant coupling constant \( J \) is
-given by
-
-$$
-\begin{align}
-    H = -J \sum_{k}^L s_k s_{k + 1},
-\tag{27}
-\end{align}
-$$
-
-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.
-
-
    
-
-
-
    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 sklearn.linear_model as skl
-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))
-
    
 
    
 A more general form for the one-dimensional Ising model is
 
 $$
 \begin{align}
     H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\tag{28}
+\tag{22}
 \end{align}
 $$
 
 
    
-Here we allow for interactions beyond the nearest neighbors and a more
-adaptive coupling matrix. This latter expression can be formulated as
-a matrix-product on the form
+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
 $$
 \begin{align}
-    H = X J,
-\tag{29}
+    \boldsymbol{H} = \boldsymbol{X} J,
+\tag{23}
 \end{align}
 $$
 
 
    
-where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
+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, viz.
+with the form utilized in linear regression, that is
+
 $$
 \begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
-\tag{30}
+    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\tag{24}
 \end{align}
 $$
 
-We organize the data as we did above
+
    
+We split the data in training and test data as discussed in the previous example
+
 
    
 
 
@@ -481,50 +451,8 @@ We organize the data as we did above
 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=0.96)
-
-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
-)
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
 
    
-
    
-We will do all fitting with Scikit-Learn,
-
-
    
-
-
-
    clf = skl.LinearRegression().fit(X_train, y_train)
-
    
-
    
-When  extracting the \( J \)-matrix we make sure to remove the intercept
-
    
-
-
-
    J_sk = clf.coef_.reshape(L, L)
-
    
-
    
-And then we plot the results
-
    
-
-
-
    fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_sk, **cmap_args)
-plt.title("LinearRegression from Scikit-learn", 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()
-
    
-
    
-The results perfectly with our previous discussion where we used our own code.
-
 
    
 
    
 
@@ -545,6 +473,9 @@ The results perfectly with our previous discussion where we used our own code.
   
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diff --git a/doc/pub/Regression/html/._Regression-bs104.html b/doc/pub/Regression/html/._Regression-bs104.html
index c4b147185..07dacee55 100644
--- a/doc/pub/Regression/html/._Regression-bs104.html
+++ b/doc/pub/Regression/html/._Regression-bs104.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -394,37 +406,50 @@ MathJax.Hub.Config({
 
 
 
-
    Ridge regression 
    
+
    Linear regression 
    
 
 
    
-Having explored the ordinary least squares we move on to ridge
-regression. In ridge regression we include a regularizer. This
-involves a new cost function which leads to a new estimate for the
-weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
-cost function is given by
+In the ordinary least squares method we choose the cost function
 
 $$
 \begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
-\tag{31}
+    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\tag{25}
 \end{align}
 $$
 
+
    
+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
+
+$$
+    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+$$
+
+
    
+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
+
 
    
 
 
-
    _lambda = 0.1
-clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
-J_ridge_sk = clf_ridge.coef_.reshape(L, L)
-fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_ridge_sk, **cmap_args)
-plt.title("Ridge from Scikit-learn", fontsize=18)
-plt.xticks(fontsize=18)
-plt.yticks(fontsize=18)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
    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
+)
+
    
+
    
 
-plt.show()
+
+
    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)
 
    
 
    
 
    
@@ -445,6 +470,9 @@ plt.show()
   
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diff --git a/doc/pub/Regression/html/._Regression-bs105.html b/doc/pub/Regression/html/._Regression-bs105.html
index f0aff16dd..231875d36 100644
--- a/doc/pub/Regression/html/._Regression-bs105.html
+++ b/doc/pub/Regression/html/._Regression-bs105.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -394,40 +406,89 @@ MathJax.Hub.Config({
 
 
 
-
    LASSO regression 
    
+
    Singular Value decomposition 
    
 
 
    
-In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
+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 singular
+value decomposition. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
 
+$$
+    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+
+
    
+where the pseudoinverse of \( \boldsymbol{X} \) is given by
+
+$$
+    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+
+
    
+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
 $$
 \begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
-\tag{32}
+    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\tag{26}
 \end{align}
 $$
 
 
    
-Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-Learn.
+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.
 
 
    
 
 
-
    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
-J_lasso_sk = clf_lasso.coef_.reshape(L, L)
-fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_lasso_sk, **cmap_args)
-plt.title("Lasso from Scikit-learn", fontsize=18)
+
    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
+
    
+
    
+
+
+
    beta = ols_svd(X_train_own,y_train)
+
    
+
    
+When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
+
+
    
+
+
+
    J = beta[1:].reshape(L, L)
+
    
+
    
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
+
+
    
+
+
+
    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()
 
    
 
    
-It is quite striking how LASSO breaks the symmetry of the coupling
-constant as opposed to ridge and OLS. We get a sparse solution with
-\( J_{j, j + 1} = -1 \).
+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?
 
 
    
 
    
@@ -447,6 +508,9 @@ constant as opposed to ridge and OLS. We get a sparse solution with
   
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diff --git a/doc/pub/Regression/html/._Regression-bs106.html b/doc/pub/Regression/html/._Regression-bs106.html
index 66446c383..72508153f 100644
--- a/doc/pub/Regression/html/._Regression-bs106.html
+++ b/doc/pub/Regression/html/._Regression-bs106.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -394,56 +406,136 @@ MathJax.Hub.Config({
 
 
 
-
    Performance as  function of the regularization parameter 
    
+
    The one-dimensional Ising model 
    
 
 
    
-We see how the different models perform for a different set of values for \( \lambda \).
+Let us bring back the Ising model again, but now with an additional
+focus on Ridge and Lasso regression as well. We repeat some of the
+basic parts of the Ising model and the setup of the training and test
+data.  The one-dimensional Ising model with nearest neighbor
+interaction, no external field and a constant coupling constant \( J \) is
+given by
+
+$$
+\begin{align}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{27}
+\end{align}
+$$
+
+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.
 
