From 331bb78d0ae3567f2e5b3a83c92dab7ed7c543c8 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Mon, 6 Sep 2021 22:28:54 +0200 Subject: [PATCH] updating week 36 --- doc/pub/week36/html/week36-bs.html | 199 ++++--- doc/pub/week36/html/week36-reveal.html | 406 +++++++------- doc/pub/week36/html/week36-solarized.html | 537 ++++++++++--------- doc/pub/week36/html/week36.html | 535 +++++++++--------- doc/pub/week36/ipynb/ipynb-week36-src.tar.gz | Bin 197 -> 192 bytes doc/pub/week36/ipynb/week36.ipynb | 260 ++++----- doc/src/week36/week36.do.txt | 205 ++++--- 7 files changed, 1110 insertions(+), 1032 deletions(-) diff --git a/doc/pub/week36/html/week36-bs.html b/doc/pub/week36/html/week36-bs.html index 1725b8031..abca03fd0 100644 --- a/doc/pub/week36/html/week36-bs.html +++ b/doc/pub/week36/html/week36-bs.html @@ -1,15 +1,15 @@ - + - + -Week 36: Resampling techniques and Ordinary Least Square +Week 36: Statistical interpretation of Linear Regression and Resampling techniques @@ -41,72 +41,111 @@ Automatically generated HTML file from DocOnce source @@ -136,7 +175,7 @@ MathJax.Hub.Config({ - Week 36: Resampling techniques and Ordinary Least Square + Week 36: Statistical interpretation of Linear Regression and Resampling techniques
@@ -144,40 +183,40 @@ MathJax.Hub.Config({
  • Contents
    • @@ -196,7 +235,7 @@ MathJax.Hub.Config({
    -

    Week 36: Resampling techniques and Ordinary Least Square

    +

    Week 36: Statistical interpretation of Linear Regression and Resampling techniques

    @@ -212,7 +251,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 16, 2020

    +

    Sep 6, 2021


    @@ -248,13 +287,13 @@ MathJax.Hub.Config({

    - © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + © 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
    diff --git a/doc/pub/week36/html/week36-reveal.html b/doc/pub/week36/html/week36-reveal.html index b2ba4614f..4bba4c2a4 100644 --- a/doc/pub/week36/html/week36-reveal.html +++ b/doc/pub/week36/html/week36-reveal.html @@ -1,11 +1,11 @@ - + - + -Week 36: Resampling techniques and Ordinary Least Square +Week 36: Statistical interpretation of Linear Regression and Resampling techniques @@ -13,7 +13,7 @@ - + @@ -76,10 +76,10 @@ document.getElementsByTagName( 'head' )[0].appendChild( link ); .reveal .alert-block > p, .alert-block > ul {margin-bottom:1em} /*.reveal .alert li {margin-top: 1em}*/ .reveal .alert-block p+p {margin-top:5px} - /*.reveal .alert-notice { background-image: url(http://hplgit.github.io/doconce/bundled/html_images/small_gray_notice.png); } - .reveal .alert-summary { background-image:url(http://hplgit.github.io/doconce/bundled/html_images/small_gray_summary.png); } - .reveal .alert-warning { background-image: url(http://hplgit.github.io/doconce/bundled/html_images/small_gray_warning.png); } - .reveal .alert-question {background-image:url(http://hplgit.github.io/doconce/bundled/html_images/small_gray_question.png); } */ + /*.reveal .alert-notice { background-image: url(https://hplgit.github.io/doconce/bundled/html_images/small_gray_notice.png); } + .reveal .alert-summary { background-image:url(https://hplgit.github.io/doconce/bundled/html_images/small_gray_summary.png); } + .reveal .alert-warning { background-image: url(https://hplgit.github.io/doconce/bundled/html_images/small_gray_warning.png); } + .reveal .alert-question {background-image:url(https://hplgit.github.io/doconce/bundled/html_images/small_gray_question.png); } */ @@ -132,7 +132,7 @@ MathJax.Hub.Config({ -

    Week 36: Resampling techniques and Ordinary Least Square

    +

    Week 36: Statistical interpretation of Linear Regression and Resampling techniques

    @@ -148,149 +148,33 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Sep 16, 2020

    +

    Sep 6, 2021


    - © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + © 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
    -

    Plans for week 36

    +

    Plans for week 36

      -

    • Thursday: Statistics, probability theory and resampling methods
      • -

    • Friday: Resampling methods and motivation for Ridge Regression
      • +

    • Thursday: Statistics, probability theory and linear regression
      • +

    • Friday: Resampling methods
    -

    Thursday September 3

    - -

    -Video of Lecture and handwritten notes - -

    -More material will be added here, see handwritten notes also. +

    Thursday September 9

    -

    Why resampling methods

    - -

    -Before we proceed, we need to rethink what we have been doing. In our -eager to fit the data, we have omitted several important elements in -our regression analysis. In what follows we will - -

      -

    1. look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
      2. -

    2. introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
      3. -
    -

    - -This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods. -

    - - -
    -

    Resampling methods

    -
    - -

    -Resampling methods are an indispensable tool in modern -statistics. They involve repeatedly drawing samples from a training -set and refitting a model of interest on each sample in order to -obtain additional information about the fitted model. For example, in -order to estimate the variability of a linear regression fit, we can -repeatedly draw different samples from the training data, fit a linear -regression to each new sample, and then examine the extent to which -the resulting fits differ. Such an approach may allow us to obtain -information that would not be available from fitting the model only -once using the original training sample. - -

    -Two resampling methods are often used in Machine Learning analyses, - -

      -

    1. The bootstrap method
      2. -

    2. and Cross-Validation
      3. -
    -

    - -In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular -cross-validation and the bootstrap method. - - -

    -
    - - -
    -

    Resampling approaches can be computationally expensive

    -
    - -

    -Resampling approaches can be computationally expensive, because they -involve fitting the same statistical method multiple times using -different subsets of the training data. However, due to recent -advances in computing power, the computational requirements of -resampling methods generally are not prohibitive. In this chapter, we -discuss two of the most commonly used resampling methods, -cross-validation and the bootstrap. Both methods are important tools -in the practical application of many statistical learning -procedures. For example, cross-validation can be used to estimate the -test error associated with a given statistical learning method in -order to evaluate its performance, or to select the appropriate level -of flexibility. The process of evaluating a model’s performance is -known as model assessment, whereas the process of selecting the proper -level of flexibility for a model is known as model selection. The -bootstrap is widely used. - - -

    -
    - - -
    -

    Why resampling methods ?

    -
    -Statistical analysis. -
      -

    • Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods
      • -

    • The results can be analysed with the same statistical tools as we would use analysing experimental data.
      • -

    • As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
      • -
    -
    -
    - - -
    -

    Statistical analysis

    -
    - -
      -

    • As in other experiments, many numerical experiments have two classes of errors:
      • - -
        - -

      • Statistical errors
        • - -

      • Systematical errors
        • -
      -

    • Statistical errors can be estimated using standard tools from statistics
      • -

    • Systematical errors are method specific and must be treated differently from case to case.
      • -
    -
    -
    - - -
    -

    Linking the regression analysis with a statistical interpretation

    +

    Linking the regression analysis with a statistical interpretation

    The @@ -329,7 +213,7 @@ row number \( i \) and perform a sum over all values \( p \).

    -

    Assumptions made

    +

    Assumptions made

    The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off) @@ -353,7 +237,7 @@ $$

    -

    Expectation value and variance

    +

    Expectation value and variance

    We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) @@ -392,7 +276,7 @@ mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (n

    -

    Expectation value and variance for \( \boldsymbol{\beta} \)

    +

    Expectation value and variance for \( \boldsymbol{\beta} \)

    With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value @@ -487,7 +371,122 @@ This means the variance we obtain with the standard OLS will always for \( \lamb

    -

    Resampling methods

    +

    Friday September 10

    +
    + + +
    +

    Why resampling methods

    + +

    +Before we proceed, we need to rethink what we have been doing. In our +eager to fit the data, we have omitted several important elements in +our regression analysis. In what follows we will + +

      +

    1. look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
      2. +

    2. introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
      3. +
    +

    + +This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods. +

    + + +
    +

    Resampling methods

    +
    + +

    +Resampling methods are an indispensable tool in modern +statistics. They involve repeatedly drawing samples from a training +set and refitting a model of interest on each sample in order to +obtain additional information about the fitted model. For example, in +order to estimate the variability of a linear regression fit, we can +repeatedly draw different samples from the training data, fit a linear +regression to each new sample, and then examine the extent to which +the resulting fits differ. Such an approach may allow us to obtain +information that would not be available from fitting the model only +once using the original training sample. + +

    +Two resampling methods are often used in Machine Learning analyses, + +

      +

    1. The bootstrap method
      2. +

    2. and Cross-Validation
      3. +
    +

    + +In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular +cross-validation and the bootstrap method. + + +

    +
    + + +
    +

    Resampling approaches can be computationally expensive

    +
    + +

    +Resampling approaches can be computationally expensive, because they +involve fitting the same statistical method multiple times using +different subsets of the training data. However, due to recent +advances in computing power, the computational requirements of +resampling methods generally are not prohibitive. In this chapter, we +discuss two of the most commonly used resampling methods, +cross-validation and the bootstrap. Both methods are important tools +in the practical application of many statistical learning +procedures. For example, cross-validation can be used to estimate the +test error associated with a given statistical learning method in +order to evaluate its performance, or to select the appropriate level +of flexibility. The process of evaluating a model’s performance is +known as model assessment, whereas the process of selecting the proper +level of flexibility for a model is known as model selection. The +bootstrap is widely used. + + +

    +
    + + +
    +

    Why resampling methods ?

