852 lines
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HTML
852 lines
58 KiB
HTML
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Plans for week 36', 2, None, 'plans-for-week-36'),
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('Material for lecture Monday September 2',
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2,
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None,
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'material-for-lecture-monday-september-2'),
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('Important technicalities: More on Rescaling data',
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2,
|
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None,
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'important-technicalities-more-on-rescaling-data'),
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('Test Function for what happens with OLS, Ridge and Lasso',
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2,
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None,
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('Linking the regression analysis with a statistical '
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'interpretation',
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2,
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None,
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'linking-the-regression-analysis-with-a-statistical-interpretation'),
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('Assumptions made', 2, None, 'assumptions-made'),
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('Expectation value and variance',
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2,
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None,
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'expectation-value-and-variance'),
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('Expectation value and variance for $\\boldsymbol{\\beta}$',
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2,
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None,
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'expectation-value-and-variance-for-boldsymbol-beta'),
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('Deriving OLS from a probability distribution',
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2,
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None,
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'deriving-ols-from-a-probability-distribution'),
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('Independent and Identically Distributed (iid)',
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2,
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None,
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'independent-and-identically-distributed-iid'),
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('Maximum Likelihood Estimation (MLE)',
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2,
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None,
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'maximum-likelihood-estimation-mle'),
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('A new Cost Function', 2, None, 'a-new-cost-function'),
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("More basic Statistics and Bayes' theorem",
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2,
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None,
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'more-basic-statistics-and-bayes-theorem'),
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('Marginal Probability', 2, None, 'marginal-probability'),
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('Conditional Probability', 2, None, 'conditional-probability'),
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("Bayes' Theorem", 2, None, 'bayes-theorem'),
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("Interpretations of Bayes' Theorem",
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2,
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None,
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'interpretations-of-bayes-theorem'),
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("Example of Usage of Bayes' theorem",
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2,
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None,
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'example-of-usage-of-bayes-theorem'),
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('Doing it correctly', 2, None, 'doing-it-correctly'),
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("Bayes' Theorem and Ridge and Lasso Regression",
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2,
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None,
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'bayes-theorem-and-ridge-and-lasso-regression'),
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('Ridge and Bayes', 2, None, 'ridge-and-bayes'),
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('Lasso and Bayes', 2, None, 'lasso-and-bayes'),
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('Material for the active learning sessions Tuesday and '
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'Wednesday',
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2,
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None,
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'material-for-the-active-learning-sessions-tuesday-and-wednesday'),
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('Linear Regression and the SVD',
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2,
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None,
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'linear-regression-and-the-svd'),
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('What does it mean?', 2, None, 'what-does-it-mean'),
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('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$',
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2,
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None,
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'and-finally-boldsymbol-x-boldsymbol-x-t'),
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('Code for SVD and Inversion of Matrices',
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2,
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None,
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'code-for-svd-and-inversion-of-matrices'),
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('Inverse of Rectangular Matrix',
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2,
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None,
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'inverse-of-rectangular-matrix'),
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('Ridge and LASSO Regression',
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2,
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None,
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'ridge-and-lasso-regression'),
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('From OLS to Ridge and Lasso',
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2,
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None,
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'from-ols-to-ridge-and-lasso'),
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('Deriving the Ridge Regression Equations',
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2,
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None,
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'deriving-the-ridge-regression-equations'),
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('Note on Scikit-Learn', 2, None, 'note-on-scikit-learn'),
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('Comparison with OLS', 2, None, 'comparison-with-ols'),
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('SVD analysis', 2, None, 'svd-analysis'),
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('Interpreting the Ridge results',
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2,
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None,
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'interpreting-the-ridge-results'),
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('More interpretations', 2, None, 'more-interpretations'),
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('Deriving the Lasso Regression Equations',
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2,
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None,