 
    
 
 
-
    lambdas = np.logspace(-4, 5, 10)
+
    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 sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
 
-train_errors = {
-    "ols_sk": np.zeros(lambdas.size),
-    "ridge_sk": np.zeros(lambdas.size),
-    "lasso_sk": np.zeros(lambdas.size)
-}
+L = 40
+n = int(1e4)
 
-test_errors = {
-    "ols_sk": np.zeros(lambdas.size),
-    "ridge_sk": np.zeros(lambdas.size),
-    "lasso_sk": np.zeros(lambdas.size)
-}
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
 
-plot_counter = 1
+energies = np.zeros(n)
 
-fig = plt.figure(figsize=(32, 54))
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
    
+
    
+A more general form for the one-dimensional Ising model is
 
-for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
-    for key, method in zip(
-        ["ols_sk", "ridge_sk", "lasso_sk"],
-        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
-    ):
-        method = method.fit(X_train, y_train)
+$$
+\begin{align}
+    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\tag{28}
+\end{align}
+$$
 
-        train_errors[key][i] = method.score(X_train, y_train)
-        test_errors[key][i] = method.score(X_test, y_test)
+
    
+Here we allow for interactions beyond the nearest neighbors and a more
+adaptive coupling matrix. This latter expression can be formulated as
+a matrix-product on the form
+$$
+\begin{align}
+    H = X J,
+\tag{29}
+\end{align}
+$$
 
-        omega = method.coef_.reshape(L, L)
+
    
+where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, viz.
+$$
+\begin{align}
+    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
+\tag{30}
+\end{align}
+$$
 
-        plt.subplot(10, 5, plot_counter)
-        plt.imshow(omega, **cmap_args)
-        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
-        plot_counter += 1
+We organize the data as we did above
+
    
 
+
+
    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=0.96)
+
+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
+)
+
    
+
    
+We will do all fitting with Scikit-Learn,
+
+
    
+
+
+
    clf = skl.LinearRegression().fit(X_train, y_train)
+
    
+
    
+When  extracting the \( J \)-matrix we make sure to remove the intercept
+
    
+
+
+
    J_sk = clf.coef_.reshape(L, L)
+
    
+
    
+And then we plot the results
+
    
+
+
+
    fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", 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()
 
    
 
    
-We see that LASSO reaches a good solution for low
-values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
-much. Ridge is more stable over a larger range of values for
-\( \lambda \), but eventually also fades away.
+The results perfectly with our previous discussion where we used our own code.
 
 
    
 
    
@@ -462,6 +554,9 @@ much. Ridge is more stable over a larger range of values for
   
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diff --git a/doc/pub/Regression/html/._Regression-bs107.html b/doc/pub/Regression/html/._Regression-bs107.html
index db39ec215..b269ceee1 100644
--- a/doc/pub/Regression/html/._Regression-bs107.html
+++ b/doc/pub/Regression/html/._Regression-bs107.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -394,57 +406,39 @@ MathJax.Hub.Config({
 
 
 
-
    Finding the optimal value of \( \lambda \) 
    
+
    Ridge regression 
    
 
 
    
-To determine which value of \( \lambda \) is best we plot the accuracy of
-the models when predicting the training and the testing set. We expect
-the accuracy of the training set to be quite good, but if the accuracy
-of the testing set is much lower this tells us that we might be
-subject to an overfit model. The ideal scenario is an accuracy on the
-testing set that is close to the accuracy of the training set.
+Having explored the ordinary least squares we move on to ridge
+regression. In ridge regression we include a regularizer. This
+involves a new cost function which leads to a new estimate for the
+weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
+cost function is given by
+
+$$
+\begin{align}
+    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
+\tag{31}
+\end{align}
+$$
 
 
    
 
 
-
    fig = plt.figure(figsize=(20, 14))
+
    _lambda = 0.1
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
-colors = {
-    "ols_sk": "r",
-    "ridge_sk": "y",
-    "lasso_sk": "c"
-}
-
-for key in train_errors:
-    plt.semilogx(
-        lambdas,
-        train_errors[key],
-        colors[key],
-        label="Train {0}".format(key),
-        linewidth=4.0
-    )
-
-for key in test_errors:
-    plt.semilogx(
-        lambdas,
-        test_errors[key],
-        colors[key] + "--",
-        label="Test {0}".format(key),
-        linewidth=4.0
-    )
-plt.legend(loc="best", fontsize=18)
-plt.xlabel(r"$\lambda$", fontsize=18)
-plt.ylabel(r"$R^2$", fontsize=18)
-plt.tick_params(labelsize=18)
 plt.show()
 
    
 
    
-From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
-achieves a very good accuracy on the test set. This by far surpasses the
-other models for all values of \( \lambda \).
-
-
    
-
 
    
 
 
      
@@ -460,6 +454,10 @@ other models for all values of \( \lambda \).
   
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diff --git a/doc/pub/Regression/html/._Regression-bs108.html b/doc/pub/Regression/html/._Regression-bs108.html
index e2fd2c209..2379a00c2 100644
--- a/doc/pub/Regression/html/._Regression-bs108.html
+++ b/doc/pub/Regression/html/._Regression-bs108.html
@@ -6,6 +6,7 @@ Automatically generated HTML file from DocOnce source
 
 
 
+
 
 
 Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis
@@ -40,215 +41,209 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -286,119 +281,116 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -414,53 +406,40 @@ MathJax.Hub.Config({
 
 
 
-
    Fitting with scikit-learn 
    
+
    LASSO regression 
    
 
 
    
-Next we fit a LinearRegression-model from Scikit-learn for comparison.
+In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
+
+$$
+\begin{align}
+    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
+\tag{32}
+\end{align}
+$$
+
+
    
+Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-Learn.
 
