diff --git a/doc/pub/week36/html/._week36-bs000.html b/doc/pub/week36/html/._week36-bs000.html index 50b40fae8..01b1be8f0 100644 --- a/doc/pub/week36/html/._week36-bs000.html +++ b/doc/pub/week36/html/._week36-bs000.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
@@ -320,7 +224,7 @@ MathJax.Hub.Config({ -->
- © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + © 1999-2025, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
diff --git a/doc/pub/week36/html/._week36-bs001.html b/doc/pub/week36/html/._week36-bs001.html index df6bbd8d4..8610f4799 100644 --- a/doc/pub/week36/html/._week36-bs001.html +++ b/doc/pub/week36/html/._week36-bs001.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs002.html b/doc/pub/week36/html/._week36-bs002.html index 0a2d04790..e3025a845 100644 --- a/doc/pub/week36/html/._week36-bs002.html +++ b/doc/pub/week36/html/._week36-bs002.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs003.html b/doc/pub/week36/html/._week36-bs003.html index 1ec9f00f0..9ff3d12e7 100644 --- a/doc/pub/week36/html/._week36-bs003.html +++ b/doc/pub/week36/html/._week36-bs003.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs004.html b/doc/pub/week36/html/._week36-bs004.html index 4e9835f0f..d90fc340d 100644 --- a/doc/pub/week36/html/._week36-bs004.html +++ b/doc/pub/week36/html/._week36-bs004.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs005.html b/doc/pub/week36/html/._week36-bs005.html index 4588482ac..c21f50cd8 100644 --- a/doc/pub/week36/html/._week36-bs005.html +++ b/doc/pub/week36/html/._week36-bs005.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs006.html b/doc/pub/week36/html/._week36-bs006.html index 1decce8ef..4bb2149e6 100644 --- a/doc/pub/week36/html/._week36-bs006.html +++ b/doc/pub/week36/html/._week36-bs006.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs007.html b/doc/pub/week36/html/._week36-bs007.html index 46ad1e3ad..60f5dfed2 100644 --- a/doc/pub/week36/html/._week36-bs007.html +++ b/doc/pub/week36/html/._week36-bs007.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs008.html b/doc/pub/week36/html/._week36-bs008.html index 2195033bb..8beff4e29 100644 --- a/doc/pub/week36/html/._week36-bs008.html +++ b/doc/pub/week36/html/._week36-bs008.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs009.html b/doc/pub/week36/html/._week36-bs009.html index 45094a0d9..8c2077a21 100644 --- a/doc/pub/week36/html/._week36-bs009.html +++ b/doc/pub/week36/html/._week36-bs009.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs010.html b/doc/pub/week36/html/._week36-bs010.html index 74c2c2208..175c8f046 100644 --- a/doc/pub/week36/html/._week36-bs010.html +++ b/doc/pub/week36/html/._week36-bs010.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs011.html b/doc/pub/week36/html/._week36-bs011.html index 2fa906caa..fa1c8d661 100644 --- a/doc/pub/week36/html/._week36-bs011.html +++ b/doc/pub/week36/html/._week36-bs011.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs012.html b/doc/pub/week36/html/._week36-bs012.html index 294cdd878..9f0c68ea3 100644 --- a/doc/pub/week36/html/._week36-bs012.html +++ b/doc/pub/week36/html/._week36-bs012.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs013.html b/doc/pub/week36/html/._week36-bs013.html index baadc87e8..5db86fafe 100644 --- a/doc/pub/week36/html/._week36-bs013.html +++ b/doc/pub/week36/html/._week36-bs013.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs014.html b/doc/pub/week36/html/._week36-bs014.html index 46c8355fe..97f4b8528 100644 --- a/doc/pub/week36/html/._week36-bs014.html +++ b/doc/pub/week36/html/._week36-bs014.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs015.html b/doc/pub/week36/html/._week36-bs015.html index 675ca3af2..2d963a125 100644 --- a/doc/pub/week36/html/._week36-bs015.html +++ b/doc/pub/week36/html/._week36-bs015.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs016.html b/doc/pub/week36/html/._week36-bs016.html index aa3175dda..a98a7925a 100644 --- a/doc/pub/week36/html/._week36-bs016.html +++ b/doc/pub/week36/html/._week36-bs016.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs017.html b/doc/pub/week36/html/._week36-bs017.html index 4a2e75ecd..3aeef120a 100644 --- a/doc/pub/week36/html/._week36-bs017.html +++ b/doc/pub/week36/html/._week36-bs017.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs018.html b/doc/pub/week36/html/._week36-bs018.html index b61b27893..393fa8b72 100644 --- a/doc/pub/week36/html/._week36-bs018.html +++ b/doc/pub/week36/html/._week36-bs018.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs019.html b/doc/pub/week36/html/._week36-bs019.html index e701d6aad..0d2d5eaa0 100644 --- a/doc/pub/week36/html/._week36-bs019.html +++ b/doc/pub/week36/html/._week36-bs019.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs020.html b/doc/pub/week36/html/._week36-bs020.html index df3e9d9a4..32d11c90d 100644 --- a/doc/pub/week36/html/._week36-bs020.html +++ b/doc/pub/week36/html/._week36-bs020.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs021.html b/doc/pub/week36/html/._week36-bs021.html index 241e2b578..fe4239146 100644 --- a/doc/pub/week36/html/._week36-bs021.html +++ b/doc/pub/week36/html/._week36-bs021.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs022.html b/doc/pub/week36/html/._week36-bs022.html index a50f7359e..181d08254 100644 --- a/doc/pub/week36/html/._week36-bs022.html +++ b/doc/pub/week36/html/._week36-bs022.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs024.html b/doc/pub/week36/html/._week36-bs024.html index 021a791bd..c94b26ffe 100644 --- a/doc/pub/week36/html/._week36-bs024.html +++ b/doc/pub/week36/html/._week36-bs024.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs025.html b/doc/pub/week36/html/._week36-bs025.html index c79e5128d..f5fad4a3a 100644 --- a/doc/pub/week36/html/._week36-bs025.html +++ b/doc/pub/week36/html/._week36-bs025.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/._week36-bs026.html b/doc/pub/week36/html/._week36-bs026.html index 2a0d16e3c..a9e89457a 100644 --- a/doc/pub/week36/html/._week36-bs026.html +++ b/doc/pub/week36/html/._week36-bs026.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
diff --git a/doc/pub/week36/html/week36-bs.html b/doc/pub/week36/html/week36-bs.html index 50b40fae8..01b1be8f0 100644 --- a/doc/pub/week36/html/week36-bs.html +++ b/doc/pub/week36/html/week36-bs.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -41,85 +41,15 @@ doconce format html week36.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -196,7 +126,7 @@ MathJax.Hub.Config({ - Week 36: Linear Regression and Statistical interpretations + Week 36: Linear Regression and Gradient descent
@@ -320,7 +224,7 @@ MathJax.Hub.Config({ -->
- © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + © 1999-2025, 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 0e059dc5b..98d949b3a 100644 --- a/doc/pub/week36/html/week36-reveal.html +++ b/doc/pub/week36/html/week36-reveal.html @@ -9,8 +9,8 @@ doconce format html week36-reveal.html week36-reveal reveal --html_slide_theme=b - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -168,29 +168,26 @@ MathJax.Hub.Config({
-

Week 36: Linear Regression and Statistical interpretations

+

Week 36: Linear Regression and Gradient descent

-Morten Hjorth-Jensen [1, 2] +Morten Hjorth-Jensen
- +
-[1] Department of Physics, University of Oslo -
-
-[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University +Department of Physics, University of Oslo, Norway

-

September 2-6, 2024

+

September 1-5, 2025


- © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + © 1999-2025, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
@@ -198,1352 +195,40 @@ MathJax.Hub.Config({

Plans for week 36

Material for lecture Monday September 2

-

Important technicalities: More on Rescaling data

- -

When you are comparing your own code with for example Scikit-Learn's -library, there are some technicalities to keep in mind. The examples -here demonstrate some of these aspects with potential pitfalls. -

- -

The discussion here focuses on the role of the intercept, how we can -set up the design matrix, what scaling we should use and other topics -which tend confuse us. -

- -

The intercept can be interpreted as the expected value of our -target/output variables when all other predictors are set to zero. -Thus, if we cannot assume that the expected outputs/targets are zero -when all predictors are zero (the columns in the design matrix), it -may be a bad idea to implement a model which penalizes the intercept. -Furthermore, in for example Ridge and Lasso regression, the default solutions -from the library Scikit-Learn (when not shrinking \( \beta_0 \)) for the unknown parameters -\( \boldsymbol{\beta} \), are derived under the assumption that both \( \boldsymbol{y} \) and -\( \boldsymbol{X} \) are zero centered, that is we subtract the mean values. -

- -

If our predictors represent different scales, then it is important to -standardize the design matrix \( \boldsymbol{X} \) by subtracting the mean of each -column from the corresponding column and dividing the column with its -standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library, -the results may differ. -

- -

The -Standardscaler -function in Scikit-Learn does this for us. For the data sets we -have been studying in our various examples, the data are in many cases -already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a -survey of your data, with a critical assessment of them in case you need to scale the data. -

- -

If you need to scale the data, not doing so will give an unfair -penalization of the parameters since their magnitude depends on the -scale of their corresponding predictor. -

- -

The Scikit-Learn site https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section has a good discussion of different ways of preprocessing data.