    +
    +Statistical analysis +
      +

    • Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods
      • +

    • The results can be analysed with the same statistical tools as we would use analysing experimental data.
      • +

    • As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
      • +
    +
    +
    + + +
    +

    Statistical analysis

    +
    + +
      +

    • As in other experiments, many numerical experiments have two classes of errors:
      • + +
        + +

      • Statistical errors
        • + +

      • Systematical errors
        • +
      +

    • Statistical errors can be estimated using standard tools from statistics
      • +

    • Systematical errors are method specific and must be treated differently from case to case.
      • +
    +
    +
    + + +
    +

    Resampling methods

    With all these analytical equations for both the OLS and Ridge @@ -517,7 +516,7 @@ training error reaches a saturation.

    -

    Resampling methods: Jackknife and Bootstrap

    +

    Resampling methods: Jackknife and Bootstrap

    Two famous @@ -540,7 +539,7 @@ need for bootstrapping.

    -

    Resampling methods: Jackknife

    +

    Resampling methods: Jackknife

    The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). @@ -561,11 +560,11 @@ number \( i \) is left out. Using this notation, define

    -

    Jackknife code example

    +

    Jackknife code example

    -

    from numpy import *
+
    from numpy import *
 from numpy.random import randint, randn
 from time import time
 
@@ -576,9 +575,9 @@ number \( i \) is left out. Using this notation, define
         t[i] = stat(delete(data,i) )
 
     # analysis                                                                                                                                                                     
-    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
-    print("original           bias      std. error")
-    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
+    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
+    print("original           bias      std. error")
+    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
 
     return t
 
@@ -598,7 +597,7 @@ t = jackknife(x, stat)
 
 
 
    
-
    Resampling methods: Bootstrap 
    
+
    Resampling methods: Bootstrap 
    
 
    
 
 
    
@@ -620,7 +619,7 @@ advantages:
 
 
 
    
-
    Resampling methods: Bootstrap background 
    
+
    Resampling methods: Bootstrap background 
    
 
 
    
 Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
@@ -636,7 +635,7 @@ estimators.
 
 
 
    
-
    Resampling methods: More Bootstrap background 
    
+
    Resampling methods: More Bootstrap background 
    
 
 
    
 In the case that \( \widehat{\theta} \) has
@@ -659,7 +658,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
 
 
 
    
-
    Resampling methods: Bootstrap approach 
    
+
    Resampling methods: Bootstrap approach 
    
 
 
    
 But
@@ -680,7 +679,7 @@ frequency of the observation \( X_i \), just draw the values
 
 
 
    
-
    Resampling methods: Bootstrap steps 
    
+
    Resampling methods: Bootstrap steps 
    
 
 
    
 The independent bootstrap works like this: 
@@ -706,7 +705,7 @@ example, if you are interested in estimating the variance of \( \widehat
 
 
 
    
-
    Code example for the Bootstrap method 
    
+
    Code example for the Bootstrap method 
    
 
 
    
 The following code starts with a Gaussian distribution with mean value
@@ -725,7 +724,7 @@ theorem.
 
    
 
 
-
    from numpy import *
+
    from numpy import *
 from numpy.random import randint, randn
 from time import time
 import matplotlib.mlab as mlab
@@ -743,9 +742,9 @@ theorem.
         t[i] = statistic(data[randint(0,n,n)])
 
     # analysis    
-    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
-    print("original           bias      std. error")
-    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
+    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
+    print("original           bias      std. error")
+    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
     return t
 
 
@@ -763,7 +762,7 @@ lt = plt.plot(binsboot, y, 'r--', li
 plt.xlabel('Smarts')
 plt.ylabel('Probability')
 plt.axis([99.5, 100.6, 0, 3.0])
-plt.grid(True)
+plt.grid(True)
 
 plt.show()
 
    
@@ -771,7 +770,7 @@ plt.show()
 
 
 
    
-
    Various steps in cross-validation 
    
+
    Various steps in cross-validation 
    
 
 
    
 When the repetitive splitting of the data set is done randomly,
@@ -792,7 +791,7 @@ cross-validation (LOOCV).
 
 
 
    
-
    How to set up the cross-validation for Ridge and/or Lasso 
    
+
    How to set up the cross-validation for Ridge and/or Lasso 
    
 
 
      
 
    •  Define a range of interest for the penalty parameter.
      • 
@@ -826,7 +825,7 @@ $$
 
 
 
      
-
      Cross-validation in brief 
      
+
      Cross-validation in brief 
      
 
 
      
 For the various values of \( k \)
@@ -848,14 +847,14 @@ For the various values of \( k \)
 
 
 
      
-
      Code Example for Cross-validation and \( k \)-fold Cross-validation 
      
+
      Code Example for Cross-validation and \( k \)-fold Cross-validation 
      
 
 
      
 The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
 
      
 
 
-
      import numpy as np
+
      import numpy as np
 import matplotlib.pyplot as plt
 from sklearn.model_selection import KFold
 from sklearn.linear_model import Ridge
@@ -949,7 +948,7 @@ plt.show()
 
 
 
      
-
      The bias-variance tradeoff 
      
+
      The bias-variance tradeoff 
      
 
 
      
 We will discuss the bias-variance tradeoff in the context of
@@ -1027,11 +1026,11 @@ that is the rewriting in terms of the so-called bias, the variance of the model
 
 
 
      
-
      Example code for Bias-Variance tradeoff 
      
+
      Example code for Bias-Variance tradeoff 
      
 
      
 
 
-
      import matplotlib.pyplot as plt
+
      import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
@@ -1055,7 +1054,7 @@ x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=# Combine x transformation and model into one operation.
 # Not neccesary, but convenient.
-model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
 
 # The following (m x n_bootstraps) matrix holds the column vectors y_pred
 # for each bootstrap iteration.
@@ -1072,13 +1071,13 @@ y_pred = np.empty((y_test.shape[0], n_boostr
 # calculated per data point in the test set.
 # Note 2: The use of keepdims=True is important in the calculation of bias as this 
 # maintains the column vector form. Dropping this yields very unexpected results.
-error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-print('Error:', error)
-print('Bias^2:', bias)
-print('Var:', variance)
-print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
+error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+print('Error:', error)
+print('Bias^2:', bias)
+print('Var:', variance)
+print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
 
 plt.plot(x[::5, :], y[::5, :], label='f(x)')
 plt.scatter(x_test, y_test, label='Data points')
@@ -1090,11 +1089,11 @@ plt.show()
 
 
 
      
-
      Understanding what happens 
      
+
      Understanding what happens 
      
 
      
 
 
-
      import matplotlib.pyplot as plt
+
      import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
@@ -1119,21 +1118,21 @@ polydegree = np.zeros(maxdegree)
 x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
 
 for degree in range(maxdegree):
-    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
     y_pred = np.empty((y_test.shape[0], n_boostraps))
     for i in range(n_boostraps):
         x_, y_ = resample(x_train, y_train)
         y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
 
     polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+    print('Polynomial degree:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
 
 plt.plot(polydegree, error, label='Error')
 plt.plot(polydegree, bias, label='bias')
@@ -1145,7 +1144,7 @@ plt.show()
 
 
 
      
-
      Summing up 
      
+
      Summing up 
      
 
 
      
 The bias-variance tradeoff summarizes the fundamental tension in
@@ -1182,11 +1181,11 @@ You may also find this recent Another Example from Scikit-Learn's Repository 
+
      Another Example from Scikit-Learn's Repository 
      
 
      
 
 
-
      """
+
      """
 ============================
 Underfitting vs. Overfitting
 ============================
@@ -1208,7 +1207,7 @@ You may also find this recent training data.
 """
 
-print(__doc__)
+print(__doc__)
 
 import numpy as np
 import matplotlib.pyplot as plt
@@ -1235,7 +1234,7 @@ plt.figure(figsize=(14, False)
+                                             include_bias=False)
     linear_regression = LinearRegression()
     pipeline = Pipeline([("polynomial_features", polynomial_features),
                          ("linear_regression", linear_regression)])
@@ -1262,12 +1261,12 @@ plt.show()
 
 
 
      
-
      More examples on bootstrap and cross-validation and errors 
      
+
      More examples on bootstrap and cross-validation and errors 
      
 
 