 
    
 
 
-
    clf = skl.LinearRegression().fit(X_train, y_train)
-
    
-
    
-Extracting the \( J \)-matrix from both our own method and the Scikit-learn model where we make sure to remove the intercept.
-
-
    
-
-
-
    J_own = omega[1:].reshape(L, L)
-J_sk = clf.coef_.reshape(L, L)
-
    
-
    
-A way of looking at the coefficients in \( J \) is to plot the matrices as images.
-
-
    
-
-
-
    fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_own, **cmap_args)
-plt.title("Home-made 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)
-
+
    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
 fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_sk, **cmap_args)
-plt.title("LinearRegression from Scikit-learn", fontsize=18)
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", 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()
 
    
 
    
-We can see that our model for the least squares method performes close
-to the benchmark from Scikit-learn. 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 \( J \).
+It is quite striking how LASSO breaks the symmetry of the coupling
+constant as opposed to ridge and OLS. We get a sparse solution with
+\( J_{j, j + 1} = -1 \).
 
 
    
 
    
@@ -480,9 +459,6 @@ valid matrix elements for \( J \).
   
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diff --git a/doc/pub/Regression/html/._Regression-bs109.html b/doc/pub/Regression/html/._Regression-bs109.html
index 59012db88..96de61f77 100644
--- a/doc/pub/Regression/html/._Regression-bs109.html
+++ b/doc/pub/Regression/html/._Regression-bs109.html
@@ -6,6 +6,7 @@ Automatically generated HTML file from DocOnce source
 
 
 
+
 
 
 Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis
@@ -40,215 +41,209 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -286,119 +281,116 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -414,68 +406,57 @@ MathJax.Hub.Config({
 
 
 
-
    Ridge regression 
    
+
    Performance as  function of the regularization parameter 
    
 
 
    
-Having explored the ordinary least squares we move on to ridge
-regression. In ridge regression we include a regularizer. This
-involves a new cost function which leads to a new estimate for the
-weights \( \omega \). This results in a penalized regression problem. The
-cost function is given by
-
-$$
-\begin{align}
-    C(X, \omega; \lambda) = ||X\omega - y||^2 + \lambda ||\omega||^2
-    = (X\omega - y)^T(X\omega - y) + \lambda \omega^T\omega.
-\tag{42}
-\end{align}
-$$
-
-Finding the extremum of this function yields the weights
-
-$$
-\begin{align}
-    \omega(\lambda) = \frac{X^Ty}{X^TX + \lambda} \to \frac{\omega_{\text{LS}}}{1 + \lambda},
-\tag{43}
-\end{align}
-$$
-
-
    
-where \( \omega_{\text{LS}} \) is the weights from ordinary least
-squares. The last assumption assumes that \( X \) is orthogonal, which it
-is not. We will therefore resort to solving the equation as it stands
-on the left hand side.
+We see how the different models perform for a different set of values for \( \lambda \).
 
 
    
 
 
-
    def get_ridge_weights(x: np.ndarray, y: np.ndarray, _lambda: float) -> np.ndarray:
-    return x.T @ y @ scl.inv(
-        x.T @ x + np.eye(x.shape[1], x.shape[1]) * _lambda
-    )
-_lambda = 0.1
-omega_ridge = get_ridge_weights(X_train_own, y_train, np.array([_lambda]))
-clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
-J_ridge_own = omega_ridge[1:].reshape(L, L)
-J_ridge_sk = clf_ridge.coef_.reshape(L, L)
-fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_ridge_own, **cmap_args)
-plt.title("Home-made ridge regression", fontsize=18)
-plt.xticks(fontsize=18)
-plt.yticks(fontsize=18)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
    lambdas = np.logspace(-4, 5, 10)
 
-fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_ridge_sk, **cmap_args)
-plt.title("Ridge from Scikit-learn", fontsize=18)
-plt.xticks(fontsize=18)
-plt.yticks(fontsize=18)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+train_errors = {
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+    for key, method in zip(
+        ["ols_sk", "ridge_sk", "lasso_sk"],
+        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+    ):
+        method = method.fit(X_train, y_train)
+
+        train_errors[key][i] = method.score(X_train, y_train)
+        test_errors[key][i] = method.score(X_test, y_test)
+
+        omega = method.coef_.reshape(L, L)
+
+        plt.subplot(10, 5, plot_counter)
+        plt.imshow(omega, **cmap_args)
+        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+        plot_counter += 1
 
 plt.show()
 
    
+
    
+We see that LASSO reaches a good solution for low
+values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
+much. Ridge is more stable over a larger range of values for
+\( \lambda \), but eventually also fades away.
+
 
    
 
    
 
@@ -493,9 +474,6 @@ plt.show()
   
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diff --git a/doc/pub/Regression/html/._Regression-bs110.html b/doc/pub/Regression/html/._Regression-bs110.html
index a1e0184e7..62e2a2964 100644
--- a/doc/pub/Regression/html/._Regression-bs110.html
+++ b/doc/pub/Regression/html/._Regression-bs110.html
@@ -6,6 +6,7 @@ Automatically generated HTML file from DocOnce source
 
 
 
+
 
 
 Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis
@@ -40,215 +41,209 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -286,119 +281,116 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -414,42 +406,57 @@ MathJax.Hub.Config({
 
 
 
-
    LASSO regression 
    
+
    Finding the optimal value of \( \lambda \) 
    
 
 
    
-In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
-$$
-\begin{align}
-    C(X, \omega; \lambda) =
-    ||X\omega - y||^2 + \lambda ||\omega||
-    = (X\omega - y)^T(X\omega - y) + \lambda \sqrt{\omega^T\omega}.
-\tag{44}
-\end{align}
-$$
-
-Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-learn.
+To determine which value of \( \lambda \) is best we plot the accuracy of
+the models when predicting the training and the testing set. We expect
+the accuracy of the training set to be quite good, but if the accuracy
+of the testing set is much lower this tells us that we might be
+subject to an overfit model. The ideal scenario is an accuracy on the
+testing set that is close to the accuracy of the training set.
 