- -

Suppose as an example that you -you have an input variable given by the heights of different persons. -Human height might be measured in inches or meters or -kilometers. If measured in kilometers, a standard linear regression -model with this predictor would probably give a much bigger -coefficient term, than if measured in millimeters. -This can clearly lead to problems in evaluating the cost/loss functions. -

- -

Keep in mind that when you transform your data set before training a model, the same transformation needs to be done -on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as -

- - - -
-
-
-
-
- 
"""
-#Model training, we compute the mean value of y and X
-y_train_mean = np.mean(y_train)
-X_train_mean = np.mean(X_train,axis=0)
-X_train = X_train - X_train_mean
-y_train = y_train - y_train_mean
-
-# The we fit our model with the training data
-trained_model = some_model.fit(X_train,y_train)
-
-
-#Model prediction, we need also to transform our data set used for the prediction.
-X_test = X_test - X_train_mean #Use mean from training data
-y_pred = trained_model(X_test)
-y_pred = y_pred + y_train_mean
-"""
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

Let us try to understand what this may imply mathematically when we -subtract the mean values, also known as zero centering. For -simplicity, we will focus on ordinary regression, as done in the above example. -

- -

The cost/loss function for regression is

-

 
-$$ -C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,. -$$ -

 
- -

Recall also that we use the squared value. This expression can lead to an -increased penalty for higher differences between predicted and -output/target values. -

- -

What we have done is to single out the \( \beta_0 \) term in the -definition of the mean squared error (MSE). The design matrix \( X \) -does in this case not contain any intercept column. When we take the -derivative with respect to \( \beta_0 \), we want the derivative to obey -

- -

 
-$$ -\frac{\partial C}{\partial \beta_j} = 0, -$$ -

 
- -

for all \( j \). For \( \beta_0 \) we have

- -

 
-$$ -\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right). -$$ -

 
- -

Multiplying away the constant \( 2/n \), we obtain

-

 
-$$ -\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j. -$$ -

 
- -

Let us specialize first to the case where we have only two parameters \( \beta_0 \) and \( \beta_1 \). -Our result for \( \beta_0 \) simplifies then to -

-

 
-$$ -n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1. -$$ -

 
- -

We obtain then

-

 
-$$ -\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}. -$$ -

 
- -

If we define

-

 
-$$ -\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}, -$$ -

 
- -

and the mean value of the outputs as

-

 
-$$ -\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i, -$$ -

 
- -

we have

-

 
-$$ -\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}. -$$ -

 
- -

In the general case with more parameters than \( \beta_0 \) and \( \beta_1 \), we have

-

 
-$$ -\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j. -$$ -

 
- -

We can rewrite the latter equation as

-

 
-$$ -\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j, -$$ -

 
- -

where we have defined

-

 
-$$ -\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij}, -$$ -

 
- -

the mean value for all elements of the column vector \( \boldsymbol{x}_j \).

- -

Replacing \( y_i \) with \( y_i - y_i - \overline{\boldsymbol{y}} \) and centering also our design matrix results in a cost function (in vector-matrix disguise)

-

 
-$$ -C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}). -$$ -

 
- -

If we minimize with respect to \( \boldsymbol{\beta} \) we have then

- -

 
-$$ -\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}, -$$ -

 
- -

where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}} \) -and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj} \). -

- -

For Ridge regression we need to add \( \lambda \boldsymbol{\beta}^T\boldsymbol{\beta} \) to the cost function and get then

-

 
-$$ -\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}. -$$ -

 
- -

What does this mean? And why do we insist on all this? Let us look at some examples.

- -

This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (code example thanks to Øyvind Sigmundson Schøyen). Here our scaling of the data is done by subtracting the mean values only. -Note also that we do not split the data into training and test. -

- - - -
-
-
-
-
- 
import numpy as np
-import matplotlib.pyplot as plt
-
-from sklearn.linear_model import LinearRegression
-
-
-np.random.seed(2021)
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-def fit_beta(X, y):
-    return np.linalg.pinv(X.T @ X) @ X.T @ y
-
-
-true_beta = [2, 0.5, 3.7]
-
-x = np.linspace(0, 1, 11)
-y = np.sum(
-    np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0
-) + 0.1 * np.random.normal(size=len(x))
-
-degree = 3
-X = np.zeros((len(x), degree))
-
-# Include the intercept in the design matrix
-for p in range(degree):
-    X[:, p] = x ** p
-
-beta = fit_beta(X, y)
-
-# Intercept is included in the design matrix
-skl = LinearRegression(fit_intercept=False).fit(X, y)
-
-print(f"True beta: {true_beta}")
-print(f"Fitted beta: {beta}")
-print(f"Sklearn fitted beta: {skl.coef_}")
-ypredictOwn = X @ beta
-ypredictSKL = skl.predict(X)
-print(f"MSE with intercept column")
-print(MSE(y,ypredictOwn))
-print(f"MSE with intercept column from SKL")
-print(MSE(y,ypredictSKL))
-
-
-plt.figure()
-plt.scatter(x, y, label="Data")
-plt.plot(x, X @ beta, label="Fit")
-plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)")
-
-
-# Do not include the intercept in the design matrix
-X = np.zeros((len(x), degree - 1))
-
-for p in range(degree - 1):
-    X[:, p] = x ** (p + 1)
-
-# Intercept is not included in the design matrix
-skl = LinearRegression(fit_intercept=True).fit(X, y)
-
-# Use centered values for X and y when computing coefficients
-y_offset = np.average(y, axis=0)
-X_offset = np.average(X, axis=0)
-
-beta = fit_beta(X - X_offset, y - y_offset)
-intercept = np.mean(y_offset - X_offset @ beta)
-
-print(f"Manual intercept: {intercept}")
-print(f"Fitted beta (without intercept): {beta}")
-print(f"Sklearn intercept: {skl.intercept_}")
-print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
-ypredictOwn = X @ beta
-ypredictSKL = skl.predict(X)
-print(f"MSE with Manual intercept")
-print(MSE(y,ypredictOwn+intercept))
-print(f"MSE with Sklearn intercept")
-print(MSE(y,ypredictSKL))
-
-plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
-plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)")
-plt.grid()
-plt.legend()
-
-plt.show()
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

The intercept is the value of our output/target variable -when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case). -

- -

Printing the MSE, we see first that both methods give the same MSE, as -they should. However, when we move to for example Ridge regression, -the way we treat the intercept may give a larger or smaller MSE, -meaning that the MSE can be penalized by the value of the -intercept. Not including the intercept in the fit, means that the -regularization term does not include \( \beta_0 \). For different values -of \( \lambda \), this may lead to different MSE values. -

- -

To remind the reader, the regularization term, with the intercept in Ridge regression, is given by

-

 
-$$ -\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2, -$$ -

 
- -

but when we take out the intercept, this equation becomes

-

 
-$$ -\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2. -$$ -

 
- -

For Lasso regression we have

-

 
-$$ -\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert. -$$ -

 
- -

It means that, when scaling the design matrix and the outputs/targets, -by subtracting the mean values, we have an optimization problem which -is not penalized by the intercept. The MSE value can then be smaller -since it focuses only on the remaining quantities. If we however bring -back the intercept, we will get a MSE which then contains the -intercept. -

- -

Armed with this wisdom, we attempt first to simply set the intercept equal to False in our implementation of Ridge regression for our well-known vanilla data set.

- - - -
-
-
-
-
- 
import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-n = 100
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-
-Maxpolydegree = 20
-X = np.zeros((n,Maxpolydegree))
-#We include explicitely the intercept column
-for degree in range(Maxpolydegree):
-    X[:,degree] = x**degree
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-p = Maxpolydegree
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 6
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-lambdas = np.logspace(-4, 2, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
-    # Note: we include the intercept column and no scaling
-    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
-    RegRidge.fit(X_train,y_train)
-    # and then make the prediction
-    ytildeOwnRidge = X_train @ OwnRidgeBeta
-    ypredictOwnRidge = X_test @ OwnRidgeBeta
-    ytildeRidge = RegRidge.predict(X_train)
-    ypredictRidge = RegRidge.predict(X_test)
-    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    print("Beta values for own Ridge implementation")
-    print(OwnRidgeBeta)
-    print("Beta values for Scikit-Learn Ridge implementation")
-    print(RegRidge.coef_)
-    print("MSE values for own Ridge implementation")
-    print(MSEOwnRidgePredict[i])
-    print("MSE values for Scikit-Learn Ridge implementation")
-    print(MSERidgePredict[i])
-
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
-
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

The results here agree when we force Scikit-Learn's Ridge function to include the first column in our design matrix. -We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix. -What happens if we do not include the intercept in our fit? -Let us see how we can change this code by zero centering. -

- - - -
-
-
-
-
- 
import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-from sklearn.preprocessing import StandardScaler
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(315)
-
-n = 100
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-
-Maxpolydegree = 20
-X = np.zeros((n,Maxpolydegree-1))
-
-for degree in range(1,Maxpolydegree): #No intercept column
-    X[:,degree-1] = x**(degree)
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
-X_train_mean = np.mean(X_train,axis=0)
-#Center by removing mean from each feature
-X_train_scaled = X_train - X_train_mean 
-X_test_scaled = X_test - X_train_mean
-#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
-#Remove the intercept from the training data.
-y_scaler = np.mean(y_train)           
-y_train_scaled = y_train - y_scaler   
-
-p = Maxpolydegree-1
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 6
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-
-lambdas = np.logspace(-4, 2, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
-    intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
-    #Add intercept to prediction
-    ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler 
-    RegRidge = linear_model.Ridge(lmb)
-    RegRidge.fit(X_train,y_train)
-    ypredictRidge = RegRidge.predict(X_test)
-    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    print("Beta values for own Ridge implementation")
-    print(OwnRidgeBeta) #Intercept is given by mean of target variable
-    print("Beta values for Scikit-Learn Ridge implementation")
-    print(RegRidge.coef_)
-    print('Intercept from own implementation:')
-    print(intercept_)
-    print('Intercept from Scikit-Learn Ridge implementation')
-    print(RegRidge.intercept_)
-    print("MSE values for own Ridge implementation")
-    print(MSEOwnRidgePredict[i])
-    print("MSE values for Scikit-Learn Ridge implementation")
-    print(MSERidgePredict[i])
-
-
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

We see here, when compared to the code which includes explicitely the -intercept column, that our MSE value is actually smaller. This is -because the regularization term does not include the intercept value -\( \beta_0 \) in the fitting. This applies to Lasso regularization as -well. It means that our optimization is now done only with the -centered matrix and/or vector that enter the fitting procedure. -

-

Test Function for what happens with OLS, Ridge and Lasso

+

Material for lab sessions sessions Tuesday and Wednesday

-

Hitherto we have discussed Ridge and Lasso regression in terms of a -linear analysis. This may to many of you feel rather technical and -perhaps not that intuitive. The question is whether we can develop a -more intuitive way of understanding what Ridge and Lasso express. -

- -

Before we proceed let us perform a Ridge, Lasso and OLS analysis of a polynomial fit.