      
 
 
-
      # Common imports
+
      # Common imports
 import os
 import numpy as np
 import pandas as pd
@@ -1297,7 +1296,7 @@ DATA_ID = "DataFiles/"
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("EoS.csv"),'r')
 
@@ -1327,7 +1326,7 @@ trials = 100
     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)
+        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)
@@ -1335,9 +1334,9 @@ trials = 100
 
     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])
+    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')
@@ -1350,12 +1349,12 @@ plt.show()
 
 
 
      
-
      The same example but now with cross-validation 
      
+
      The same example but now with cross-validation 
      
 
 
      
 
 
-
      # Common imports
+
      # Common imports
 import os
 import numpy as np
 import pandas as pd
@@ -1387,7 +1386,7 @@ DATA_ID = "DataFiles/"
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("EoS.csv"),'r')
 
@@ -1427,11 +1426,11 @@ plt.show()
 
 
 
      
-
      Cross-validation with Ridge 
      
+
      Cross-validation with Ridge 
      
 
      
 
 
-
      import numpy as np
+
      import numpy as np
 import matplotlib.pyplot as plt
 from sklearn.model_selection import KFold
 from sklearn.linear_model import Ridge
@@ -1470,17 +1469,6 @@ plt.show()
 
      
 
 
-
      
-
      Friday September 4  
      
-
-
      
-Video of Lecture and handwritten notes
-
-
      
-More material will be added here, see handwritten notes also.
-
      
-
-
 
 
       
 
       
diff --git a/doc/pub/week36/html/week36-solarized.html b/doc/pub/week36/html/week36-solarized.html
index b5f639b18..d94b64180 100644
--- a/doc/pub/week36/html/week36-solarized.html
+++ b/doc/pub/week36/html/week36-solarized.html
@@ -1,19 +1,19 @@
 
 
 
 
-
+
 
-
+
 
-Week 36: Resampling techniques and Ordinary Least Square
+Week 36: Statistical interpretation of Linear Regression and Resampling techniques
 
 
-
-
+
+
 
 
 
@@ -48,10 +48,10 @@ pre {
 .alert-block > p, .alert-block > ul {margin-bottom:1em}
 .alert li {margin-top: 1em}
 .alert-block p+p {margin-top:5px}
-.alert-notice { background-image: url(https://cdn.rawgit.com/hplgit/doconce/master/bundled/html_images/small_yellow_notice.png); }
-.alert-summary  { background-image:url(https://cdn.rawgit.com/hplgit/doconce/master/bundled/html_images/small_yellow_summary.png); }
-.alert-warning { background-image: url(https://cdn.rawgit.com/hplgit/doconce/master/bundled/html_images/small_yellow_warning.png); }
-.alert-question {background-image:url(https://cdn.rawgit.com/hplgit/doconce/master/bundled/html_images/small_yellow_question.png); }
+.alert-notice { background-image: url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_yellow_notice.png); }
+.alert-summary  { background-image:url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_yellow_summary.png); }
+.alert-warning { background-image: url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_yellow_warning.png); }
+.alert-question {background-image:url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_yellow_question.png); }
 
 div { text-align: justify; text-justify: inter-word; }
 
@@ -61,72 +61,111 @@ div { text-align: justify; text-justify: inter-word; }
 
 
 
 
@@ -152,7 +191,7 @@ MathJax.Hub.Config({
 
 
 
-
      Week 36: Resampling techniques and Ordinary Least Square
        
+
      Week 36: Statistical interpretation of Linear Regression and Resampling techniques
        
 
 
      
 
@@ -168,149 +207,26 @@ MathJax.Hub.Config({
 
      [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
      
 
      
 
      
-
      Sep 16, 2020
       
+
      Sep 6, 2021
       
 
      
 
      
 









      
 
-
      Plans for week 36 
      
+
      Plans for week 36 
      
 
 
        
-
      •  Thursday: Statistics, probability theory and resampling methods
        • 
-
      •  Friday: Resampling methods and motivation for Ridge Regression
        • 
+
      •  Thursday: Statistics, probability theory and linear regression
        • 
+
      •  Friday: Resampling methods
        • 
 
      
 
 









      
 
-
      Thursday September 3  
      
-
-
      
-Video of Lecture and handwritten notes
-
-
      
-More material will be added here, see handwritten notes also.
-
-
      
-









      
-
-
      Why resampling methods 
      
-
-
      
-Before we proceed, we need to rethink what we have been doing. In our
-eager to fit the data, we have omitted several important elements in
-our regression analysis. In what follows we will
-
-
        
-
      1.  look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
        2. 
-
      2.  introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
        3. 
-
      
-
-This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
-
-
      
-









      
-
-
      Resampling methods 
      
-
      
-
-
      
-Resampling methods are an indispensable tool in modern
-statistics. They involve repeatedly drawing samples from a training
-set and refitting a model of interest on each sample in order to
-obtain additional information about the fitted model. For example, in
-order to estimate the variability of a linear regression fit, we can
-repeatedly draw different samples from the training data, fit a linear
-regression to each new sample, and then examine the extent to which
-the resulting fits differ. Such an approach may allow us to obtain
-information that would not be available from fitting the model only
-once using the original training sample.
-
-
      
-Two resampling methods are often used in Machine Learning analyses,
-
-
        
-
      1.  The bootstrap method
        2. 
-
      2.  and Cross-Validation
        3. 
-
      
-
-In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
-cross-validation and the bootstrap method.
-
-
-
      
-
-
-
      
-









      
-
-
      Resampling approaches can be computationally expensive 
      
-
      
-
-
      
-
-
      
-Resampling approaches can be computationally expensive, because they
-involve fitting the same statistical method multiple times using
-different subsets of the training data. However, due to recent
-advances in computing power, the computational requirements of
-resampling methods generally are not prohibitive. In this chapter, we
-discuss two of the most commonly used resampling methods,
-cross-validation and the bootstrap. Both methods are important tools
-in the practical application of many statistical learning
-procedures. For example, cross-validation can be used to estimate the
-test error associated with a given statistical learning method in
-order to evaluate its performance, or to select the appropriate level
-of flexibility. The process of evaluating a model’s performance is
-known as model assessment, whereas the process of selecting the proper
-level of flexibility for a model is known as model selection. The
-bootstrap is widely used.
-
-
-
      
-
-
-
      
-









      
-
-
      Why resampling methods ? 
      
-
      
-Statistical analysis.
-
      
-
-
        
-
      •  Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods
        • 
-
      •  The results can be analysed with the same statistical tools as we would use analysing experimental data.
        • 
-
      •  As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
        • 
-
      
-
      
-
-
-
      
-









      
-
-
      Statistical analysis 
      
-
      
-
-
      
-
-
        
-
      •  As in other experiments, many numerical  experiments have two classes of errors:
        • 
-
-
          
-  
        •  Statistical errors
          • 
-  
        •  Systematical errors
          • 
-
        
-
-
      •  Statistical errors can be estimated using standard tools from statistics
        • 
-
      •  Systematical errors are method specific and must be treated differently from case to case.
        •  
-
      
-
      
-
+
      Thursday September 9  
      
 
 
      
 
 
-
      Linking the regression analysis with a statistical interpretation 
      
+
      Linking the regression analysis with a statistical interpretation 
      
 
 
      
 The
@@ -347,7 +263,7 @@ row number \( i \) and perform a sum over all values \( p \).
 
      
 









      
 
-
      Assumptions made 
      
+
      Assumptions made 
      
 
 
      
 The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)
@@ -367,7 +283,7 @@ $$
 
      
 









      
 
-
      Expectation value and variance 
      
+
      Expectation value and variance 
      
 
 
      
 We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) 
@@ -402,7 +318,7 @@ mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (n
 
      
 









      
 
-
      Expectation value and variance for \( \boldsymbol{\beta} \) 
      
+
      Expectation value and variance for \( \boldsymbol{\beta} \) 
      
 
 
      
 With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
@@ -487,7 +403,129 @@ This means the variance we obtain with the standard OLS will always for \( \lamb
 
      
 









      
 
-
      Resampling methods 
      
+
      Friday September 10  
      
+
+
      
+









      
+
+
      Why resampling methods 
      
+
+
      
+Before we proceed, we need to rethink what we have been doing. In our
+eager to fit the data, we have omitted several important elements in
+our regression analysis. In what follows we will
+
+
        
+
      1.  look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
        2. 
+
      2.  introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
        3. 
+
      
+
+This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
+
+
      
+









      
+
+
      Resampling methods 
      
+
      
+
+
      
+Resampling methods are an indispensable tool in modern
+statistics. They involve repeatedly drawing samples from a training
+set and refitting a model of interest on each sample in order to
+obtain additional information about the fitted model. For example, in
+order to estimate the variability of a linear regression fit, we can
+repeatedly draw different samples from the training data, fit a linear
+regression to each new sample, and then examine the extent to which
+the resulting fits differ. Such an approach may allow us to obtain
+information that would not be available from fitting the model only
+once using the original training sample.
+
+
      
+Two resampling methods are often used in Machine Learning analyses,
+
+
        
+
      1.  The bootstrap method
        2. 
+
      2.  and Cross-Validation
        3. 
+
      
+
+In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
+cross-validation and the bootstrap method.
+
+
+
      
+
+
+
      
+









      
+
+
      Resampling approaches can be computationally expensive 
      
+
      
+
+
      
+
+
      
+Resampling approaches can be computationally expensive, because they
+involve fitting the same statistical method multiple times using
+different subsets of the training data. However, due to recent
+advances in computing power, the computational requirements of
+resampling methods generally are not prohibitive. In this chapter, we
+discuss two of the most commonly used resampling methods,
+cross-validation and the bootstrap. Both methods are important tools
+in the practical application of many statistical learning
+procedures. For example, cross-validation can be used to estimate the
+test error associated with a given statistical learning method in
+order to evaluate its performance, or to select the appropriate level
+of flexibility. The process of evaluating a model’s performance is
+known as model assessment, whereas the process of selecting the proper
+level of flexibility for a model is known as model selection. The
+bootstrap is widely used.
+
+
+
      
+
+
+
      
+









      
+
+
      Why resampling methods ? 
      