 
    
 
 
-
    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
-J_lasso_sk = clf_lasso.coef_.reshape(L, L)
-fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_lasso_sk, **cmap_args)
-plt.title("Lasso from Scikit-learn", fontsize=18)
-plt.xticks(fontsize=18)
-plt.yticks(fontsize=18)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
    fig = plt.figure(figsize=(20, 14))
 
+colors = {
+    "ols_sk": "r",
+    "ridge_sk": "y",
+    "lasso_sk": "c"
+}
+
+for key in train_errors:
+    plt.semilogx(
+        lambdas,
+        train_errors[key],
+        colors[key],
+        label="Train {0}".format(key),
+        linewidth=4.0
+    )
+
+for key in test_errors:
+    plt.semilogx(
+        lambdas,
+        test_errors[key],
+        colors[key] + "--",
+        label="Test {0}".format(key),
+        linewidth=4.0
+    )
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
 plt.show()
 
    
 
    
-It is quite striking how LASSO breaks the symmetry of the coupling
-constant as opposed to ridge and OLS. We get a sparse solution with
-\( J_{j, j + 1} = -1 \).
+From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
+achieves a very good accuracy on the test set. This by far surpasses the
+other models for all values of \( \lambda \).
 
 
    
+
 
    
 
 
      
@@ -465,10 +472,6 @@ constant as opposed to ridge and OLS. We get a sparse solution with
   
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diff --git a/doc/pub/Regression/html/Regression-bs.html b/doc/pub/Regression/html/Regression-bs.html
index 9fe8c4983..6bf1b9e0f 100644
--- a/doc/pub/Regression/html/Regression-bs.html
+++ b/doc/pub/Regression/html/Regression-bs.html
@@ -217,24 +217,33 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -370,15 +379,18 @@ MathJax.Hub.Config({
       
  • Understanding what happens
    • 
       
  • Summing up
    • 
       
  • Another Example from Scikit-Learn's Repository
    • 
-      
  • The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • More examples on bootstrap and cross-validation and errors
    • 
+      
  • The same example but now with cross-validation
    • 
+      
  • Cross-validation with Ridge
    • 
+      
  • The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
 
         
       
@@ -413,7 +425,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Sep 10, 2019
     
+
    Sep 12, 2019
     
 
    
 
    
 
@@ -437,7 +449,7 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/Regression/html/Regression-reveal.html b/doc/pub/Regression/html/Regression-reveal.html
index a54774d74..209131d74 100644
--- a/doc/pub/Regression/html/Regression-reveal.html
+++ b/doc/pub/Regression/html/Regression-reveal.html
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
     
    
-
    Sep 10, 2019
     
+
    Sep 12, 2019
     
 
    
 
    
 
@@ -4000,7 +4000,216 @@ plt.show()
 
 
 
    
-
    The Ising model 
    
+
    More examples on bootstrap and cross-validation and errors 
    
+
+
    
+
+
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+testerror = np.zeros(Maxpolydegree)
+trainingerror = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+
+trials = 100
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+
+# loop over trials in order to estimate the expectation value of the MSE
+    testerror[polydegree] = 0.0
+    trainingerror[polydegree] = 0.0
+    for samples in range(trials):
+        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
+        ypred = model.predict(x_train)
+        ytilde = model.predict(x_test)
+        testerror[polydegree] += mean_squared_error(y_test, ytilde)
+        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
+
+    testerror[polydegree] /= trials
+    trainingerror[polydegree] /= trials
+    print("Degree of polynomial: %3d"% polynomial[polydegree])
+    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
+    print("Mean squared error on test data: %.8f" % testerror[polydegree])
+
+plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
+plt.plot(polynomial, np.log10(testerror), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
    
+
    
+
+
+
    
+
    The same example but now with cross-validation 
    
+
+
    
+
+
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import KFold
+from sklearn.model_selection import cross_val_score
+
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+estimated_mse_sklearn = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+k =5
+kfold = KFold(n_splits = k)
+
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+        OLS = LinearRegression()
+# loop over trials in order to estimate the expectation value of the MSE
+    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
+#[:, np.newaxis]
+    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
+
+plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
    
+
    
+
+
+
    
+
    Cross-validation with Ridge 
    
+
    
+
+
+
    import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+np.random.seed(3155)
+# Generate the data.
+n = 100
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 10)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+    i += 1
+plt.figure()
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
    
+
    
+
+
+
    
+
    The Ising model 
    
 
 
    
 The one-dimensional Ising model with nearest neighbor interaction, no
@@ -4059,7 +4268,7 @@ the coupling constant to achieve this.
 
 
 
    
-
    Reformulating the problem to suit regression 
    
+
    Reformulating the problem to suit regression 
    
 
 
    
 A more general form for the one-dimensional Ising model is
@@ -4116,7 +4325,7 @@ X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=
-
    Linear regression 
    
+
    Linear regression 
    
 
 
    
 In the ordinary least squares method we choose the cost function
@@ -4169,7 +4378,7 @@ beta = ols_inv(X_train_own, y_train)
 
 
 
    
-
    Singular Value decomposition 
    
+
    Singular Value decomposition 
    
 
 
    
 Doing the inversion directly turns out to be a bad idea since the matrix
@@ -4262,7 +4471,7 @@ In this case our matrix inversion was actually possible. The obvious question no
 
 
 
    
-
    The one-dimensional Ising model 
    
+
    The one-dimensional Ising model 
    
 
 
    
 Let us bring back the Ising model again, but now with an additional
@@ -4404,7 +4613,7 @@ The results perfectly with our previous discussion where we used our own code.
 