- -

We will play around with a study of the values for the optimal -parameters \( \boldsymbol{\beta} \) using OLS, Ridge and Lasso regression. For -OLS, you will notice as function of the noise and polynomial degree, -that the parameters \( \beta \) will fluctuate from order to order in the -polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS. -

- -

For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one.

- - - -
-
-
-
-
- 
import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-# Make data set.
-n = 10000
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)
-
-Maxpolydegree = 5
-X = np.zeros((len(x),Maxpolydegree))
-X[:,0] = 1.0
-
-
-for polydegree in range(1,Maxpolydegree):
-    X[:,polydegree] = x**(polydegree)
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# matrix inversion to find beta
-OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-# Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn
-# Decide which values of lambda to use
-nlambdas = 4
-MSERidgePredict = np.zeros(nlambdas)
-MSELassoPredict = np.zeros(nlambdas)
-lambdas = np.logspace(-3, 1, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    # Make the fit using Ridge and Lasso
-    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
-    RegRidge.fit(X_train,y_train)
-    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
-    RegLasso.fit(X_train,y_train)
-    # and then make the prediction
-    ypredictRidge = RegRidge.predict(X_test)
-    ypredictLasso = RegLasso.predict(X_test)
-    # Compute the MSE and print it
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    MSELassoPredict[i] = MSE(y_test,ypredictLasso)
-    print(lmb,RegRidge.coef_)
-    print(lmb,RegLasso.coef_)
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSERidgePredict, 'b', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSELassoPredict, 'r', label = 'MSE Lasso Test')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

How can we understand this?

-
- -
-

Linking the regression analysis with a statistical interpretation

- -

We will now couple the discussions of ordinary least squares, Ridge -and Lasso regression with a statistical interpretation, that is we -move from a linear algebra analysis to a statistical analysis. In -particular, we will focus on what the regularization terms can result -in. We will amongst other things show that the regularization -parameter can reduce considerably the variance of the parameters -\( \beta \). -

- -

The -advantage of doing linear regression is that we actually end up with -analytical expressions for several statistical quantities. -Standard least squares and Ridge regression allow us to -derive quantities like the variance and other expectation values in a -rather straightforward way. -

- -

It is assumed that \( \varepsilon_i -\sim \mathcal{N}(0, \sigma^2) \) and the \( \varepsilon_{i} \) are -independent, i.e.: -

-

 
-$$ -\begin{align*} -\mbox{Cov}(\varepsilon_{i_1}, -\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if} -& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right. -\end{align*} -$$ -

 
- -

The randomness of \( \varepsilon_i \) implies that -\( \mathbf{y}_i \) is also a random variable. In particular, -\( \mathbf{y}_i \) is normally distributed, because \( \varepsilon_i \sim -\mathcal{N}(0, \sigma^2) \) and \( \mathbf{X}_{i,\ast} \, \boldsymbol{\beta} \) is a -non-random scalar. To specify the parameters of the distribution of -\( \mathbf{y}_i \) we need to calculate its first two moments. -

- -

Recall that \( \boldsymbol{X} \) is a matrix of dimensionality \( n\times p \). The -notation above \( \mathbf{X}_{i,\ast} \) means that we are looking at the -row number \( i \) and perform a sum over all values \( p \). -

-
- -
-

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) -that there exists a function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \) -which describe our data -

-

 
-$$ -\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} -$$ -

 
- -

We approximate this function with our model from the solution of the linear regression equations, that is our -function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we want to minimize \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), our MSE, with -

-

 
-$$ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. -$$ -

 
-

- -
-

Expectation value and variance

- -

We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \)

-

 
-$$ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -$$ -

 
- -

while -its variance is -

-

 
-$$ -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -$$ -

 
- -

Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with -mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (not be confused with the singular values of the SVD). -

-
- -
-

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

- -

With the OLS expressions for the optimal parameters \( \boldsymbol{\hat{\beta}} \) we can evaluate the expectation value

-

 
-$$ -\mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. -$$ -

 
- -

This means that the estimator of the regression parameters is unbiased.

- -

We can also calculate the variance

- -

The variance of the optimal value \( \boldsymbol{\hat{\beta}} \) is

-

 
-$$ -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\hat{\beta}}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T -\\ -& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} -$$ -

 
- -

where we have used that \( \mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn} \). From \( \mbox{Var}(\boldsymbol{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1} \), one obtains an estimate of the -variance of the estimate of the \( j \)-th regression coefficient: -\( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \). This may be used to -construct a confidence interval for the estimates. -

- -

In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters \( \boldsymbol{\beta} \) and their variance -when we employ Ridge regression, allowing us again to define a confidence interval. -

- -

It is rather straightforward to show that

-

 
-$$ -\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}. -$$ -

 
- -

We see clearly that -\( \mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}} \) for any \( \lambda > 0 \). We say then that the ridge estimator is biased. -

- -

We can also compute the variance as

- -

 
-$$ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, -$$ -

 
- -

and it is easy to see that if the parameter \( \lambda \) goes to infinity then the variance of Ridge parameters \( \boldsymbol{\beta} \) goes to zero.

- -

With this, we can compute the difference

- -

 
-$$ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. -$$ -

 
- -

The difference is non-negative definite since each component of the -matrix product is non-negative definite. -This means the variance we obtain with the standard OLS will always for \( \lambda > 0 \) be larger than the variance of \( \boldsymbol{\beta} \) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. -

-
- -
-

Deriving OLS from a probability distribution

- -

Our basic assumption when we derived the OLS equations was to assume -that our output is determined by a given continuous function -\( f(\boldsymbol{x}) \) and a random noise \( \boldsymbol{\epsilon} \) given by the normal -distribution with zero mean value and an undetermined variance -\( \sigma^2 \). -

- -

We found above that the outputs \( \boldsymbol{y} \) have a mean value given by -\( \boldsymbol{X}\hat{\boldsymbol{\beta}} \) and variance \( \sigma^2 \). Since the entries to -the design matrix are not stochastic variables, we can assume that the -probability distribution of our targets is also a normal distribution -but now with mean value \( \boldsymbol{X}\hat{\boldsymbol{\beta}} \). This means that a -single output \( y_i \) is given by the Gaussian distribution -

- -

 
-$$ -y_i\sim \mathcal{N}(\boldsymbol{X}_{i,*}\boldsymbol{\beta}, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}. -$$ -

 
-

- -
-

Independent and Identically Distributed (iid)

- -

We assume now that the various \( y_i \) values are stochastically distributed according to the above Gaussian distribution. -We define this distribution as -

-

 
-$$ -p(y_i, \boldsymbol{X}\vert\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}, -$$ -

 
- -

which reads as finding the likelihood of an event \( y_i \) with the input variables \( \boldsymbol{X} \) given the parameters (to be determined) \( \boldsymbol{\beta} \).

- -

Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event \( \boldsymbol{y} \) as the product of the single events, that is we have

- -

 
-$$ -p(\boldsymbol{y},\boldsymbol{X}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}=\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta}). -$$ -

 
- -

We will write this in a more compact form reserving \( \boldsymbol{D} \) for the domain of events, including the ouputs (targets) and the inputs. That is -in case we have a simple one-dimensional input and output case -

-

 
-$$ -\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})]. -$$ -

 
- -

In the more general case the various inputs should be replaced by the possible features represented by the input data set \( \boldsymbol{X} \). -We can now rewrite the above probability as -

-

 
-$$ -p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}. -$$ -

 
- -

It is a conditional probability (see below) and reads as the likelihood of a domain of events \( \boldsymbol{D} \) given a set of parameters \( \boldsymbol{\beta} \).

-
- -
-

Maximum Likelihood Estimation (MLE)

- -

In statistics, maximum likelihood estimation (MLE) is a method of -estimating the parameters of an assumed probability distribution, -given some observed data. This is achieved by maximizing a likelihood -function so that, under the assumed statistical model, the observed -data is the most probable. -

- -

We will assume here that our events are given by the above Gaussian -distribution and we will determine the optimal parameters \( \beta \) by -maximizing the above PDF. However, computing the derivatives of a -product function is cumbersome and can easily lead to overflow and/or -underflowproblems, with potentials for loss of numerical precision. -

- -

In practice, it is more convenient to maximize the logarithm of the -PDF because it is a monotonically increasing function of the argument. -Alternatively, and this will be our option, we will minimize the -negative of the logarithm since this is a monotonically decreasing -function. -

- -

Note also that maximization/minimization of the logarithm of the PDF -is equivalent to the maximization/minimization of the function itself. -

-
- -
-

A new Cost Function

- -

We could now define a new cost function to minimize, namely the negative logarithm of the above PDF

- -

 
-$$ -C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}, -$$ -

 
- -

which becomes

-

 
-$$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}. -$$ -

 
- -

Taking the derivative of the new cost function with respect to the parameters \( \beta \) we recognize our familiar OLS equation, namely

- -

 
-$$ -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0, -$$ -

 
- -

which leads to the well-known OLS equation for the optimal paramters \( \beta \)

-

 
-$$ -\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}! -$$ -

 
- -

Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics.

-
- -
-

More basic Statistics and Bayes' theorem

- -

A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry. -Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics. -

- -

Assume we have two domains of events \( X=[x_0,x_1,\dots,x_{n-1}] \) and \( Y=[y_0,y_1,\dots,y_{n-1}] \).

- -

We define also the likelihood for \( X \) and \( Y \) as \( p(X) \) and \( p(Y) \) respectively. -The likelihood of a specific event \( x_i \) (or \( y_i \)) is then written as \( p(X=x_i) \) or just \( p(x_i)=p_i \). -

- -
-Union of events is given by -

-

 
-$$ -p(X \cup Y)= p(X)+p(Y)-p(X \cap Y). -$$ -

 
-

- - -
-The product rule (aka joint probability) is given by -

-

 
-$$ -p(X \cup Y)= p(X,Y)= p(X\vert Y)p(Y)=p(Y\vert X)p(X), -$$ -

 
- -

where we read \( p(X\vert Y) \) as the likelihood of obtaining \( X \) given \( Y \).

-
- -

If we have independent events then \( p(X,Y)=p(X)p(Y) \).