+
      
+Statistical analysis
+
      
+
+
        
+
      •  Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods
        • 
+
      •  The results can be analysed with the same statistical tools as we would use analysing experimental data.
        • 
+
      •  As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
        • 
+
      
+
      
+
+
+
      
+









      
+
+
      Statistical analysis 
      
+
      
+
+
      
+
+
        
+
      •  As in other experiments, many numerical  experiments have two classes of errors:
        • 
+
+
          
+  
        •  Statistical errors
          • 
+  
        •  Systematical errors
          • 
+
        
+
+
      •  Statistical errors can be estimated using standard tools from statistics
        • 
+
      •  Systematical errors are method specific and must be treated differently from case to case.
        •  
+
      
+
      
+
+
+
      
+









      
+
+
      Resampling methods 
      
 
 
      
 With all these analytical equations for both the OLS and Ridge
@@ -516,7 +554,7 @@ training error reaches a saturation.
 
      
 









      
 
-
      Resampling methods: Jackknife and Bootstrap 
      
+
      Resampling methods: Jackknife and Bootstrap 
      
 
 
      
 Two famous
@@ -539,7 +577,7 @@ need for bootstrapping.
 
      
 









      
 
-
      Resampling methods: Jackknife 
      
+
      Resampling methods: Jackknife 
      
 
 
      
 The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). 
@@ -558,11 +596,11 @@ number \( i \) is left out. Using this notation, define
 
      
 









      
 
-
      Jackknife code example 
      
+
      Jackknife code example 
      
 
      
 
 
-
      from numpy import *
+
      from numpy import *
 from numpy.random import randint, randn
 from time import time
 
@@ -573,9 +611,9 @@ number \( i \) is left out. Using this notation, define
         t[i] = stat(delete(data,i) )
 
     # analysis                                                                                                                                                                     
-    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
-    print("original           bias      std. error")
-    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
+    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
+    print("original           bias      std. error")
+    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
 
     return t
 
@@ -594,7 +632,7 @@ t = jackknife(x, stat)
 
      
 









      
 
-
      Resampling methods: Bootstrap 
      
+
      Resampling methods: Bootstrap 
      
 
      
 
 
      
@@ -615,7 +653,7 @@ advantages:
 
      
 









      
 
-
      Resampling methods: Bootstrap background 
      
+
      Resampling methods: Bootstrap background 
      
 
 
      
 Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
@@ -631,7 +669,7 @@ estimators.
 
      
 









      
 
-
      Resampling methods: More Bootstrap background 
      
+
      Resampling methods: More Bootstrap background 
      
 
 
      
 In the case that \( \widehat{\theta} \) has
@@ -653,7 +691,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
 
      
 









      
 
-
      Resampling methods: Bootstrap approach 
      
+
      Resampling methods: Bootstrap approach 
      
 
 
      
 But
@@ -674,7 +712,7 @@ frequency of the observation \( X_i \), just draw the values
 
      
 









      
 
-
      Resampling methods: Bootstrap steps 
      
+
      Resampling methods: Bootstrap steps 
      
 
 
      
 The independent bootstrap works like this: 
@@ -699,7 +737,7 @@ example, if you are interested in estimating the variance of \( \widehat
 
      
 









      
 
-
      Code example for the Bootstrap method 
      
+
      Code example for the Bootstrap method 
      
 
 
      
 The following code starts with a Gaussian distribution with mean value
@@ -718,7 +756,7 @@ theorem.
 
      
 
 
-
      from numpy import *
+
      from numpy import *
 from numpy.random import randint, randn
 from time import time
 import matplotlib.mlab as mlab
@@ -736,9 +774,9 @@ theorem.
         t[i] = statistic(data[randint(0,n,n)])
 
     # analysis    
-    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
-    print("original           bias      std. error")
-    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
+    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
+    print("original           bias      std. error")
+    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
     return t
 
 
@@ -756,14 +794,14 @@ lt = plt.plot(binsboot, y, 'r--', li
 plt.xlabel('Smarts')
 plt.ylabel('Probability')
 plt.axis([99.5, 100.6, 0, 3.0])
-plt.grid(True)
+plt.grid(True)
 
 plt.show()
 
      
 
      
 
 
-
      Various steps in cross-validation 
      
+
      Various steps in cross-validation 
      
 
 
      
 When the repetitive splitting of the data set is done randomly,
@@ -784,7 +822,7 @@ cross-validation (LOOCV).
 
      
 
 
-
      How to set up the cross-validation for Ridge and/or Lasso 
      
+
      How to set up the cross-validation for Ridge and/or Lasso 
      
 
 
        
 
      •  Define a range of interest for the penalty parameter.
        • 
@@ -816,7 +854,7 @@ $$
 
        
 









        
 
-
        Cross-validation in brief 
        
+
        Cross-validation in brief 
        
 
 
        
 For the various values of \( k \)
@@ -838,14 +876,14 @@ For the various values of \( k \)
 
 









        
 
-
        Code Example for Cross-validation and \( k \)-fold Cross-validation 
        
+
        Code Example for Cross-validation and \( k \)-fold Cross-validation 
        
 
 
        
 The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
 
        
 
 
-
        import numpy as np
+
        import numpy as np
 import matplotlib.pyplot as plt
 from sklearn.model_selection import KFold
 from sklearn.linear_model import Ridge
@@ -938,7 +976,7 @@ plt.show()
 
        
 









        
 
-
        The bias-variance tradeoff 
        
+
        The bias-variance tradeoff 
        
 
 
        
 We will discuss the bias-variance tradeoff in the context of
@@ -1004,11 +1042,11 @@ that is the rewriting in terms of the so-called bias, the variance of the model
 
        
 









        
 
-
        Example code for Bias-Variance tradeoff 
        
+
        Example code for Bias-Variance tradeoff 
        
 
        
 
 
-
        import matplotlib.pyplot as plt
+
        import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
@@ -1032,7 +1070,7 @@ x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=# Combine x transformation and model into one operation.
 # Not neccesary, but convenient.
-model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
 
 # The following (m x n_bootstraps) matrix holds the column vectors y_pred
 # for each bootstrap iteration.
@@ -1049,13 +1087,13 @@ y_pred = np.empty((y_test.shape[0], n_boostr
 # calculated per data point in the test set.
 # Note 2: The use of keepdims=True is important in the calculation of bias as this 
 # maintains the column vector form. Dropping this yields very unexpected results.
-error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-print('Error:', error)
-print('Bias^2:', bias)
-print('Var:', variance)
-print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
+error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+print('Error:', error)
+print('Bias^2:', bias)
+print('Var:', variance)
+print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
 
 plt.plot(x[::5, :], y[::5, :], label='f(x)')
 plt.scatter(x_test, y_test, label='Data points')
@@ -1066,11 +1104,11 @@ plt.show()
 
        
 









        
 
-
        Understanding what happens 
        
+
        Understanding what happens 
        
 
        
 
 
-
        import matplotlib.pyplot as plt
+
        import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
@@ -1095,21 +1133,21 @@ polydegree = np.zeros(maxdegree)
 x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
 
 for degree in range(maxdegree):
-    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
     y_pred = np.empty((y_test.shape[0], n_boostraps))
     for i in range(n_boostraps):
         x_, y_ = resample(x_train, y_train)
         y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
 
     polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+    print('Polynomial degree:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
 
 plt.plot(polydegree, error, label='Error')
 plt.plot(polydegree, bias, label='bias')
@@ -1120,7 +1158,7 @@ plt.show()
 
        
 
 
-
        Summing up 
        
+
        Summing up 
        
 
 
        
 The bias-variance tradeoff summarizes the fundamental tension in
@@ -1157,11 +1195,11 @@ You may also find this recent Another Example from Scikit-Learn's Repository 
+
        Another Example from Scikit-Learn's Repository 
        