 
 
    
-
    Ridge regression 
    
+
    Ridge regression 
    
 
 
    
 Having explored the ordinary least squares we move on to ridge
@@ -4442,7 +4651,7 @@ plt.show()
 
 
 
    
-
    LASSO regression 
    
+
    LASSO regression 
    
 
 
    
 In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
@@ -4482,7 +4691,7 @@ constant as opposed to ridge and OLS. We get a sparse solution with
 
 
 
    
-
    Performance as  function of the regularization parameter 
    
+
    Performance as  function of the regularization parameter 
    
 
 
    
 We see how the different models perform for a different set of values for \( \lambda \).
@@ -4536,7 +4745,7 @@ much. Ridge is more stable over a larger range of values for
 
 
 
    
-
    Finding the optimal value of \( \lambda \) 
    
+
    Finding the optimal value of \( \lambda \) 
    
 
 
    
 To determine which value of \( \lambda \) is best we plot the accuracy of
diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html
index 4e8543b0f..69e7d4aa5 100644
--- a/doc/pub/Regression/html/Regression-solarized.html
+++ b/doc/pub/Regression/html/Regression-solarized.html
@@ -237,24 +237,33 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -296,7 +305,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Sep 10, 2019
     
+
    Sep 12, 2019
     
 
    
 
    
 









    
@@ -3900,7 +3909,213 @@ plt.show()
 
    
 









    
 
-
    The Ising model 
    
+
    More examples on bootstrap and cross-validation and errors 
    
+
+
    
+
+
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+testerror = np.zeros(Maxpolydegree)
+trainingerror = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+
+trials = 100
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+
+# loop over trials in order to estimate the expectation value of the MSE
+    testerror[polydegree] = 0.0
+    trainingerror[polydegree] = 0.0
+    for samples in range(trials):
+        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
+        ypred = model.predict(x_train)
+        ytilde = model.predict(x_test)
+        testerror[polydegree] += mean_squared_error(y_test, ytilde)
+        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
+
+    testerror[polydegree] /= trials
+    trainingerror[polydegree] /= trials
+    print("Degree of polynomial: %3d"% polynomial[polydegree])
+    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
+    print("Mean squared error on test data: %.8f" % testerror[polydegree])
+
+plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
+plt.plot(polynomial, np.log10(testerror), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
    
+
    
+
+
+
    The same example but now with cross-validation 
    
+
+
    
+
+
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import KFold
+from sklearn.model_selection import cross_val_score
+
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+estimated_mse_sklearn = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+k =5
+kfold = KFold(n_splits = k)
+
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+        OLS = LinearRegression()
+# loop over trials in order to estimate the expectation value of the MSE
+    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
+#[:, np.newaxis]
+    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
+
+plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
    
+
    
+









    
+
+
    Cross-validation with Ridge 
    
+
    
+
+
+
    import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+np.random.seed(3155)
+# Generate the data.
+n = 100
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 10)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+    i += 1
+plt.figure()
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
    
+
    
+









    
+
+
    The Ising model 
    
 
 
    
 The one-dimensional Ising model with nearest neighbor interaction, no
@@ -3957,7 +4172,7 @@ the coupling constant to achieve this.
 
    
 









    
 
-
    Reformulating the problem to suit regression 
    
+
    Reformulating the problem to suit regression 
    
 
 
    
 A more general form for the one-dimensional Ising model is
@@ -4007,7 +4222,7 @@ X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=
 









    
 
-
    Linear regression 
    
+
    Linear regression 
    
 
 
    
 In the ordinary least squares method we choose the cost function
@@ -4055,7 +4270,7 @@ beta = ols_inv(X_train_own, y_train)
 
    
 









    
 
-
    Singular Value decomposition 
    
+
    Singular Value decomposition 
    
 
 
    
 Doing the inversion directly turns out to be a bad idea since the matrix
@@ -4142,7 +4357,7 @@ In this case our matrix inversion was actually possible. The obvious question no
 
    
 









    
 
-
    The one-dimensional Ising model 
    
+
    The one-dimensional Ising model 
    
 
 
    
 Let us bring back the Ising model again, but now with an additional
@@ -4276,7 +4491,7 @@ The results perfectly with our previous discussion where we used our own code.
 
    
 









    
 
-
    Ridge regression 
    
+
    Ridge regression 
    
 
 
    
 Having explored the ordinary least squares we move on to ridge
@@ -4311,7 +4526,7 @@ plt.show()
 
    
 









    
 
-
    LASSO regression 
    
+
    LASSO regression 
    
 
 
    
 In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
@@ -4349,7 +4564,7 @@ constant as opposed to ridge and OLS. We get a sparse solution with
 
    
 









    
 
-
    Performance as  function of the regularization parameter 
    
+
    Performance as  function of the regularization parameter 
    
 
 
    
 We see how the different models perform for a different set of values for \( \lambda \).
@@ -4403,7 +4618,7 @@ much. Ridge is more stable over a larger range of values for
 
    
 









    
 
-
    Finding the optimal value of \( \lambda \) 
    
+
    Finding the optimal value of \( \lambda \) 
    
 
 