-
- -
-

Marginal Probability

- -

The marginal probability is defined in terms of only one of the set of variables \( X,Y \). For a discrete probability we have

-
- -

-

 
-$$ -p(X)=\sum_{i=0}^{n-1}p(X,Y=y_i)=\sum_{i=0}^{n-1}p(X\vert Y=y_i)p(Y=y_i)=\sum_{i=0}^{n-1}p(X\vert y_i)p(y_i). -$$ -

 
-

-
- -
-

Conditional Probability

- -

The conditional probability, if \( p(Y) > 0 \), is

-
- -

-

 
-$$ -p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}. -$$ -

 
-

-
- -
-

Bayes' Theorem

- -

If we combine the conditional probability with the marginal probability and the standard product rule, we have

-

 
-$$ -p(X\vert Y)= \frac{p(X,Y)}{p(Y)}, -$$ -

 
- -

which we can rewrite as

- -

 
-$$ -p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}, -$$ -

 
- -

which is Bayes' theorem. It allows us to evaluate the uncertainty in in \( X \) after we have observed \( Y \). We can easily interchange \( X \) with \( Y \).

-
- -
-

Interpretations of Bayes' Theorem

- -

The quantity \( p(Y\vert X) \) on the right-hand side of the theorem is -evaluated for the observed data \( Y \) and can be viewed as a function of -the parameter space represented by \( X \). This function is not -necesseraly normalized and is normally called the likelihood function. -

- -

The function \( p(X) \) on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.

- -

Let us try to illustrate Bayes' theorem through an example.

-
- -
-

Example of Usage of Bayes' theorem

- -

Let us suppose that you are undergoing a series of mammography scans in -order to rule out possible breast cancer cases. We define the -sensitivity for a positive event by the variable \( X \). It takes binary -values with \( X=1 \) representing a positive event and \( X=0 \) being a -negative event. We reserve \( Y \) as a classification parameter for -either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a positive thing). -

- -

We let \( Y=1 \) represent the the case of having breast cancer and \( Y=0 \) as not.

- -

Let us assume that if you have breast cancer, the test will be positive with a probability of \( 0.8 \), that is we have

- -

 
-$$ -p(X=1\vert Y=1) =0.8. -$$ -

 
- -

This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer. -It is however not correct, as the following Bayesian analysis shows. -

-
- -
-

Doing it correctly

- -

If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number. -Let us assume that the prior probability in the population as a whole is -

- -

 
-$$ -p(Y=1) =0.004. -$$ -

 
- -

We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have

-

 
-$$ -p(X=1\vert Y=0) =0.1. -$$ -

 
- -

Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute

- -

 
-$$ -p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=0)p(Y=0)}=\frac{0.8\times 0.004}{0.8\times 0.004+0.1\times 0.996}=0.031. -$$ -

 
- -

That is, in case of a positive test, there is only a \( 3\% \) chance of having breast cancer!

-
- -
-

Bayes' Theorem and Ridge and Lasso Regression

- -

Using Bayes' theorem we can gain a better intuition about Ridge and Lasso regression.

- -

For ordinary least squares we postulated that the maximum likelihood for the doamin of events \( \boldsymbol{D} \) (one-dimensional case)

-

 
-$$ -\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})], -$$ -

 
- -

is given by

-

 
-$$ -p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}. -$$ -

 
- -

In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set \( \boldsymbol{\beta} \) given a domain of events \( \boldsymbol{D} \)? That is, how can we define the posterior probability

- -

 
-$$ -p(\boldsymbol{\beta}\vert\boldsymbol{D}). -$$ -

 
- -

Bayes' theorem comes to our rescue here since (omitting the normalization constant)

-

 
-$$ -p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}). -$$ -

 
- -

We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the prior \( p(\boldsymbol{\beta} \)!

-
- -
-

Ridge and Bayes

- -

With the posterior probability defined by a likelihood which we have -already modeled and an unknown prior, we are now ready to make -additional models for the prior. -

- -

We can, based on our discussions of the variance of \( \boldsymbol{\beta} \) and the mean value, assume that the prior for the values \( \boldsymbol{\beta} \) is given by a Gaussian with mean value zero and variance \( \tau^2 \), that is

- -

 
-$$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. -$$ -

 
- -

Our posterior probability becomes then (omitting the normalization factor which is just a constant)

-

 
-$$ -p(\boldsymbol{\beta\vert\boldsymbol{D})}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. -$$ -

 
- -

We can now optimize this quantity with respect to \( \boldsymbol{\beta} \). As we -did for OLS, this is most conveniently done by taking the negative -logarithm of the posterior probability. Doing so and leaving out the -constants terms that do not depend on \( \beta \), we have -

- -

 
-$$ -C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, -$$ -

 
- -

and replacing \( 1/2\tau^2 \) with \( \lambda \) we have

- -

 
-$$ -C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_2^2, -$$ -

 
- -

which is our Ridge cost function! Nice, isn't it?

-
- -
-

Lasso and Bayes

- -

To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution (Laplace in this case) with zero mean value, that is

- -

 
-$$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. -$$ -

 
- -

Our posterior probability becomes then (omitting the normalization factor which is just a constant)

-

 
-$$ -p(\boldsymbol{\beta}\vert\boldsymbol{D})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. -$$ -

 
- -

Taking the negative -logarithm of the posterior probability and leaving out the -constants terms that do not depend on \( \beta \), we have -

- -

 
-$$ -C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, -$$ -

 
- -

and replacing \( 1/\tau \) with \( \lambda \) we have

- -

 
-$$ -C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1, -$$ -

 
- -

which is our Lasso cost function!

-
- -
-

Material for the active learning sessions Tuesday and Wednesday

- -

The material here contains a summary from last week and discussion of SVD, Ridge and Lasso regression with examples

+

The material here contains a summary of the lecture on Monday and discussion of SVD, Ridge and Lasso regression with examples

@@ -1617,223 +302,6 @@ encode from the columns of \( \boldsymbol{X} \).

Note that these are also the eigenvectors and eigenvalues of the Hessian matrix.

- -

If we now recall the definition of the covariance matrix (not using -Bessel's correction) we have -

- -

 
-$$ -\boldsymbol{C}[\boldsymbol{X}]=\frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}, -$$ -

 
- -

meaning that every squared non-singular value of \( \boldsymbol{X} \) divided by \( n \) ( -the number of samples) are the eigenvalues of the covariance -matrix. Every singular value of \( \boldsymbol{X} \) is thus a positive square -root of an eigenvalue of \( \boldsymbol{X}^T\boldsymbol{X} \). If the matrix \( \boldsymbol{X} \) is -self-adjoint, the singular values of \( \boldsymbol{X} \) are equal to the -absolute value of the eigenvalues of \( \boldsymbol{X} \). -

-
- -
-

And finally \( \boldsymbol{X}\boldsymbol{X}^T \)

- -

For \( \boldsymbol{X}\boldsymbol{X}^T \) we found

- -

 
-$$ -\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T=\boldsymbol{U}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{U}^T. -$$ -

 
- -

Since the matrices here have dimension \( n\times n \), we have

-

 
-$$ -\boldsymbol{\Sigma}\boldsymbol{\Sigma}^T = \begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \\ \boldsymbol{0}\\ \end{bmatrix}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \boldsymbol{0}\\ \end{bmatrix}=\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}, -$$ -

 
- -

leading to

-

 
-$$ -\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}\boldsymbol{U}^T. -$$ -

 
- -

Multiplying with \( \boldsymbol{U} \) from the right gives us the eigenvalue problem

-

 
-$$ -(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U}=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}. -$$ -

 
- -

It means that the eigenvalues of \( \boldsymbol{X}\boldsymbol{X}^T \) are again given by -the non-zero singular values plus now a series of zeros. The column -vectors of \( \boldsymbol{U} \) are the eigenvectors of \( \boldsymbol{X}\boldsymbol{X}^T \) and -measure how much correlations are contained in the rows of \( \boldsymbol{X} \). -

- -

Since we will mainly be interested in the correlations among the features -of our data (the columns of \( \boldsymbol{X} \), the quantity of interest for us are the non-zero singular -values and the column vectors of \( \boldsymbol{V} \). -

-
- -
-

Code for SVD and Inversion of Matrices

- -

How do we use the SVD to invert a matrix \( \boldsymbol{X}^\boldsymbol{X} \) which is singular or near singular? -The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix \( \boldsymbol{X} \), that is -

- - -
-
-
-
-
- 
import numpy as np
-X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
-Xinv = np.linlag.pinv(X)
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD.

- - - -
-
-
-
-
- 
import numpy as np
-# SVD inversion
-def SVDinv(A):
-    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
-    SVD is numerically more stable than the inversion algorithms provided by
-    numpy and scipy.linalg at the cost of being slower.
-    '''
-    U, s, VT = np.linalg.svd(A)
-    print('test U')
-    print( (np.transpose(U) @ U - U @np.transpose(U)))
-    print('test VT')
-    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
-
-
-    D = np.zeros((len(U),len(VT)))
-    D = np.diag(s)
-    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
-    return np.matmul(V,np.matmul(invD,UT))
-
-
-#X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
-# Non-singular square matrix
-X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
-print(X)
-A = np.transpose(X) @ X
-# Brute force inversion
-B = np.linalg.inv(A)  # here we could use np.linalg.pinv(A)
-C = SVDinv(A)
-print(np.abs(B-C))
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -
-

Inverse of Rectangular Matrix

- -

Although our matrix to invert \( \boldsymbol{X}^T\boldsymbol{X} \) is a square matrix, our matrix may be singular.

- -

The pseudoinverse is the generalization of the matrix inverse for square matrices to -rectangular matrices where the number of rows and columns are not equal. -

- -

It is also called the the Moore-Penrose Inverse after two independent discoverers of the method or the Generalized Inverse. -It is used for the calculation of the inverse for singular or near singular matrices and for rectangular matrices. -

- -

Using the SVD we can obtain the pseudoinverse of a matrix \( \boldsymbol{A} \) (labeled here as \( \boldsymbol{A}_{\mathrm{PI}} \))

-

 
-$$ -\boldsymbol{A}_{\mathrm{PI}}= \boldsymbol{V}\boldsymbol{D}_{\mathrm{PI}}\boldsymbol{U}^T, -$$ -

 
- -

where \( \boldsymbol{D}_{\mathrm{PI}} \) can be calculated by creating a diagonal matrix from \( \boldsymbol{\Sigma} \) where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD.