 
        
 
 
-
        """
+
        """
 ============================
 Underfitting vs. Overfitting
 ============================
@@ -1183,7 +1221,7 @@ You may also find this recent training data.
 """
 
-print(__doc__)
+print(__doc__)
 
 import numpy as np
 import matplotlib.pyplot as plt
@@ -1210,7 +1248,7 @@ plt.figure(figsize=(14, False)
+                                             include_bias=False)
     linear_regression = LinearRegression()
     pipeline = Pipeline([("polynomial_features", polynomial_features),
                          ("linear_regression", linear_regression)])
@@ -1236,12 +1274,12 @@ plt.show()
 
        
 









        
 
-
        More examples on bootstrap and cross-validation and errors 
        
+
        More examples on bootstrap and cross-validation and errors 
        
 
 
        
 
 
-
        # Common imports
+
        # Common imports
 import os
 import numpy as np
 import pandas as pd
@@ -1271,7 +1309,7 @@ DATA_ID = "DataFiles/"
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("EoS.csv"),'r')
 
@@ -1301,7 +1339,7 @@ trials = 100
     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)
+        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)
@@ -1309,9 +1347,9 @@ trials = 100
 
     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])
+    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')
@@ -1323,12 +1361,12 @@ plt.show()
 
        
 
 
-
        The same example but now with cross-validation 
        
+
        The same example but now with cross-validation 
        
 
 
        
 
 
-
        # Common imports
+
        # Common imports
 import os
 import numpy as np
 import pandas as pd
@@ -1360,7 +1398,7 @@ DATA_ID = "DataFiles/"
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("EoS.csv"),'r')
 
@@ -1399,11 +1437,11 @@ plt.show()
 
        
 









        
 
-
        Cross-validation with Ridge 
        
+
        Cross-validation with Ridge 
        
 
        
 
 
-
        import numpy as np
+
        import numpy as np
 import matplotlib.pyplot as plt
 from sklearn.model_selection import KFold
 from sklearn.linear_model import Ridge
@@ -1439,24 +1477,13 @@ plt.ylabel('MSE')
 plt.legend()
 plt.show()
 
        
-
        
-









        
-
-
        Friday September 4  
        
-
-
        
-Video of Lecture and handwritten notes
-
-
        
-More material will be added here, see handwritten notes also.
-
 
        
 
 
 
 
 
        
- © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+ © 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 
        
 
 
diff --git a/doc/pub/week36/html/week36.html b/doc/pub/week36/html/week36.html
index 9f4b8c105..49f70c3af 100644
--- a/doc/pub/week36/html/week36.html
+++ b/doc/pub/week36/html/week36.html
@@ -1,15 +1,15 @@
 
 
 
 
-
+
 
-
+
 
-Week 36: Resampling techniques and Ordinary Least Square
+Week 36: Statistical interpretation of Linear Regression and Resampling techniques
 
 
 
@@ -66,72 +66,111 @@ div { text-align: justify; text-justify: inter-word; }
 
 
 
 
@@ -157,7 +196,7 @@ MathJax.Hub.Config({
 
 
 
-
        Week 36: Resampling techniques and Ordinary Least Square
          
+
        Week 36: Statistical interpretation of Linear Regression and Resampling techniques
          
 
 
        
 
@@ -173,149 +212,26 @@ MathJax.Hub.Config({
 
        [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
        
 
        
 
        
-
        Sep 16, 2020
         
+
        Sep 6, 2021
         
 
        
 
        
 









        
 
-
        Plans for week 36 
        
+
        Plans for week 36 
        
 
 
          
-
        •  Thursday: Statistics, probability theory and resampling methods
          • 
-
        •  Friday: Resampling methods and motivation for Ridge Regression
          • 
+
        •  Thursday: Statistics, probability theory and linear regression
          • 
+
        •  Friday: Resampling methods
          • 
 
        
 
 









        
 
-
        Thursday September 3  
        
-
-
        
-Video of Lecture and handwritten notes
-
-
        
-More material will be added here, see handwritten notes also.
-
-
        
-









        
-
-
        Why resampling methods 
        
-
-
        
-Before we proceed, we need to rethink what we have been doing. In our
-eager to fit the data, we have omitted several important elements in
-our regression analysis. In what follows we will
-
-
          
-
        1.  look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
          2. 
-
        2.  introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
          3. 
-
        
-
-This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
-
-
        
-









        
-
-
        Resampling methods 
        
-
        
-
-
        
-Resampling methods are an indispensable tool in modern
-statistics. They involve repeatedly drawing samples from a training
-set and refitting a model of interest on each sample in order to
-obtain additional information about the fitted model. For example, in
-order to estimate the variability of a linear regression fit, we can
-repeatedly draw different samples from the training data, fit a linear
-regression to each new sample, and then examine the extent to which
-the resulting fits differ. Such an approach may allow us to obtain
-information that would not be available from fitting the model only
-once using the original training sample.
-
-
        
-Two resampling methods are often used in Machine Learning analyses,
-
-
          
-
        1.  The bootstrap method
          2. 
-
        2.  and Cross-Validation
          3. 
-
        
-
-In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
-cross-validation and the bootstrap method.
-
-
-
        
-
-
-
        
-









        
-
-
        Resampling approaches can be computationally expensive 
        
-
        
-
-
        
-
-
        
-Resampling approaches can be computationally expensive, because they
-involve fitting the same statistical method multiple times using
-different subsets of the training data. However, due to recent
-advances in computing power, the computational requirements of
-resampling methods generally are not prohibitive. In this chapter, we
-discuss two of the most commonly used resampling methods,
-cross-validation and the bootstrap. Both methods are important tools
-in the practical application of many statistical learning
-procedures. For example, cross-validation can be used to estimate the
-test error associated with a given statistical learning method in
-order to evaluate its performance, or to select the appropriate level
-of flexibility. The process of evaluating a model’s performance is
-known as model assessment, whereas the process of selecting the proper
-level of flexibility for a model is known as model selection. The
-bootstrap is widely used.
-
-
-
        
-
-
-
        
-









        
-
-
        Why resampling methods ? 
        
-
        
-Statistical analysis.
-
        
-
-
          
-
        •  Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods
          • 
-
        •  The results can be analysed with the same statistical tools as we would use analysing experimental data.
          • 
-
        •  As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
          • 
-
        
-
        
-
-
-
        
-









        
-
-
        Statistical analysis 
        
-
        
-
-
        
-
-
          
-
        •  As in other experiments, many numerical  experiments have two classes of errors:
          • 
-
-
            
-  
          •  Statistical errors
            • 
-  
          •  Systematical errors
            • 
-
          
-
-
        •  Statistical errors can be estimated using standard tools from statistics
          • 
-
        •  Systematical errors are method specific and must be treated differently from case to case.
          •  
-
        
-
        
-
+
        Thursday September 9  
        
 
 
        
 
 
-
        Linking the regression analysis with a statistical interpretation 
        
+
        Linking the regression analysis with a statistical interpretation 
        
 
 
        
 The
@@ -352,7 +268,7 @@ row number \( i \) and perform a sum over all values \( p \).
 
        
 









        
 
-
        Assumptions made 
        
+
        Assumptions made 
        
 
 
        
 The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)
@@ -372,7 +288,7 @@ $$
 
        
 









        
 
-
        Expectation value and variance 
        
+
        Expectation value and variance 
        
 
 
        
 We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) 
@@ -407,7 +323,7 @@ mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (n
 
        
 









        
 
-
        Expectation value and variance for \( \boldsymbol{\beta} \) 
        
+
        Expectation value and variance for \( \boldsymbol{\beta} \) 
        
 
 
        
 With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
@@ -492,7 +408,129 @@ This means the variance we obtain with the standard OLS will always for \( \lamb
 
        
 









        
 
-
        Resampling methods 
        
+
        Friday September 10  
        
+
+
        
+









        
+
+
        Why resampling methods 
        
+
+
        
+Before we proceed, we need to rethink what we have been doing. In our
+eager to fit the data, we have omitted several important elements in
+our regression analysis. In what follows we will
+
+
          
+
        1.  look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
          2. 
+
        2.  introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
          3. 
+
        
+
+This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
+
+
        
+









        
+
+
        Resampling methods 
        
+
        
+
+
        
+Resampling methods are an indispensable tool in modern
+statistics. They involve repeatedly drawing samples from a training
+set and refitting a model of interest on each sample in order to
+obtain additional information about the fitted model. For example, in
+order to estimate the variability of a linear regression fit, we can
+repeatedly draw different samples from the training data, fit a linear
+regression to each new sample, and then examine the extent to which
+the resulting fits differ. Such an approach may allow us to obtain
+information that would not be available from fitting the model only
+once using the original training sample.
+
+
        
+Two resampling methods are often used in Machine Learning analyses,
+
+
          
+
        1.  The bootstrap method
          2. 
+
        2.  and Cross-Validation
          3. 
+
        
+
+In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
+cross-validation and the bootstrap method.
+
+
+
        
+
+
+
        
+









        
+
+
        Resampling approaches can be computationally expensive 
        
+
        
+
+
        
+
+
        
+Resampling approaches can be computationally expensive, because they
+involve fitting the same statistical method multiple times using
+different subsets of the training data. However, due to recent
+advances in computing power, the computational requirements of
+resampling methods generally are not prohibitive. In this chapter, we
+discuss two of the most commonly used resampling methods,
+cross-validation and the bootstrap. Both methods are important tools
+in the practical application of many statistical learning
+procedures. For example, cross-validation can be used to estimate the
+test error associated with a given statistical learning method in
+order to evaluate its performance, or to select the appropriate level
+of flexibility. The process of evaluating a model’s performance is
+known as model assessment, whereas the process of selecting the proper
+level of flexibility for a model is known as model selection. The
+bootstrap is widely used.
+
+
+
        
+
+
+
        
+









        
+
+
        Why resampling methods ? 
        