    
 To determine which value of \( \lambda \) is best we plot the accuracy of
diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html
index 1f5d99773..85e22501f 100644
--- a/doc/pub/Regression/html/Regression.html
+++ b/doc/pub/Regression/html/Regression.html
@@ -242,24 +242,33 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                '___sec97'),
-              ('The Ising model', 2, None, '___sec98'),
-              ('Reformulating the problem to suit regression',
+              ('More examples on bootstrap and cross-validation and errors',
+               2,
+               None,
+               '___sec98'),
+              ('The same example but now with cross-validation',
                2,
                None,
                '___sec99'),
-              ('Linear regression', 2, None, '___sec100'),
-              ('Singular Value decomposition', 2, None, '___sec101'),
-              ('The one-dimensional Ising model', 2, None, '___sec102'),
-              ('Ridge regression', 2, None, '___sec103'),
-              ('LASSO regression', 2, None, '___sec104'),
+              ('Cross-validation with Ridge', 2, None, '___sec100'),
+              ('The Ising model', 2, None, '___sec101'),
+              ('Reformulating the problem to suit regression',
+               2,
+               None,
+               '___sec102'),
+              ('Linear regression', 2, None, '___sec103'),
+              ('Singular Value decomposition', 2, None, '___sec104'),
+              ('The one-dimensional Ising model', 2, None, '___sec105'),
+              ('Ridge regression', 2, None, '___sec106'),
+              ('LASSO regression', 2, None, '___sec107'),
               ('Performance as  function of the regularization parameter',
                2,
                None,
-               '___sec105'),
+               '___sec108'),
               ('Finding the optimal value of $\\lambda$',
                2,
                None,
-               '___sec106')]}
+               '___sec109')]}
 end of tocinfo -->
 
 
@@ -301,7 +310,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Sep 10, 2019
     
+
    Sep 12, 2019
     
 
    
 
    
 









    
@@ -3905,7 +3914,213 @@ plt.show()
 
    
 









    
 
-
    The Ising model 
    
+
    More examples on bootstrap and cross-validation and errors 
    
+
+
    
+
+
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+testerror = np.zeros(Maxpolydegree)
+trainingerror = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+
+trials = 100
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+
+# loop over trials in order to estimate the expectation value of the MSE
+    testerror[polydegree] = 0.0
+    trainingerror[polydegree] = 0.0
+    for samples in range(trials):
+        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
+        ypred = model.predict(x_train)
+        ytilde = model.predict(x_test)
+        testerror[polydegree] += mean_squared_error(y_test, ytilde)
+        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
+
+    testerror[polydegree] /= trials
+    trainingerror[polydegree] /= trials
+    print("Degree of polynomial: %3d"% polynomial[polydegree])
+    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
+    print("Mean squared error on test data: %.8f" % testerror[polydegree])
+
+plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
+plt.plot(polynomial, np.log10(testerror), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
    
+
    
+
+
+
    The same example but now with cross-validation 
    
+
+
    
+
+
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import KFold
+from sklearn.model_selection import cross_val_score
+
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+estimated_mse_sklearn = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+k =5
+kfold = KFold(n_splits = k)
+
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+        OLS = LinearRegression()
+# loop over trials in order to estimate the expectation value of the MSE
+    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
+#[:, np.newaxis]
+    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
+
+plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
    
+
    
+









    
+
+
    Cross-validation with Ridge 
    
+
    
+
+
+
    import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+np.random.seed(3155)
+# Generate the data.
+n = 100
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 10)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+    i += 1
+plt.figure()
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
    
+
    
+









    
+
+
    The Ising model 
    
 
 
    
 The one-dimensional Ising model with nearest neighbor interaction, no
@@ -3962,7 +4177,7 @@ the coupling constant to achieve this.
 
    
 









    
 
-
    Reformulating the problem to suit regression 
    
+
    Reformulating the problem to suit regression 
    
 
 
    
 A more general form for the one-dimensional Ising model is
@@ -4012,7 +4227,7 @@ X_train, X_test, y_train, y_test = train_tes
 
    
 









    
 
-
    Linear regression 
    
+
    Linear regression 
    
 
 
    
 In the ordinary least squares method we choose the cost function
@@ -4060,7 +4275,7 @@ beta = ols_inv(X_train_own, y_train)
 
    
 









    
 
-
    Singular Value decomposition 
    
+
    Singular Value decomposition 
    
 
 
    
 Doing the inversion directly turns out to be a bad idea since the matrix
@@ -4147,7 +4362,7 @@ In this case our matrix inversion was actually possible. The obvious question no
 
    
 









    
 
-
    The one-dimensional Ising model 
    
+
    The one-dimensional Ising model 
    
 
 
    
 Let us bring back the Ising model again, but now with an additional
@@ -4281,7 +4496,7 @@ The results perfectly with our previous discussion where we used our own code.
 
    
 









    
 
-
    Ridge regression 
    
+
    Ridge regression 
    
 
 
    
 Having explored the ordinary least squares we move on to ridge
@@ -4316,7 +4531,7 @@ plt.show()
 
    
 









    
 
-
    LASSO regression 
    
+
    LASSO regression 
    
 
 
    
 In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
@@ -4354,7 +4569,7 @@ constant as opposed to ridge and OLS. We get a sparse solution with
 
    
 









    
 
-
    Performance as  function of the regularization parameter 
    
+
    Performance as  function of the regularization parameter 
    
 
 
    
 We see how the different models perform for a different set of values for \( \lambda \).
@@ -4408,7 +4623,7 @@ much. Ridge is more stable over a larger range of values for
 
    
 









    
 
-
    Finding the optimal value of \( \lambda \) 
    
+
    Finding the optimal value of \( \lambda \) 
    
 
 