- - - -
-
-
-
-
- 
import numpy as np
-# SVD inversion
-def SVDinv(A):
-    U, s, VT = np.linalg.svd(A)
-    # reciprocals of singular values of s
-    d = 1.0 / s
-    # create m x n D matrix
-    D = np.zeros(A.shape)
-    # populate D with n x n diagonal matrix
-    D[:A.shape[1], :A.shape[1]] = np.diag(d)
-    UT = np.transpose(U)
-    V = np.transpose(VT)
-    return np.matmul(V,np.matmul(D.T,UT))
-
-
-A = np.array([ [0.3, 0.4], [0.5, 0.6], [0.7, 0.8],[0.9, 1.0]])
-print(A)
-# Brute force inversion of super-collinear matrix
-B = np.linalg.pinv(A)
-print(B)
-# Compare our own algorithm with pinv
-C = SVDinv(A)
-print(np.abs(C-B))
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

As you can see from this example, our own decomposition based on the SVD agrees with the pseudoinverse algorithm provided by Numpy.

@@ -2093,7 +561,7 @@ $$ $$

 
-

This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor \( 2/n \) in a redefinition of the parameter \( \lambda \). We will solve this type of problems using libraries like scikit-learn.

+

This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor \( 2/n \) in a redefinition of the parameter \( \lambda \). We will solve this type of problems using libraries like scikit-learn and using our own gradient descent code in project 1.

diff --git a/doc/pub/week36/html/week36-solarized.html b/doc/pub/week36/html/week36-solarized.html index b377be763..c6335e3fa 100644 --- a/doc/pub/week36/html/week36-solarized.html +++ b/doc/pub/week36/html/week36-solarized.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --pygments_html_style=perldoc --html_style=sol - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent @@ -23,32 +23,6 @@ pre { border: 0pt solid #93a1a1; box-shadow: none; } -.alert-text-small { font-size: 80%; } -.alert-text-large { font-size: 130%; } -.alert-text-normal { font-size: 90%; } -.alert { - padding:8px 35px 8px 14px; margin-bottom:18px; - text-shadow:0 1px 0 rgba(255,255,255,0.5); - border:1px solid #93a1a1; - border-radius: 4px; - -webkit-border-radius: 4px; - -moz-border-radius: 4px; - color: #555; - background-color: #eee8d5; - background-position: 10px 5px; - background-repeat: no-repeat; - background-size: 38px; - padding-left: 55px; - width: 75%; - } -.alert-block {padding-top:14px; padding-bottom:14px} -.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/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; } .tab { padding-left: 1.5em; @@ -68,85 +42,15 @@ div.toc p,a { 2, None, 'material-for-lecture-monday-september-2'), - ('Important technicalities: More on Rescaling data', + ('Material for lab sessions sessions Tuesday and Wednesday', 2, None, - 'important-technicalities-more-on-rescaling-data'), - ('Test Function for what happens with OLS, Ridge and Lasso', - 2, - None, - 'test-function-for-what-happens-with-ols-ridge-and-lasso'), - ('Linking the regression analysis with a statistical ' - 'interpretation', - 2, - None, - 'linking-the-regression-analysis-with-a-statistical-interpretation'), - ('Assumptions made', 2, None, 'assumptions-made'), - ('Expectation value and variance', - 2, - None, - 'expectation-value-and-variance'), - ('Expectation value and variance for $\\boldsymbol{\\beta}$', - 2, - None, - 'expectation-value-and-variance-for-boldsymbol-beta'), - ('Deriving OLS from a probability distribution', - 2, - None, - 'deriving-ols-from-a-probability-distribution'), - ('Independent and Identically Distributed (iid)', - 2, - None, - 'independent-and-identically-distributed-iid'), - ('Maximum Likelihood Estimation (MLE)', - 2, - None, - 'maximum-likelihood-estimation-mle'), - ('A new Cost Function', 2, None, 'a-new-cost-function'), - ("More basic Statistics and Bayes' theorem", - 2, - None, - 'more-basic-statistics-and-bayes-theorem'), - ('Marginal Probability', 2, None, 'marginal-probability'), - ('Conditional Probability', 2, None, 'conditional-probability'), - ("Bayes' Theorem", 2, None, 'bayes-theorem'), - ("Interpretations of Bayes' Theorem", - 2, - None, - 'interpretations-of-bayes-theorem'), - ("Example of Usage of Bayes' theorem", - 2, - None, - 'example-of-usage-of-bayes-theorem'), - ('Doing it correctly', 2, None, 'doing-it-correctly'), - ("Bayes' Theorem and Ridge and Lasso Regression", - 2, - None, - 'bayes-theorem-and-ridge-and-lasso-regression'), - ('Ridge and Bayes', 2, None, 'ridge-and-bayes'), - ('Lasso and Bayes', 2, None, 'lasso-and-bayes'), - ('Material for the active learning sessions Tuesday and ' - 'Wednesday', - 2, - None, - 'material-for-the-active-learning-sessions-tuesday-and-wednesday'), + 'material-for-lab-sessions-sessions-tuesday-and-wednesday'), ('Linear Regression and the SVD', 2, None, 'linear-regression-and-the-svd'), ('What does it mean?', 2, None, 'what-does-it-mean'), - ('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$', - 2, - None, - 'and-finally-boldsymbol-x-boldsymbol-x-t'), - ('Code for SVD and Inversion of Matrices', - 2, - None, - 'code-for-svd-and-inversion-of-matrices'), - ('Inverse of Rectangular Matrix', - 2, - None, - 'inverse-of-rectangular-matrix'), ('Ridge and LASSO Regression', 2, None, @@ -217,23 +121,20 @@ MathJax.Hub.Config({
-

Week 36: Linear Regression and Statistical interpretations

+

Week 36: Linear Regression and Gradient descent

-Morten Hjorth-Jensen [1, 2] +Morten Hjorth-Jensen
- +
-[1] Department of Physics, University of Oslo -
-
-[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University +Department of Physics, University of Oslo, Norway

-

September 2-6, 2024

+

September 1-5, 2025


@@ -241,1205 +142,32 @@ MathJax.Hub.Config({

Plans for week 36











Material for lecture Monday September 2

-

Important technicalities: More on Rescaling data

- -

When you are comparing your own code with for example Scikit-Learn's -library, there are some technicalities to keep in mind. The examples -here demonstrate some of these aspects with potential pitfalls. -

- -

The discussion here focuses on the role of the intercept, how we can -set up the design matrix, what scaling we should use and other topics -which tend confuse us. -

- -

The intercept can be interpreted as the expected value of our -target/output variables when all other predictors are set to zero. -Thus, if we cannot assume that the expected outputs/targets are zero -when all predictors are zero (the columns in the design matrix), it -may be a bad idea to implement a model which penalizes the intercept. -Furthermore, in for example Ridge and Lasso regression, the default solutions -from the library Scikit-Learn (when not shrinking \( \beta_0 \)) for the unknown parameters -\( \boldsymbol{\beta} \), are derived under the assumption that both \( \boldsymbol{y} \) and -\( \boldsymbol{X} \) are zero centered, that is we subtract the mean values. -

- -

If our predictors represent different scales, then it is important to -standardize the design matrix \( \boldsymbol{X} \) by subtracting the mean of each -column from the corresponding column and dividing the column with its -standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library, -the results may differ. -

- -

The -Standardscaler -function in Scikit-Learn does this for us. For the data sets we -have been studying in our various examples, the data are in many cases -already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a -survey of your data, with a critical assessment of them in case you need to scale the data. -

- -

If you need to scale the data, not doing so will give an unfair -penalization of the parameters since their magnitude depends on the -scale of their corresponding predictor. -

- -

The Scikit-Learn site https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section has a good discussion of different ways of preprocessing data.

- -

Suppose as an example that you -you have an input variable given by the heights of different persons. -Human height might be measured in inches or meters or -kilometers. If measured in kilometers, a standard linear regression -model with this predictor would probably give a much bigger -coefficient term, than if measured in millimeters. -This can clearly lead to problems in evaluating the cost/loss functions. -

- -

Keep in mind that when you transform your data set before training a model, the same transformation needs to be done -on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as -

- - - -
-
-
-
-
- 
"""
-#Model training, we compute the mean value of y and X
-y_train_mean = np.mean(y_train)
-X_train_mean = np.mean(X_train,axis=0)
-X_train = X_train - X_train_mean
-y_train = y_train - y_train_mean
-
-# The we fit our model with the training data
-trained_model = some_model.fit(X_train,y_train)
-
-
-#Model prediction, we need also to transform our data set used for the prediction.
-X_test = X_test - X_train_mean #Use mean from training data
-y_pred = trained_model(X_test)
-y_pred = y_pred + y_train_mean
-"""
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

Let us try to understand what this may imply mathematically when we -subtract the mean values, also known as zero centering. For -simplicity, we will focus on ordinary regression, as done in the above example. -

- -

The cost/loss function for regression is

-$$ -C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,. -$$ - -

Recall also that we use the squared value. This expression can lead to an -increased penalty for higher differences between predicted and -output/target values. -

- -

What we have done is to single out the \( \beta_0 \) term in the -definition of the mean squared error (MSE). The design matrix \( X \) -does in this case not contain any intercept column. When we take the -derivative with respect to \( \beta_0 \), we want the derivative to obey -

- -$$ -\frac{\partial C}{\partial \beta_j} = 0, -$$ - -

for all \( j \). For \( \beta_0 \) we have

- -$$ -\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right). -$$ - -

Multiplying away the constant \( 2/n \), we obtain

-$$ -\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j. -$$ - -

Let us specialize first to the case where we have only two parameters \( \beta_0 \) and \( \beta_1 \). -Our result for \( \beta_0 \) simplifies then to -

-$$ -n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1. -$$ - -

We obtain then

-$$ -\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}. -$$ - -

If we define

-$$ -\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}, -$$ - -

and the mean value of the outputs as

-$$ -\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i, -$$ - -

we have

-$$ -\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}. -$$ - -

In the general case with more parameters than \( \beta_0 \) and \( \beta_1 \), we have

-$$ -\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j. -$$ - -

We can rewrite the latter equation as

-$$ -\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j, -$$ - -

where we have defined

-$$ -\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij}, -$$ - -

the mean value for all elements of the column vector \( \boldsymbol{x}_j \).