+
        
+Statistical analysis
+
        
+
+
          
+
        •  Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods
          • 
+
        •  The results can be analysed with the same statistical tools as we would use analysing experimental data.
          • 
+
        •  As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
          • 
+
        
+
        
+
+
+
        
+









        
+
+
        Statistical analysis 
        
+
        
+
+
        
+
+
          
+
        •  As in other experiments, many numerical  experiments have two classes of errors:
          • 
+
+
            
+  
          •  Statistical errors
            • 
+  
          •  Systematical errors
            • 
+
          
+
+
        •  Statistical errors can be estimated using standard tools from statistics
          • 
+
        •  Systematical errors are method specific and must be treated differently from case to case.
          •  
+
        
+
        
+
+
+
        
+









        
+
+
        Resampling methods 
        
 
 
        
 With all these analytical equations for both the OLS and Ridge
@@ -521,7 +559,7 @@ training error reaches a saturation.
 
        
 









        
 
-
        Resampling methods: Jackknife and Bootstrap 
        
+
        Resampling methods: Jackknife and Bootstrap 
        
 
 
        
 Two famous
@@ -544,7 +582,7 @@ need for bootstrapping.
 
        
 









        
 
-
        Resampling methods: Jackknife 
        
+
        Resampling methods: Jackknife 
        
 
 
        
 The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \). 
@@ -563,11 +601,11 @@ number \( i \) is left out. Using this notation, define
 
        
 









        
 
-
        Jackknife code example 
        
+
        Jackknife code example 
        
 
        
 
 
-
        from numpy import *
+
        from numpy import *
 from numpy.random import randint, randn
 from time import time
 
@@ -578,9 +616,9 @@ number \( i \) is left out. Using this notation, define
         t[i] = stat(delete(data,i) )
 
     # analysis                                                                                                                                                                     
-    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
-    print("original           bias      std. error")
-    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
+    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
+    print("original           bias      std. error")
+    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
 
     return t
 
@@ -599,7 +637,7 @@ t = jackknife(x, stat)
 
        
 









        
 
-
        Resampling methods: Bootstrap 
        
+
        Resampling methods: Bootstrap 
        
 
        
 
 
        
@@ -620,7 +658,7 @@ advantages:
 
        
 









        
 
-
        Resampling methods: Bootstrap background 
        
+
        Resampling methods: Bootstrap background 
        
 
 
        
 Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
@@ -636,7 +674,7 @@ estimators.
 
        
 









        
 
-
        Resampling methods: More Bootstrap background 
        
+
        Resampling methods: More Bootstrap background 
        
 
 
        
 In the case that \( \widehat{\theta} \) has
@@ -658,7 +696,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
 
        
 









        
 
-
        Resampling methods: Bootstrap approach 
        
+
        Resampling methods: Bootstrap approach 
        
 
 
        
 But
@@ -679,7 +717,7 @@ frequency of the observation \( X_i \), just draw the values
 
        
 









        
 
-
        Resampling methods: Bootstrap steps 
        
+
        Resampling methods: Bootstrap steps 
        
 
 
        
 The independent bootstrap works like this: 
@@ -704,7 +742,7 @@ example, if you are interested in estimating the variance of \( \widehat
 
        
 









        
 
-
        Code example for the Bootstrap method 
        
+
        Code example for the Bootstrap method 
        
 
 
        
 The following code starts with a Gaussian distribution with mean value
@@ -723,7 +761,7 @@ theorem.
 
        
 
 
-
        from numpy import *
+
        from numpy import *
 from numpy.random import randint, randn
 from time import time
 import matplotlib.mlab as mlab
@@ -741,9 +779,9 @@ theorem.
         t[i] = statistic(data[randint(0,n,n)])
 
     # analysis    
-    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
-    print("original           bias      std. error")
-    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
+    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
+    print("original           bias      std. error")
+    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
     return t
 
 
@@ -761,14 +799,14 @@ lt = plt..xlabel('Smarts')
 plt.ylabel('Probability')
 plt.axis([99.5, 100.6, 0, 3.0])
-plt.grid(True)
+plt.grid(True)
 
 plt.show()
 
        
 
        
 
 
-
        Various steps in cross-validation 
        
+
        Various steps in cross-validation 
        
 
 
        
 When the repetitive splitting of the data set is done randomly,
@@ -789,7 +827,7 @@ cross-validation (LOOCV).
 
        
 
 
-
        How to set up the cross-validation for Ridge and/or Lasso 
        
+
        How to set up the cross-validation for Ridge and/or Lasso 
        
 
 
          
 
        •  Define a range of interest for the penalty parameter.
          • 
@@ -821,7 +859,7 @@ $$
 
          
 









          
 
-
          Cross-validation in brief 
          
+
          Cross-validation in brief 
          
 
 
          
 For the various values of \( k \)
@@ -843,14 +881,14 @@ For the various values of \( k \)
 
 









          
 
-
          Code Example for Cross-validation and \( k \)-fold Cross-validation 
          
+
          Code Example for Cross-validation and \( k \)-fold Cross-validation 
          
 
 
          
 The code here uses Ridge regression with cross-validation (CV)  resampling and \( k \)-fold CV in order to fit a specific polynomial. 
 
          
 
 
-
          import numpy as np
+
          import numpy as np
 import matplotlib.pyplot as plt
 from sklearn.model_selection import KFold
 from sklearn.linear_model import Ridge
@@ -943,7 +981,7 @@ plt.show()
 
          
 









          
 
-
          The bias-variance tradeoff 
          
+
          The bias-variance tradeoff 
          
 
 
          
 We will discuss the bias-variance tradeoff in the context of
@@ -1009,11 +1047,11 @@ that is the rewriting in terms of the so-called bias, the variance of the model
 
          
 









          
 
-
          Example code for Bias-Variance tradeoff 
          
+
          Example code for Bias-Variance tradeoff 
          
 
          
 
 
-
          import matplotlib.pyplot as plt
+
          import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
@@ -1037,7 +1075,7 @@ x_train, x_test, y_train, y_test = train_tes
 
 # Combine x transformation and model into one operation.
 # Not neccesary, but convenient.
-model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
 
 # The following (m x n_bootstraps) matrix holds the column vectors y_pred
 # for each bootstrap iteration.
@@ -1054,13 +1092,13 @@ y_pred = np.# calculated per data point in the test set.
 # Note 2: The use of keepdims=True is important in the calculation of bias as this 
 # maintains the column vector form. Dropping this yields very unexpected results.
-error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-print('Error:', error)
-print('Bias^2:', bias)
-print('Var:', variance)
-print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
+error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+print('Error:', error)
+print('Bias^2:', bias)
+print('Var:', variance)
+print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
 
 plt.plot(x[::5, :], y[::5, :], label='f(x)')
 plt.scatter(x_test, y_test, label='Data points')
@@ -1071,11 +1109,11 @@ plt.show()
 
          
 









          
 
-
          Understanding what happens 
          
+
          Understanding what happens 
          
 
          
 
 
-
          import matplotlib.pyplot as plt
+
          import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
@@ -1100,21 +1138,21 @@ polydegree = np
 x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
 
 for degree in range(maxdegree):
-    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
     y_pred = np.empty((y_test.shape[0], n_boostraps))
     for i in range(n_boostraps):
         x_, y_ = resample(x_train, y_train)
         y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
 
     polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+    print('Polynomial degree:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
 
 plt.plot(polydegree, error, label='Error')
 plt.plot(polydegree, bias, label='bias')
@@ -1125,7 +1163,7 @@ plt.show()
 
          
 
 
-
          Summing up 
          
+
          Summing up 
          
 
 
          
 The bias-variance tradeoff summarizes the fundamental tension in
@@ -1162,11 +1200,11 @@ You may also find this recent Another Example from Scikit-Learn's Repository 
+
          Another Example from Scikit-Learn's Repository 
          
 
          
 
 