    
 To determine which value of \( \lambda \) is best we plot the accuracy of
diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb
index e2d2fc1ae..d878532e1 100644
--- a/doc/pub/Regression/ipynb/Regression.ipynb
+++ b/doc/pub/Regression/ipynb/Regression.ipynb
@@ -10,7 +10,7 @@
     "  \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: **Sep 10, 2019**\n",
+    "Date: **Sep 12, 2019**\n",
     "\n",
     "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
     "\n",
@@ -5197,6 +5197,238 @@
     "plt.show()"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## More examples on bootstrap and cross-validation and errors"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "# Common imports\n",
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n",
+    "from sklearn.model_selection import train_test_split\n",
+    "from sklearn.utils import resample\n",
+    "from sklearn.metrics import mean_squared_error\n",
+    "# Where to save the figures and data files\n",
+    "PROJECT_ROOT_DIR = \"Results\"\n",
+    "FIGURE_ID = \"Results/FigureFiles\"\n",
+    "DATA_ID = \"DataFiles/\"\n",
+    "\n",
+    "if not os.path.exists(PROJECT_ROOT_DIR):\n",
+    "    os.mkdir(PROJECT_ROOT_DIR)\n",
+    "\n",
+    "if not os.path.exists(FIGURE_ID):\n",
+    "    os.makedirs(FIGURE_ID)\n",
+    "\n",
+    "if not os.path.exists(DATA_ID):\n",
+    "    os.makedirs(DATA_ID)\n",
+    "\n",
+    "def image_path(fig_id):\n",
+    "    return os.path.join(FIGURE_ID, fig_id)\n",
+    "\n",
+    "def data_path(dat_id):\n",
+    "    return os.path.join(DATA_ID, dat_id)\n",
+    "\n",
+    "def save_fig(fig_id):\n",
+    "    plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
+    "\n",
+    "infile = open(data_path(\"EoS.csv\"),'r')\n",
+    "\n",
+    "# Read the EoS data as  csv file and organize the data into two arrays with density and energies\n",
+    "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n",
+    "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n",
+    "EoS = EoS.dropna()\n",
+    "Energies = EoS['Energy']\n",
+    "Density = EoS['Density']\n",
+    "#  The design matrix now as function of various polytrops\n",
+    "\n",
+    "Maxpolydegree = 30\n",
+    "X = np.zeros((len(Density),Maxpolydegree))\n",
+    "X[:,0] = 1.0\n",
+    "testerror = np.zeros(Maxpolydegree)\n",
+    "trainingerror = np.zeros(Maxpolydegree)\n",
+    "polynomial = np.zeros(Maxpolydegree)\n",
+    "\n",
+    "trials = 100\n",
+    "for polydegree in range(1, Maxpolydegree):\n",
+    "    polynomial[polydegree] = polydegree\n",
+    "    for degree in range(polydegree):\n",
+    "        X[:,degree] = Density**(degree/3.0)\n",
+    "\n",
+    "# loop over trials in order to estimate the expectation value of the MSE\n",
+    "    testerror[polydegree] = 0.0\n",
+    "    trainingerror[polydegree] = 0.0\n",
+    "    for samples in range(trials):\n",
+    "        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)\n",
+    "        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)\n",
+    "        ypred = model.predict(x_train)\n",
+    "        ytilde = model.predict(x_test)\n",
+    "        testerror[polydegree] += mean_squared_error(y_test, ytilde)\n",
+    "        trainingerror[polydegree] += mean_squared_error(y_train, ypred) \n",
+    "\n",
+    "    testerror[polydegree] /= trials\n",
+    "    trainingerror[polydegree] /= trials\n",
+    "    print(\"Degree of polynomial: %3d\"% polynomial[polydegree])\n",
+    "    print(\"Mean squared error on training data: %.8f\" % trainingerror[polydegree])\n",
+    "    print(\"Mean squared error on test data: %.8f\" % testerror[polydegree])\n",
+    "\n",
+    "plt.plot(polynomial, np.log10(trainingerror), label='Training Error')\n",
+    "plt.plot(polynomial, np.log10(testerror), label='Test Error')\n",
+    "plt.xlabel('Polynomial degree')\n",
+    "plt.ylabel('log10[MSE]')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## The same example but now with cross-validation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "# Common imports\n",
+    "import os\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n",
+    "from sklearn.metrics import mean_squared_error\n",
+    "from sklearn.model_selection import KFold\n",
+    "from sklearn.model_selection import cross_val_score\n",
+    "\n",
+    "\n",
+    "# Where to save the figures and data files\n",
+    "PROJECT_ROOT_DIR = \"Results\"\n",
+    "FIGURE_ID = \"Results/FigureFiles\"\n",
+    "DATA_ID = \"DataFiles/\"\n",
+    "\n",
+    "if not os.path.exists(PROJECT_ROOT_DIR):\n",
+    "    os.mkdir(PROJECT_ROOT_DIR)\n",
+    "\n",
+    "if not os.path.exists(FIGURE_ID):\n",
+    "    os.makedirs(FIGURE_ID)\n",
+    "\n",
+    "if not os.path.exists(DATA_ID):\n",
+    "    os.makedirs(DATA_ID)\n",
+    "\n",
+    "def image_path(fig_id):\n",
+    "    return os.path.join(FIGURE_ID, fig_id)\n",
+    "\n",
+    "def data_path(dat_id):\n",
+    "    return os.path.join(DATA_ID, dat_id)\n",
+    "\n",
+    "def save_fig(fig_id):\n",
+    "    plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
+    "\n",
+    "infile = open(data_path(\"EoS.csv\"),'r')\n",
+    "\n",
+    "# Read the EoS data as  csv file and organize the data into two arrays with density and energies\n",
+    "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n",
+    "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n",
+    "EoS = EoS.dropna()\n",
+    "Energies = EoS['Energy']\n",
+    "Density = EoS['Density']\n",
+    "#  The design matrix now as function of various polytrops\n",
+    "\n",
+    "Maxpolydegree = 30\n",
+    "X = np.zeros((len(Density),Maxpolydegree))\n",
+    "X[:,0] = 1.0\n",
+    "estimated_mse_sklearn = np.zeros(Maxpolydegree)\n",
+    "polynomial = np.zeros(Maxpolydegree)\n",
+    "k =5\n",
+    "kfold = KFold(n_splits = k)\n",
+    "\n",
+    "for polydegree in range(1, Maxpolydegree):\n",
+    "    polynomial[polydegree] = polydegree\n",
+    "    for degree in range(polydegree):\n",
+    "        X[:,degree] = Density**(degree/3.0)\n",
+    "        OLS = LinearRegression()\n",
+    "# loop over trials in order to estimate the expectation value of the MSE\n",
+    "    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)\n",
+    "#[:, np.newaxis]\n",
+    "    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)\n",
+    "\n",
+    "plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n",
+    "plt.xlabel('Polynomial degree')\n",
+    "plt.ylabel('log10[MSE]')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Cross-validation with Ridge"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from sklearn.model_selection import KFold\n",
+    "from sklearn.linear_model import Ridge\n",
+    "from sklearn.model_selection import cross_val_score\n",
+    "from sklearn.preprocessing import PolynomialFeatures\n",
+    "\n",
+    "# A seed just to ensure that the random numbers are the same for every run.\n",
+    "np.random.seed(3155)\n",
+    "# Generate the data.\n",
+    "n = 100\n",
+    "x = np.linspace(-3, 3, n).reshape(-1, 1)\n",
+    "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n",
+    "# Decide degree on polynomial to fit\n",
+    "poly = PolynomialFeatures(degree = 10)\n",
+    "\n",
+    "# Decide which values of lambda to use\n",
+    "nlambdas = 500\n",
+    "lambdas = np.logspace(-3, 5, nlambdas)\n",
+    "# Initialize a KFold instance\n",
+    "k = 5\n",
+    "kfold = KFold(n_splits = k)\n",
+    "estimated_mse_sklearn = np.zeros(nlambdas)\n",
+    "i = 0\n",
+    "for lmb in lambdas:\n",
+    "    ridge = Ridge(alpha = lmb)\n",
+    "    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)\n",
+    "    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)\n",
+    "    i += 1\n",
+    "plt.figure()\n",
+    "plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')\n",
+    "plt.xlabel('log10(lambda)')\n",
+    "plt.ylabel('MSE')\n",
+    "plt.legend()\n",
+    "plt.show()"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -5237,7 +5469,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 34,
    "metadata": {
     "collapsed": false
    },
@@ -5352,7 +5584,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 35,
    "metadata": {
     "collapsed": false
    },
@@ -5418,7 +5650,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 36,
    "metadata": {
     "collapsed": false
    },
@@ -5436,7 +5668,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 37,
    "metadata": {
     "collapsed": false
    },
@@ -5523,7 +5755,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 38,
    "metadata": {
     "collapsed": false
    },
@@ -5536,7 +5768,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 39,
    "metadata": {
     "collapsed": false
    },
@@ -5554,7 +5786,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 40,
    "metadata": {
     "collapsed": false
    },
@@ -5572,7 +5804,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 38,
+   "execution_count": 41,
    "metadata": {
     "collapsed": false
    },
@@ -5640,7 +5872,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 39,
+   "execution_count": 42,
    "metadata": {
     "collapsed": false
    },
@@ -5748,7 +5980,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 40,
+   "execution_count": 43,
    "metadata": {
     "collapsed": false
    },
@@ -5780,7 +6012,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 41,
+   "execution_count": 44,
    "metadata": {
     "collapsed": false
    },
@@ -5798,7 +6030,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 45,
    "metadata": {
     "collapsed": false
    },
@@ -5816,7 +6048,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 46,
    "metadata": {
     "collapsed": false
    },
@@ -5874,7 +6106,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 47,
    "metadata": {
     "collapsed": false
    },
@@ -5927,7 +6159,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 45,
+   "execution_count": 48,
    "metadata": {
     "collapsed": false
    },
@@ -5963,7 +6195,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 46,
+   "execution_count": 49,
    "metadata": {
     "collapsed": false
    },
@@ -6028,7 +6260,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 47,
+   "execution_count": 50,
    "metadata": {
     "collapsed": false
    },
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 88591ac22..c42997483 100644
--- a/doc/src/Regression/Regression.do.txt
+++ b/doc/src/Regression/Regression.do.txt
@@ -3259,6 +3259,216 @@ plt.show()
 !ec
 