- -

Replacing \( y_i \) with \( y_i - y_i - \overline{\boldsymbol{y}} \) and centering also our design matrix results in a cost function (in vector-matrix disguise)

-$$ -C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}). -$$ - -

If we minimize with respect to \( \boldsymbol{\beta} \) we have then

- -$$ -\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}, -$$ - -

where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}} \) -and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj} \). -

- -

For Ridge regression we need to add \( \lambda \boldsymbol{\beta}^T\boldsymbol{\beta} \) to the cost function and get then

-$$ -\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}. -$$ - -

What does this mean? And why do we insist on all this? Let us look at some examples.

- -

This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (code example thanks to Øyvind Sigmundson Schøyen). Here our scaling of the data is done by subtracting the mean values only. -Note also that we do not split the data into training and test. -

- - - -
-
-
-
-
- 
import numpy as np
-import matplotlib.pyplot as plt
-
-from sklearn.linear_model import LinearRegression
-
-
-np.random.seed(2021)
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-def fit_beta(X, y):
-    return np.linalg.pinv(X.T @ X) @ X.T @ y
-
-
-true_beta = [2, 0.5, 3.7]
-
-x = np.linspace(0, 1, 11)
-y = np.sum(
-    np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0
-) + 0.1 * np.random.normal(size=len(x))
-
-degree = 3
-X = np.zeros((len(x), degree))
-
-# Include the intercept in the design matrix
-for p in range(degree):
-    X[:, p] = x ** p
-
-beta = fit_beta(X, y)
-
-# Intercept is included in the design matrix
-skl = LinearRegression(fit_intercept=False).fit(X, y)
-
-print(f"True beta: {true_beta}")
-print(f"Fitted beta: {beta}")
-print(f"Sklearn fitted beta: {skl.coef_}")
-ypredictOwn = X @ beta
-ypredictSKL = skl.predict(X)
-print(f"MSE with intercept column")
-print(MSE(y,ypredictOwn))
-print(f"MSE with intercept column from SKL")
-print(MSE(y,ypredictSKL))
-
-
-plt.figure()
-plt.scatter(x, y, label="Data")
-plt.plot(x, X @ beta, label="Fit")
-plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)")
-
-
-# Do not include the intercept in the design matrix
-X = np.zeros((len(x), degree - 1))
-
-for p in range(degree - 1):
-    X[:, p] = x ** (p + 1)
-
-# Intercept is not included in the design matrix
-skl = LinearRegression(fit_intercept=True).fit(X, y)
-
-# Use centered values for X and y when computing coefficients
-y_offset = np.average(y, axis=0)
-X_offset = np.average(X, axis=0)
-
-beta = fit_beta(X - X_offset, y - y_offset)
-intercept = np.mean(y_offset - X_offset @ beta)
-
-print(f"Manual intercept: {intercept}")
-print(f"Fitted beta (without intercept): {beta}")
-print(f"Sklearn intercept: {skl.intercept_}")
-print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
-ypredictOwn = X @ beta
-ypredictSKL = skl.predict(X)
-print(f"MSE with Manual intercept")
-print(MSE(y,ypredictOwn+intercept))
-print(f"MSE with Sklearn intercept")
-print(MSE(y,ypredictSKL))
-
-plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
-plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)")
-plt.grid()
-plt.legend()
-
-plt.show()
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

The intercept is the value of our output/target variable -when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case). -

- -

Printing the MSE, we see first that both methods give the same MSE, as -they should. However, when we move to for example Ridge regression, -the way we treat the intercept may give a larger or smaller MSE, -meaning that the MSE can be penalized by the value of the -intercept. Not including the intercept in the fit, means that the -regularization term does not include \( \beta_0 \). For different values -of \( \lambda \), this may lead to different MSE values. -

- -

To remind the reader, the regularization term, with the intercept in Ridge regression, is given by

-$$ -\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2, -$$ - -

but when we take out the intercept, this equation becomes

-$$ -\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2. -$$ - -

For Lasso regression we have

-$$ -\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert. -$$ - -

It means that, when scaling the design matrix and the outputs/targets, -by subtracting the mean values, we have an optimization problem which -is not penalized by the intercept. The MSE value can then be smaller -since it focuses only on the remaining quantities. If we however bring -back the intercept, we will get a MSE which then contains the -intercept. -

- -

Armed with this wisdom, we attempt first to simply set the intercept equal to False in our implementation of Ridge regression for our well-known vanilla data set.

- - - -
-
-
-
-
- 
import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-n = 100
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-
-Maxpolydegree = 20
-X = np.zeros((n,Maxpolydegree))
-#We include explicitely the intercept column
-for degree in range(Maxpolydegree):
-    X[:,degree] = x**degree
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-p = Maxpolydegree
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 6
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-lambdas = np.logspace(-4, 2, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
-    # Note: we include the intercept column and no scaling
-    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
-    RegRidge.fit(X_train,y_train)
-    # and then make the prediction
-    ytildeOwnRidge = X_train @ OwnRidgeBeta
-    ypredictOwnRidge = X_test @ OwnRidgeBeta
-    ytildeRidge = RegRidge.predict(X_train)
-    ypredictRidge = RegRidge.predict(X_test)
-    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    print("Beta values for own Ridge implementation")
-    print(OwnRidgeBeta)
-    print("Beta values for Scikit-Learn Ridge implementation")
-    print(RegRidge.coef_)
-    print("MSE values for own Ridge implementation")
-    print(MSEOwnRidgePredict[i])
-    print("MSE values for Scikit-Learn Ridge implementation")
-    print(MSERidgePredict[i])
-
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
-
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- -

The results here agree when we force Scikit-Learn's Ridge function to include the first column in our design matrix. -We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix. -What happens if we do not include the intercept in our fit? -Let us see how we can change this code by zero centering. -

- - - -
-
-
-
-
- 
import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-from sklearn.preprocessing import StandardScaler
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(315)
-
-n = 100
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-
-Maxpolydegree = 20
-X = np.zeros((n,Maxpolydegree-1))
-
-for degree in range(1,Maxpolydegree): #No intercept column
-    X[:,degree-1] = x**(degree)
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
-X_train_mean = np.mean(X_train,axis=0)
-#Center by removing mean from each feature
-X_train_scaled = X_train - X_train_mean 
-X_test_scaled = X_test - X_train_mean
-#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
-#Remove the intercept from the training data.
-y_scaler = np.mean(y_train)           
-y_train_scaled = y_train - y_scaler   
-
-p = Maxpolydegree-1
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 6
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-
-lambdas = np.logspace(-4, 2, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
-    intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
-    #Add intercept to prediction
-    ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler 
-    RegRidge = linear_model.Ridge(lmb)
-    RegRidge.fit(X_train,y_train)
-    ypredictRidge = RegRidge.predict(X_test)
-    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    print("Beta values for own Ridge implementation")
-    print(OwnRidgeBeta) #Intercept is given by mean of target variable
-    print("Beta values for Scikit-Learn Ridge implementation")
-    print(RegRidge.coef_)
-    print('Intercept from own implementation:')
-    print(intercept_)
-    print('Intercept from Scikit-Learn Ridge implementation')
-    print(RegRidge.intercept_)
-    print("MSE values for own Ridge implementation")
-    print(MSEOwnRidgePredict[i])
-    print("MSE values for Scikit-Learn Ridge implementation")
-    print(MSERidgePredict[i])
-
-
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
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-
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-
-
-
-
- -

We see here, when compared to the code which includes explicitely the -intercept column, that our MSE value is actually smaller. This is -because the regularization term does not include the intercept value -\( \beta_0 \) in the fitting. This applies to Lasso regularization as -well. It means that our optimization is now done only with the -centered matrix and/or vector that enter the fitting procedure. -











-

Test Function for what happens with OLS, Ridge and Lasso

+

Material for lab sessions sessions Tuesday and Wednesday

-

Hitherto we have discussed Ridge and Lasso regression in terms of a -linear analysis. This may to many of you feel rather technical and -perhaps not that intuitive. The question is whether we can develop a -more intuitive way of understanding what Ridge and Lasso express. -

- -

Before we proceed let us perform a Ridge, Lasso and OLS analysis of a polynomial fit.

- -

We will play around with a study of the values for the optimal -parameters \( \boldsymbol{\beta} \) using OLS, Ridge and Lasso regression. For -OLS, you will notice as function of the noise and polynomial degree, -that the parameters \( \beta \) will fluctuate from order to order in the -polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS. -

- -

For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one.

- - - -
-
-
-
-
- 
import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-# Make data set.
-n = 10000
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)
-
-Maxpolydegree = 5
-X = np.zeros((len(x),Maxpolydegree))
-X[:,0] = 1.0
-
-
-for polydegree in range(1,Maxpolydegree):
-    X[:,polydegree] = x**(polydegree)
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# matrix inversion to find beta
-OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-# Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn
-# Decide which values of lambda to use
-nlambdas = 4
-MSERidgePredict = np.zeros(nlambdas)
-MSELassoPredict = np.zeros(nlambdas)
-lambdas = np.logspace(-3, 1, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    # Make the fit using Ridge and Lasso
-    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
-    RegRidge.fit(X_train,y_train)
-    RegLasso = linear_model.Lasso(lmb,fit_intercept=False)
-    RegLasso.fit(X_train,y_train)
-    # and then make the prediction
-    ypredictRidge = RegRidge.predict(X_test)
-    ypredictLasso = RegLasso.predict(X_test)
-    # Compute the MSE and print it
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    MSELassoPredict[i] = MSE(y_test,ypredictLasso)
-    print(lmb,RegRidge.coef_)
-    print(lmb,RegLasso.coef_)
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSERidgePredict, 'b', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSELassoPredict, 'r', label = 'MSE Lasso Test')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-
-
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-
-
-
-
-
-
-
-
-
-
- -

How can we understand this?