-
          """
+
          """
 ============================
 Underfitting vs. Overfitting
 ============================
@@ -1188,7 +1226,7 @@ You may also find this recent training data.
 """
 
-print(__doc__)
+print(__doc__)
 
 import numpy as np
 import matplotlib.pyplot as plt
@@ -1215,7 +1253,7 @@ plt.figure(figsize.setp(ax, xticks=(), yticks=())
 
     polynomial_features = PolynomialFeatures(degree=degrees[i],
-                                             include_bias=False)
+                                             include_bias=False)
     linear_regression = LinearRegression()
     pipeline = Pipeline([("polynomial_features", polynomial_features),
                          ("linear_regression", linear_regression)])
@@ -1234,19 +1272,19 @@ plt.figure(figsize.xlim((0, 1))
     plt.ylim((-2, 2))
     plt.legend(loc="best")
-    plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
+    plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
         degrees[i], -scores.mean(), scores.std()))
 plt.show()
 
          
 
          
 









          
 
-
          More examples on bootstrap and cross-validation and errors 
          
+
          More examples on bootstrap and cross-validation and errors 
          
 
 
          
 
 
-
          # Common imports
+
          # Common imports
 import os
 import numpy as np
 import pandas as pd
@@ -1276,7 +1314,7 @@ DATA_ID = "
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("EoS.csv"),'r')
 
@@ -1306,7 +1344,7 @@ trials = 100= 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)
+        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)
@@ -1314,9 +1352,9 @@ trials = 100/= 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])
+    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')
@@ -1328,12 +1366,12 @@ plt.show()
 
          
 
 
-
          The same example but now with cross-validation 
          
+
          The same example but now with cross-validation 
          
 
 
          
 
 
-
          # Common imports
+
          # Common imports
 import os
 import numpy as np
 import pandas as pd
@@ -1365,7 +1403,7 @@ DATA_ID = "
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("EoS.csv"),'r')
 
@@ -1404,11 +1442,11 @@ plt.show()
 
          
 









          
 
-
          Cross-validation with Ridge 
          
+
          Cross-validation with Ridge 
          
 
          
 
 
-
          import numpy as np
+
          import numpy as np
 import matplotlib.pyplot as plt
 from sklearn.model_selection import KFold
 from sklearn.linear_model import Ridge
@@ -1444,24 +1482,13 @@ plt.ylabel('
 plt.legend()
 plt.show()
 
          
-
          
-









          
-
-
          Friday September 4  
          
-
-
          
-Video of Lecture and handwritten notes
-
-
          
-More material will be added here, see handwritten notes also.
-
 
          
 
 
 
 
 
          
- © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+ © 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 
          
 
 
diff --git a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz
index 30952a3e0b798eae07e8a7d7362cf0e6291d1071..a55297b25bc3bc3b6764afc0f5c9a8b7d6f7e2ba 100644
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diff --git a/doc/pub/week36/ipynb/week36.ipynb b/doc/pub/week36/ipynb/week36.ipynb
index 4be21a655..d21821d08 100644
--- a/doc/pub/week36/ipynb/week36.ipynb
+++ b/doc/pub/week36/ipynb/week36.ipynb
@@ -4,118 +4,26 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "\n",
-    "# Week 36: Resampling techniques and Ordinary Least Square\n",
+    "\n",
+    "# Week 36: Statistical interpretation of Linear Regression and Resampling techniques\n",
     "\n",
     "  \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 16, 2020**\n",
+    "Date: **Sep 6, 2021**\n",
     "\n",
-    "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
+    "Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
     "\n",
     "\n",
     "\n",
     "\n",
     "## Plans for week 36\n",
     "\n",
-    "* Thursday: Statistics, probability theory and resampling methods\n",
-    "\n",
-    "* Friday: Resampling methods and motivation for Ridge Regression\n",
-    "\n",
-    "## Thursday September 3\n",
-    "\n",
-    "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSept3.mp4?vrtx=view-as-webpage) and [handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember3.pdf)\n",
-    "\n",
-    "More material will be added here, see handwritten notes also.\n",
-    "\n",
-    "\n",
-    "## Why resampling methods\n",
-    "\n",
-    "Before we proceed, we need to rethink what we have been doing. In our\n",
-    "eager to fit the data, we have omitted several important elements in\n",
-    "our regression analysis. In what follows we will\n",
-    "1. look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff\n",
-    "\n",
-    "2. introduce resampling techniques like cross-validation, bootstrapping and jackknife and more\n",
-    "\n",
-    "This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods. \n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "## Resampling methods\n",
-    "Resampling methods are an indispensable tool in modern\n",
-    "statistics. They involve repeatedly drawing samples from a training\n",
-    "set and refitting a model of interest on each sample in order to\n",
-    "obtain additional information about the fitted model. For example, in\n",
-    "order to estimate the variability of a linear regression fit, we can\n",
-    "repeatedly draw different samples from the training data, fit a linear\n",
-    "regression to each new sample, and then examine the extent to which\n",
-    "the resulting fits differ. Such an approach may allow us to obtain\n",
-    "information that would not be available from fitting the model only\n",
-    "once using the original training sample.\n",
-    "\n",
-    "Two resampling methods are often used in Machine Learning analyses,\n",
-    "1. The **bootstrap method**\n",
-    "\n",
-    "2. and **Cross-Validation**\n",
-    "\n",
-    "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n",
-    "cross-validation and the bootstrap method.\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "## Resampling approaches can be computationally expensive\n",
-    "\n",
-    "Resampling approaches can be computationally expensive, because they\n",
-    "involve fitting the same statistical method multiple times using\n",
-    "different subsets of the training data. However, due to recent\n",
-    "advances in computing power, the computational requirements of\n",
-    "resampling methods generally are not prohibitive. In this chapter, we\n",
-    "discuss two of the most commonly used resampling methods,\n",
-    "cross-validation and the bootstrap. Both methods are important tools\n",
-    "in the practical application of many statistical learning\n",
-    "procedures. For example, cross-validation can be used to estimate the\n",
-    "test error associated with a given statistical learning method in\n",
-    "order to evaluate its performance, or to select the appropriate level\n",
-    "of flexibility. The process of evaluating a model’s performance is\n",
-    "known as model assessment, whereas the process of selecting the proper\n",
-    "level of flexibility for a model is known as model selection. The\n",
-    "bootstrap is widely used.\n",
-    "\n",
-    "\n",
-    "\n",
-    "## Why resampling methods ?\n",
-    "**Statistical analysis.**\n",
-    "\n",
-    "\n",
-    "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n",
-    "\n",
-    "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n",
-    "\n",
-    "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n",
-    "\n",
-    "    \n",
-    "\n",
-    "## Statistical analysis\n",
-    "\n",
-    "* As in other experiments, many numerical  experiments have two classes of errors:\n",
-    "\n",
-    "  * Statistical errors\n",
-    "\n",
-    "  * Systematical errors\n",
-    "\n",
-    "\n",
-    "* Statistical errors can be estimated using standard tools from statistics\n",
-    "\n",
-    "* Systematical errors are method specific and must be treated differently from case to case.\n",
-    "\n",
-    "    \n",
+    "* Thursday: Statistics, probability theory and linear regression\n",
     "\n",
+    "* Friday: Resampling methods\n",
     "\n",
+    "## Thursday September 9\n",
     "\n",
     "\n",
     "\n",
@@ -383,6 +291,102 @@
     "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. \n",
     "\n",
     "\n",
+    "\n",
+    "\n",
+    "\n",
+    "## Friday September 10\n",
+    "\n",
+    "\n",
+    "\n",
+    "## Why resampling methods\n",
+    "\n",
+    "Before we proceed, we need to rethink what we have been doing. In our\n",
+    "eager to fit the data, we have omitted several important elements in\n",
+    "our regression analysis. In what follows we will\n",
+    "1. look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff\n",
+    "\n",
+    "2. introduce resampling techniques like cross-validation, bootstrapping and jackknife and more\n",
+    "\n",
+    "This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods. \n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "## Resampling methods\n",
+    "Resampling methods are an indispensable tool in modern\n",
+    "statistics. They involve repeatedly drawing samples from a training\n",
+    "set and refitting a model of interest on each sample in order to\n",
+    "obtain additional information about the fitted model. For example, in\n",
+    "order to estimate the variability of a linear regression fit, we can\n",
+    "repeatedly draw different samples from the training data, fit a linear\n",
+    "regression to each new sample, and then examine the extent to which\n",
+    "the resulting fits differ. Such an approach may allow us to obtain\n",
+    "information that would not be available from fitting the model only\n",
+    "once using the original training sample.\n",
+    "\n",
+    "Two resampling methods are often used in Machine Learning analyses,\n",
+    "1. The **bootstrap method**\n",
+    "\n",
+    "2. and **Cross-Validation**\n",
+    "\n",
+    "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n",
+    "cross-validation and the bootstrap method.\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "## Resampling approaches can be computationally expensive\n",
+    "\n",
+    "Resampling approaches can be computationally expensive, because they\n",
+    "involve fitting the same statistical method multiple times using\n",
+    "different subsets of the training data. However, due to recent\n",
+    "advances in computing power, the computational requirements of\n",
+    "resampling methods generally are not prohibitive. In this chapter, we\n",
+    "discuss two of the most commonly used resampling methods,\n",
+    "cross-validation and the bootstrap. Both methods are important tools\n",
+    "in the practical application of many statistical learning\n",
+    "procedures. For example, cross-validation can be used to estimate the\n",
+    "test error associated with a given statistical learning method in\n",
+    "order to evaluate its performance, or to select the appropriate level\n",
+    "of flexibility. The process of evaluating a model’s performance is\n",
+    "known as model assessment, whereas the process of selecting the proper\n",
+    "level of flexibility for a model is known as model selection. The\n",
+    "bootstrap is widely used.\n",
+    "\n",
+    "\n",
+    "\n",
+    "## Why resampling methods ?\n",
+    "**Statistical analysis.**\n",
+    "\n",
+    "\n",
+    "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n",
+    "\n",
+    "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n",
+    "\n",
+    "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n",
+    "\n",
+    "    \n",
+    "\n",
+    "## Statistical analysis\n",
+    "\n",
+    "* As in other experiments, many numerical  experiments have two classes of errors:\n",
+    "\n",
+    "  * Statistical errors\n",
+    "\n",
+    "  * Systematical errors\n",
+    "\n",
+    "\n",
+    "* Statistical errors can be estimated using standard tools from statistics\n",
+    "\n",
+    "* Systematical errors are method specific and must be treated differently from case to case.\n",
+    "\n",
+    "    \n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
     "## Resampling methods\n",
     "\n",
     "With all these analytical equations for both the OLS and Ridge\n",
@@ -455,9 +459,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -598,9 +603,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -748,9 +754,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -979,9 +986,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1050,9 +1058,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1151,9 +1160,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1239,9 +1249,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1335,9 +1346,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1419,9 +1431,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1461,20 +1474,9 @@
     "plt.legend()\n",
     "plt.show()"
    ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Friday September 4\n",
-    "\n",
-    "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember4.mp4?vrtx=view-as-webpage) and [handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember4.pdf)\n",
-    "\n",
-    "More material will be added here, see handwritten notes also."
-   ]
   }
  ],
  "metadata": {},
  "nbformat": 4,
- "nbformat_minor": 2
+ "nbformat_minor": 4
 }
diff --git a/doc/src/week36/week36.do.txt b/doc/src/week36/week36.do.txt
index be6831a85..c2277695f 100644
--- a/doc/src/week36/week36.do.txt
+++ b/doc/src/week36/week36.do.txt
@@ -1,4 +1,4 @@
-TITLE: Week 36: Resampling techniques and Ordinary Least Square
+TITLE: Week 36: Statistical interpretation of Linear Regression and Resampling techniques
 AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
 DATE: today
 