 
+!split
+===== More examples on bootstrap and cross-validation and errors =====
+
+!bc pycod
+# Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+testerror = np.zeros(Maxpolydegree)
+trainingerror = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+
+trials = 100
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+
+# loop over trials in order to estimate the expectation value of the MSE
+    testerror[polydegree] = 0.0
+    trainingerror[polydegree] = 0.0
+    for samples in range(trials):
+        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
+        ypred = model.predict(x_train)
+        ytilde = model.predict(x_test)
+        testerror[polydegree] += mean_squared_error(y_test, ytilde)
+        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
+
+    testerror[polydegree] /= trials
+    trainingerror[polydegree] /= trials
+    print("Degree of polynomial: %3d"% polynomial[polydegree])
+    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
+    print("Mean squared error on test data: %.8f" % testerror[polydegree])
+
+plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
+plt.plot(polynomial, np.log10(testerror), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
+!ec
+
+
+!split 
+=====  The same example but now with cross-validation =====
+
+!bc pycod
+# Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import KFold
+from sklearn.model_selection import cross_val_score
+
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+estimated_mse_sklearn = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+k =5
+kfold = KFold(n_splits = k)
+
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+        OLS = LinearRegression()
+# loop over trials in order to estimate the expectation value of the MSE
+    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
+#[:, np.newaxis]
+    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
+
+plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
+!ec
+
+!split
+=====  Cross-validation with Ridge =====
+!bc pycod 
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+np.random.seed(3155)
+# Generate the data.
+n = 100
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 10)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+    i += 1
+plt.figure()
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+!ec
+
+
+
+
+
+
+
+
+
 
 !split
 =====  The Ising model =====
diff --git a/doc/src/Regression/Results/FigureFiles/EoSfitting.png b/doc/src/Regression/Results/FigureFiles/EoSfitting.png
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