- - -

Linking the regression analysis with a statistical interpretation

- -

We will now couple the discussions of ordinary least squares, Ridge -and Lasso regression with a statistical interpretation, that is we -move from a linear algebra analysis to a statistical analysis. In -particular, we will focus on what the regularization terms can result -in. We will amongst other things show that the regularization -parameter can reduce considerably the variance of the parameters -\( \beta \). -

- -

The -advantage of doing linear regression is that we actually end up with -analytical expressions for several statistical quantities. -Standard least squares and Ridge regression allow us to -derive quantities like the variance and other expectation values in a -rather straightforward way. -

- -

It is assumed that \( \varepsilon_i -\sim \mathcal{N}(0, \sigma^2) \) and the \( \varepsilon_{i} \) are -independent, i.e.: -

-$$ -\begin{align*} -\mbox{Cov}(\varepsilon_{i_1}, -\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if} -& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right. -\end{align*} -$$ - -

The randomness of \( \varepsilon_i \) implies that -\( \mathbf{y}_i \) is also a random variable. In particular, -\( \mathbf{y}_i \) is normally distributed, because \( \varepsilon_i \sim -\mathcal{N}(0, \sigma^2) \) and \( \mathbf{X}_{i,\ast} \, \boldsymbol{\beta} \) is a -non-random scalar. To specify the parameters of the distribution of -\( \mathbf{y}_i \) we need to calculate its first two moments. -

- -

Recall that \( \boldsymbol{X} \) is a matrix of dimensionality \( n\times p \). The -notation above \( \mathbf{X}_{i,\ast} \) means that we are looking at the -row number \( i \) and perform a sum over all values \( p \). -

- -









-

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) -that there exists a function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \) -which describe our data -

-$$ -\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} -$$ - -

We approximate this function with our model from the solution of the linear regression equations, that is our -function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we want to minimize \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), our MSE, with -

-$$ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. -$$ - - -









-

Expectation value and variance

- -

We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \)

-$$ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -$$ - -

while -its variance is -

-$$ -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -$$ - -

Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with -mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (not be confused with the singular values of the SVD). -

- -









-

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

- -

With the OLS expressions for the optimal parameters \( \boldsymbol{\hat{\beta}} \) we can evaluate the expectation value

-$$ -\mathbb{E}(\boldsymbol{\hat{\beta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. -$$ - -

This means that the estimator of the regression parameters is unbiased.

- -

We can also calculate the variance

- -

The variance of the optimal value \( \boldsymbol{\hat{\beta}} \) is

-$$ -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\hat{\beta}}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T -\\ -& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} -$$ - -

where we have used that \( \mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn} \). From \( \mbox{Var}(\boldsymbol{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1} \), one obtains an estimate of the -variance of the estimate of the \( j \)-th regression coefficient: -\( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \). This may be used to -construct a confidence interval for the estimates. -

- -

In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters \( \boldsymbol{\beta} \) and their variance -when we employ Ridge regression, allowing us again to define a confidence interval. -

- -

It is rather straightforward to show that

-$$ -\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}. -$$ - -

We see clearly that -\( \mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}} \) for any \( \lambda > 0 \). We say then that the ridge estimator is biased. -

- -

We can also compute the variance as

- -$$ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, -$$ - -

and it is easy to see that if the parameter \( \lambda \) goes to infinity then the variance of Ridge parameters \( \boldsymbol{\beta} \) goes to zero.

- -

With this, we can compute the difference

- -$$ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. -$$ - -

The difference is non-negative definite since each component of the -matrix product is non-negative definite. -This means the variance we obtain with the standard OLS will always for \( \lambda > 0 \) be larger than the variance of \( \boldsymbol{\beta} \) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. -

- -









-

Deriving OLS from a probability distribution

- -

Our basic assumption when we derived the OLS equations was to assume -that our output is determined by a given continuous function -\( f(\boldsymbol{x}) \) and a random noise \( \boldsymbol{\epsilon} \) given by the normal -distribution with zero mean value and an undetermined variance -\( \sigma^2 \). -

- -

We found above that the outputs \( \boldsymbol{y} \) have a mean value given by -\( \boldsymbol{X}\hat{\boldsymbol{\beta}} \) and variance \( \sigma^2 \). Since the entries to -the design matrix are not stochastic variables, we can assume that the -probability distribution of our targets is also a normal distribution -but now with mean value \( \boldsymbol{X}\hat{\boldsymbol{\beta}} \). This means that a -single output \( y_i \) is given by the Gaussian distribution -

- -$$ -y_i\sim \mathcal{N}(\boldsymbol{X}_{i,*}\boldsymbol{\beta}, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}. -$$ - - -









-

Independent and Identically Distributed (iid)

- -

We assume now that the various \( y_i \) values are stochastically distributed according to the above Gaussian distribution. -We define this distribution as -

-$$ -p(y_i, \boldsymbol{X}\vert\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}, -$$ - -

which reads as finding the likelihood of an event \( y_i \) with the input variables \( \boldsymbol{X} \) given the parameters (to be determined) \( \boldsymbol{\beta} \).

- -

Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event \( \boldsymbol{y} \) as the product of the single events, that is we have

- -$$ -p(\boldsymbol{y},\boldsymbol{X}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}=\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta}). -$$ - -

We will write this in a more compact form reserving \( \boldsymbol{D} \) for the domain of events, including the ouputs (targets) and the inputs. That is -in case we have a simple one-dimensional input and output case -

-$$ -\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})]. -$$ - -

In the more general case the various inputs should be replaced by the possible features represented by the input data set \( \boldsymbol{X} \). -We can now rewrite the above probability as -

-$$ -p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}. -$$ - -

It is a conditional probability (see below) and reads as the likelihood of a domain of events \( \boldsymbol{D} \) given a set of parameters \( \boldsymbol{\beta} \).

- -









-

Maximum Likelihood Estimation (MLE)

- -

In statistics, maximum likelihood estimation (MLE) is a method of -estimating the parameters of an assumed probability distribution, -given some observed data. This is achieved by maximizing a likelihood -function so that, under the assumed statistical model, the observed -data is the most probable. -

- -

We will assume here that our events are given by the above Gaussian -distribution and we will determine the optimal parameters \( \beta \) by -maximizing the above PDF. However, computing the derivatives of a -product function is cumbersome and can easily lead to overflow and/or -underflowproblems, with potentials for loss of numerical precision. -

- -

In practice, it is more convenient to maximize the logarithm of the -PDF because it is a monotonically increasing function of the argument. -Alternatively, and this will be our option, we will minimize the -negative of the logarithm since this is a monotonically decreasing -function. -

- -

Note also that maximization/minimization of the logarithm of the PDF -is equivalent to the maximization/minimization of the function itself. -

- -









-

A new Cost Function

- -

We could now define a new cost function to minimize, namely the negative logarithm of the above PDF

- -$$ -C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}, -$$ - -

which becomes

-$$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}. -$$ - -

Taking the derivative of the new cost function with respect to the parameters \( \beta \) we recognize our familiar OLS equation, namely

- -$$ -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0, -$$ - -

which leads to the well-known OLS equation for the optimal paramters \( \beta \)

-$$ -\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}! -$$ - -

Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics.

- -









-

More basic Statistics and Bayes' theorem

- -

A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry. -Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics. -

- -

Assume we have two domains of events \( X=[x_0,x_1,\dots,x_{n-1}] \) and \( Y=[y_0,y_1,\dots,y_{n-1}] \).

- -

We define also the likelihood for \( X \) and \( Y \) as \( p(X) \) and \( p(Y) \) respectively. -The likelihood of a specific event \( x_i \) (or \( y_i \)) is then written as \( p(X=x_i) \) or just \( p(x_i)=p_i \). -

- -
-Union of events is given by -

-$$ -p(X \cup Y)= p(X)+p(Y)-p(X \cap Y). -$$ -

- - -
-The product rule (aka joint probability) is given by -

-$$ -p(X \cup Y)= p(X,Y)= p(X\vert Y)p(Y)=p(Y\vert X)p(X), -$$ - -

where we read \( p(X\vert Y) \) as the likelihood of obtaining \( X \) given \( Y \).

-
- - -

If we have independent events then \( p(X,Y)=p(X)p(Y) \).

- -









-

Marginal Probability

- -

The marginal probability is defined in terms of only one of the set of variables \( X,Y \). For a discrete probability we have

-
- -

-$$ -p(X)=\sum_{i=0}^{n-1}p(X,Y=y_i)=\sum_{i=0}^{n-1}p(X\vert Y=y_i)p(Y=y_i)=\sum_{i=0}^{n-1}p(X\vert y_i)p(y_i). -$$ -

- - -









-

Conditional Probability

- -

The conditional probability, if \( p(Y) > 0 \), is

-
- -

-$$ -p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}. -$$ -

- - -









-

Bayes' Theorem

- -

If we combine the conditional probability with the marginal probability and the standard product rule, we have

-$$ -p(X\vert Y)= \frac{p(X,Y)}{p(Y)}, -$$ - -

which we can rewrite as

- -$$ -p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}, -$$ - -

which is Bayes' theorem. It allows us to evaluate the uncertainty in in \( X \) after we have observed \( Y \). We can easily interchange \( X \) with \( Y \).

- -









-

Interpretations of Bayes' Theorem

- -

The quantity \( p(Y\vert X) \) on the right-hand side of the theorem is -evaluated for the observed data \( Y \) and can be viewed as a function of -the parameter space represented by \( X \). This function is not -necesseraly normalized and is normally called the likelihood function. -

- -

The function \( p(X) \) on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.

- -

Let us try to illustrate Bayes' theorem through an example.

- -









-

Example of Usage of Bayes' theorem

- -

Let us suppose that you are undergoing a series of mammography scans in -order to rule out possible breast cancer cases. We define the -sensitivity for a positive event by the variable \( X \). It takes binary -values with \( X=1 \) representing a positive event and \( X=0 \) being a -negative event. We reserve \( Y \) as a classification parameter for -either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a positive thing). -

- -

We let \( Y=1 \) represent the the case of having breast cancer and \( Y=0 \) as not.