@@ -6,103 +6,12 @@ DATE: today
 !split
 ===== Plans for week 36 =====
 
-* Thursday: Statistics, probability theory and resampling methods
-* Friday: Resampling methods and motivation for Ridge Regression
+* Thursday: Statistics, probability theory and linear regression
+* Friday: Resampling methods
 
 
 !split
-===== Thursday September 3  =====
-
-"Video of Lecture":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSept3.mp4?vrtx=view-as-webpage" and "handwritten notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember3.pdf"
-
-More material will be added here, see handwritten notes also.
-
-
-!split
-===== Why resampling methods =====
-
-Before we proceed, we need to rethink what we have been doing. In our
-eager to fit the data, we have omitted several important elements in
-our regression analysis. In what follows we will
-o look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
-o introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
-
-This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods. 
-
-
-
-
-
-!split
-===== Resampling methods =====
-!bblock
-Resampling methods are an indispensable tool in modern
-statistics. They involve repeatedly drawing samples from a training
-set and refitting a model of interest on each sample in order to
-obtain additional information about the fitted model. For example, in
-order to estimate the variability of a linear regression fit, we can
-repeatedly draw different samples from the training data, fit a linear
-regression to each new sample, and then examine the extent to which
-the resulting fits differ. Such an approach may allow us to obtain
-information that would not be available from fitting the model only
-once using the original training sample.
-
-Two resampling methods are often used in Machine Learning analyses,
-o The _bootstrap method_
-o and _Cross-Validation_
-
-In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
-cross-validation and the bootstrap method. 
-
-
-!eblock
-
-
-!split
-===== Resampling approaches can be computationally expensive =====
-!bblock
-
-Resampling approaches can be computationally expensive, because they
-involve fitting the same statistical method multiple times using
-different subsets of the training data. However, due to recent
-advances in computing power, the computational requirements of
-resampling methods generally are not prohibitive. In this chapter, we
-discuss two of the most commonly used resampling methods,
-cross-validation and the bootstrap. Both methods are important tools
-in the practical application of many statistical learning
-procedures. For example, cross-validation can be used to estimate the
-test error associated with a given statistical learning method in
-order to evaluate its performance, or to select the appropriate level
-of flexibility. The process of evaluating a model’s performance is
-known as model assessment, whereas the process of selecting the proper
-level of flexibility for a model is known as model selection. The
-bootstrap is widely used.
-
-!eblock
-
-!split
-===== Why resampling methods ? ===== 
-!bblock Statistical analysis
-    
-* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods
-* The results can be analysed with the same statistical tools as we would use analysing experimental data.
-* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
-
-
-!eblock    
-
-!split
-=====  Statistical analysis ===== 
-!bblock 
-
-* As in other experiments, many numerical  experiments have two classes of errors:
-  * Statistical errors
-  * Systematical errors
-* Statistical errors can be estimated using standard tools from statistics
-* Systematical errors are method specific and must be treated differently from case to case. 
-!eblock    
-
-
+===== Thursday September 9  =====
 
 
 !split 
@@ -270,6 +179,102 @@ matrix product is non-negative definite.
 This means the variance we obtain with the standard OLS will always for $\lambda > 0$ be larger than the variance of $\bm{\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. 
 
 
+
+
+
+!split
+===== Friday September 10  =====
+
+
+
+!split
+===== Why resampling methods =====
+
+Before we proceed, we need to rethink what we have been doing. In our
+eager to fit the data, we have omitted several important elements in
+our regression analysis. In what follows we will
+o look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
+o introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
+
+This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods. 
+
+
+
+
+
+!split
+===== Resampling methods =====
+!bblock
+Resampling methods are an indispensable tool in modern
+statistics. They involve repeatedly drawing samples from a training
+set and refitting a model of interest on each sample in order to
+obtain additional information about the fitted model. For example, in
+order to estimate the variability of a linear regression fit, we can
+repeatedly draw different samples from the training data, fit a linear
+regression to each new sample, and then examine the extent to which
+the resulting fits differ. Such an approach may allow us to obtain
+information that would not be available from fitting the model only
+once using the original training sample.
+
+Two resampling methods are often used in Machine Learning analyses,
+o The _bootstrap method_
+o and _Cross-Validation_
+
+In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
+cross-validation and the bootstrap method. 
+
+
+!eblock
+
+
+!split
+===== Resampling approaches can be computationally expensive =====
+!bblock
+
+Resampling approaches can be computationally expensive, because they
+involve fitting the same statistical method multiple times using
+different subsets of the training data. However, due to recent
+advances in computing power, the computational requirements of
+resampling methods generally are not prohibitive. In this chapter, we
+discuss two of the most commonly used resampling methods,
+cross-validation and the bootstrap. Both methods are important tools
+in the practical application of many statistical learning
+procedures. For example, cross-validation can be used to estimate the
+test error associated with a given statistical learning method in
+order to evaluate its performance, or to select the appropriate level
+of flexibility. The process of evaluating a model’s performance is
+known as model assessment, whereas the process of selecting the proper
+level of flexibility for a model is known as model selection. The
+bootstrap is widely used.
+
+!eblock
+
+!split
+===== Why resampling methods ? ===== 
+!bblock Statistical analysis
+    
+* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods
+* The results can be analysed with the same statistical tools as we would use analysing experimental data.
+* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
+
+
+!eblock    
+
+!split
+=====  Statistical analysis ===== 
+!bblock 
+
+* As in other experiments, many numerical  experiments have two classes of errors:
+  * Statistical errors
+  * Systematical errors
+* Statistical errors can be estimated using standard tools from statistics
+* Systematical errors are method specific and must be treated differently from case to case. 
+!eblock    
+
+
+
+
+
 !split
 ===== Resampling methods =====
 
@@ -1174,13 +1179,3 @@ plt.show()
 !ec
 
 
-!split
-===== Friday September 4  =====
-
-"Video of Lecture":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember4.mp4?vrtx=view-as-webpage" and "handwritten notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember4.pdf"
-
-More material will be added here, see handwritten notes also.
-
-
-
-