- -

Let us assume that if you have breast cancer, the test will be positive with a probability of \( 0.8 \), that is we have

- -$$ -p(X=1\vert Y=1) =0.8. -$$ - -

This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer. -It is however not correct, as the following Bayesian analysis shows. -

- -









-

Doing it correctly

- -

If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number. -Let us assume that the prior probability in the population as a whole is -

- -$$ -p(Y=1) =0.004. -$$ - -

We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have

-$$ -p(X=1\vert Y=0) =0.1. -$$ - -

Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute

- -$$ -p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=0)p(Y=0)}=\frac{0.8\times 0.004}{0.8\times 0.004+0.1\times 0.996}=0.031. -$$ - -

That is, in case of a positive test, there is only a \( 3\% \) chance of having breast cancer!

- -









-

Bayes' Theorem and Ridge and Lasso Regression

- -

Using Bayes' theorem we can gain a better intuition about Ridge and Lasso regression.

- -

For ordinary least squares we postulated that the maximum likelihood for the doamin of events \( \boldsymbol{D} \) (one-dimensional case)

-$$ -\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})], -$$ - -

is given by

-$$ -p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}. -$$ - -

In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set \( \boldsymbol{\beta} \) given a domain of events \( \boldsymbol{D} \)? That is, how can we define the posterior probability

- -$$ -p(\boldsymbol{\beta}\vert\boldsymbol{D}). -$$ - -

Bayes' theorem comes to our rescue here since (omitting the normalization constant)

-$$ -p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}). -$$ - -

We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the prior \( p(\boldsymbol{\beta} \)!

- -









-

Ridge and Bayes

- -

With the posterior probability defined by a likelihood which we have -already modeled and an unknown prior, we are now ready to make -additional models for the prior. -

- -

We can, based on our discussions of the variance of \( \boldsymbol{\beta} \) and the mean value, assume that the prior for the values \( \boldsymbol{\beta} \) is given by a Gaussian with mean value zero and variance \( \tau^2 \), that is

- -$$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. -$$ - -

Our posterior probability becomes then (omitting the normalization factor which is just a constant)

-$$ -p(\boldsymbol{\beta\vert\boldsymbol{D})}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. -$$ - -

We can now optimize this quantity with respect to \( \boldsymbol{\beta} \). As we -did for OLS, this is most conveniently done by taking the negative -logarithm of the posterior probability. Doing so and leaving out the -constants terms that do not depend on \( \beta \), we have -

- -$$ -C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, -$$ - -

and replacing \( 1/2\tau^2 \) with \( \lambda \) we have

- -$$ -C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_2^2, -$$ - -

which is our Ridge cost function! Nice, isn't it?

- -









-

Lasso and Bayes

- -

To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution (Laplace in this case) with zero mean value, that is

- -$$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. -$$ - -

Our posterior probability becomes then (omitting the normalization factor which is just a constant)

-$$ -p(\boldsymbol{\beta}\vert\boldsymbol{D})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. -$$ - -

Taking the negative -logarithm of the posterior probability and leaving out the -constants terms that do not depend on \( \beta \), we have -

- -$$ -C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, -$$ - -

and replacing \( 1/\tau \) with \( \lambda \) we have

- -$$ -C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1, -$$ - -

which is our Lasso cost function!

- -









-

Material for the active learning sessions Tuesday and Wednesday

- -

The material here contains a summary from last week and discussion of SVD, Ridge and Lasso regression with examples

+

The material here contains a summary of the lecture on Monday and discussion of SVD, Ridge and Lasso regression with examples











Linear Regression and the SVD

@@ -1501,209 +229,6 @@ encode from the columns of \( \boldsymbol{X} \). Hessian matrix.

-

If we now recall the definition of the covariance matrix (not using -Bessel's correction) we have -

- -$$ -\boldsymbol{C}[\boldsymbol{X}]=\frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}, -$$ - -

meaning that every squared non-singular value of \( \boldsymbol{X} \) divided by \( n \) ( -the number of samples) are the eigenvalues of the covariance -matrix. Every singular value of \( \boldsymbol{X} \) is thus a positive square -root of an eigenvalue of \( \boldsymbol{X}^T\boldsymbol{X} \). If the matrix \( \boldsymbol{X} \) is -self-adjoint, the singular values of \( \boldsymbol{X} \) are equal to the -absolute value of the eigenvalues of \( \boldsymbol{X} \). -

- -









-

And finally \( \boldsymbol{X}\boldsymbol{X}^T \)

- -

For \( \boldsymbol{X}\boldsymbol{X}^T \) we found

- -$$ -\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T=\boldsymbol{U}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{U}^T. -$$ - -

Since the matrices here have dimension \( n\times n \), we have

-$$ -\boldsymbol{\Sigma}\boldsymbol{\Sigma}^T = \begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \\ \boldsymbol{0}\\ \end{bmatrix}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} \boldsymbol{0}\\ \end{bmatrix}=\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}, -$$ - -

leading to

-$$ -\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}\boldsymbol{U}^T. -$$ - -

Multiplying with \( \boldsymbol{U} \) from the right gives us the eigenvalue problem

-$$ -(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U}=\boldsymbol{U}\begin{bmatrix} \tilde{\boldsymbol{\Sigma}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0}\\ \end{bmatrix}. -$$ - -

It means that the eigenvalues of \( \boldsymbol{X}\boldsymbol{X}^T \) are again given by -the non-zero singular values plus now a series of zeros. The column -vectors of \( \boldsymbol{U} \) are the eigenvectors of \( \boldsymbol{X}\boldsymbol{X}^T \) and -measure how much correlations are contained in the rows of \( \boldsymbol{X} \). -

- -

Since we will mainly be interested in the correlations among the features -of our data (the columns of \( \boldsymbol{X} \), the quantity of interest for us are the non-zero singular -values and the column vectors of \( \boldsymbol{V} \). -

- -









-

Code for SVD and Inversion of Matrices

- -

How do we use the SVD to invert a matrix \( \boldsymbol{X}^\boldsymbol{X} \) which is singular or near singular? -The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix \( \boldsymbol{X} \), that is -

- - -
-
-
-
-
- 
import numpy as np
-X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
-Xinv = np.linlag.pinv(X)
-
-
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- -

Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD.

- - - -
-
-
-
-
- 
import numpy as np
-# SVD inversion
-def SVDinv(A):
-    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
-    SVD is numerically more stable than the inversion algorithms provided by
-    numpy and scipy.linalg at the cost of being slower.
-    '''
-    U, s, VT = np.linalg.svd(A)
-    print('test U')
-    print( (np.transpose(U) @ U - U @np.transpose(U)))
-    print('test VT')
-    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
-
-
-    D = np.zeros((len(U),len(VT)))
-    D = np.diag(s)
-    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
-    return np.matmul(V,np.matmul(invD,UT))
-
-
-#X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
-# Non-singular square matrix
-X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
-print(X)
-A = np.transpose(X) @ X
-# Brute force inversion
-B = np.linalg.inv(A)  # here we could use np.linalg.pinv(A)
-C = SVDinv(A)
-print(np.abs(B-C))
-
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- - -









-

Inverse of Rectangular Matrix

- -

Although our matrix to invert \( \boldsymbol{X}^T\boldsymbol{X} \) is a square matrix, our matrix may be singular.

- -

The pseudoinverse is the generalization of the matrix inverse for square matrices to -rectangular matrices where the number of rows and columns are not equal. -

- -

It is also called the the Moore-Penrose Inverse after two independent discoverers of the method or the Generalized Inverse. -It is used for the calculation of the inverse for singular or near singular matrices and for rectangular matrices. -

- -

Using the SVD we can obtain the pseudoinverse of a matrix \( \boldsymbol{A} \) (labeled here as \( \boldsymbol{A}_{\mathrm{PI}} \))

-$$ -\boldsymbol{A}_{\mathrm{PI}}= \boldsymbol{V}\boldsymbol{D}_{\mathrm{PI}}\boldsymbol{U}^T, -$$ - -

where \( \boldsymbol{D}_{\mathrm{PI}} \) can be calculated by creating a diagonal matrix from \( \boldsymbol{\Sigma} \) where we only keep the singular values (the non-zero values). The following code computes the pseudoinvers of the matrix based on the SVD.

- - - -
-
-
-
-
- 
import numpy as np
-# SVD inversion
-def SVDinv(A):
-    U, s, VT = np.linalg.svd(A)
-    # reciprocals of singular values of s
-    d = 1.0 / s
-    # create m x n D matrix
-    D = np.zeros(A.shape)
-    # populate D with n x n diagonal matrix
-    D[:A.shape[1], :A.shape[1]] = np.diag(d)
-    UT = np.transpose(U)
-    V = np.transpose(VT)
-    return np.matmul(V,np.matmul(D.T,UT))
-
-
-A = np.array([ [0.3, 0.4], [0.5, 0.6], [0.7, 0.8],[0.9, 1.0]])
-print(A)
-# Brute force inversion of super-collinear matrix
-B = np.linalg.pinv(A)
-print(B)
-# Compare our own algorithm with pinv
-C = SVDinv(A)
-print(np.abs(C-B))
-
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As you can see from this example, our own decomposition based on the SVD agrees with the pseudoinverse algorithm provided by Numpy.

-









Ridge and LASSO Regression

@@ -1911,7 +436,7 @@ $$ \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}+\lambda sgn(\boldsymbol{\beta})=\boldsymbol{X}^T\boldsymbol{y}. $$ -

This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor \( 2/n \) in a redefinition of the parameter \( \lambda \). We will solve this type of problems using libraries like scikit-learn.

+

This equation does not lead to a nice analytical equation as in Ridge regression or ordinary least squares. We have absorbed the factor \( 2/n \) in a redefinition of the parameter \( \lambda \). We will solve this type of problems using libraries like scikit-learn and using our own gradient descent code in project 1.











Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression

@@ -2372,7 +897,7 @@ plt.show()
- © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + © 1999-2025, 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 c9dded8bb..85d40c677 100644 --- a/doc/pub/week36/html/week36.html +++ b/doc/pub/week36/html/week36.html @@ -8,8 +8,8 @@ doconce format html week36.do.txt --pygments_html_style=default --html_style=blo - -Week 36: Linear Regression and Statistical interpretations + +Week 36: Linear Regression and Gradient descent