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'deriving-the-lasso-regression-equations'),
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('Simple example to illustrate Ordinary Least Squares, Ridge and '
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'Lasso Regression',
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2,
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None,
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'simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression'),
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('Ridge Regression', 2, None, 'ridge-regression'),
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('Lasso Regression', 2, None, 'lasso-regression'),
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('Yet another Example', 2, None, 'yet-another-example'),
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('The OLS case', 2, None, 'the-ols-case'),
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('The Ridge case', 2, None, 'the-ridge-case'),
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('Writing the Cost Function',
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2,
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None,
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'writing-the-cost-function'),
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('Lasso case', 2, None, 'lasso-case'),
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('The first Case', 2, None, 'the-first-case'),
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('Simple code for solving the above problem',
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2,
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None,
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'simple-code-for-solving-the-above-problem'),
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('With Lasso Regression', 2, None, 'with-lasso-regression'),
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('Another Example, now with a polynomial fit',
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2,
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None,
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'another-example-now-with-a-polynomial-fit')]}
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<a class="navbar-brand" href="week36-bs.html">Week 36: Linear Regression and Statistical interpretations</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week36-bs001.html#plans-for-week-36" style="font-size: 80%;">Plans for week 36</a></li>
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<!-- navigation toc: --> <li><a href="#material-for-lecture-monday-september-2" style="font-size: 80%;">Material for lecture Monday September 2</a></li>
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<!-- navigation toc: --> <li><a href="#important-technicalities-more-on-rescaling-data" style="font-size: 80%;">Important technicalities: More on Rescaling data</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs003.html#test-function-for-what-happens-with-ols-ridge-and-lasso" style="font-size: 80%;">Test Function for what happens with OLS, Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.html#linking-the-regression-analysis-with-a-statistical-interpretation" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs005.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs006.html#expectation-value-and-variance" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#expectation-value-and-variance-for-boldsymbol-beta" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs008.html#deriving-ols-from-a-probability-distribution" style="font-size: 80%;">Deriving OLS from a probability distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs009.html#independent-and-identically-distributed-iid" style="font-size: 80%;">Independent and Identically Distributed (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs010.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs011.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#more-basic-statistics-and-bayes-theorem" style="font-size: 80%;">More basic Statistics and Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.html#interpretations-of-bayes-theorem" style="font-size: 80%;">Interpretations of Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.html#example-of-usage-of-bayes-theorem" style="font-size: 80%;">Example of Usage of Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#bayes-theorem-and-ridge-and-lasso-regression" style="font-size: 80%;">Bayes' Theorem and Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.html#material-for-the-active-learning-sessions-tuesday-and-wednesday" style="font-size: 80%;">Material for the active learning sessions Tuesday and Wednesday</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs023.html#linear-regression-and-the-svd" style="font-size: 80%;">Linear Regression and the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#what-does-it-mean" style="font-size: 80%;">What does it mean?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;">And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#code-for-svd-and-inversion-of-matrices" style="font-size: 80%;">Code for SVD and Inversion of Matrices</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.html#inverse-of-rectangular-matrix" style="font-size: 80%;">Inverse of Rectangular Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.html#from-ols-to-ridge-and-lasso" style="font-size: 80%;">From OLS to Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs030.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#note-on-scikit-learn" style="font-size: 80%;">Note on Scikit-Learn</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#comparison-with-ols" style="font-size: 80%;">Comparison with OLS</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs036.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs037.html#simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression" style="font-size: 80%;">Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs038.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs039.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs040.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs041.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs042.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs043.html#writing-the-cost-function" style="font-size: 80%;">Writing the Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs045.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs046.html#simple-code-for-solving-the-above-problem" style="font-size: 80%;">Simple code for solving the above problem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs047.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.html#another-example-now-with-a-polynomial-fit" style="font-size: 80%;">Another Example, now with a polynomial fit</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0002"></a>
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<!-- !split -->
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<h2 id="material-for-lecture-monday-september-2" class="anchor">Material for lecture Monday September 2 </h2>
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<h2 id="important-technicalities-more-on-rescaling-data" class="anchor">Important technicalities: More on Rescaling data </h2>
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<p>When you are comparing your own code with for example <b>Scikit-Learn</b>'s
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library, there are some technicalities to keep in mind. The examples
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here demonstrate some of these aspects with potential pitfalls.
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</p>
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<p>The discussion here focuses on the role of the intercept, how we can
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set up the design matrix, what scaling we should use and other topics
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which tend confuse us.
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</p>
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<p>The intercept can be interpreted as the expected value of our
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target/output variables when all other predictors are set to zero.
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Thus, if we cannot assume that the expected outputs/targets are zero
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when all predictors are zero (the columns in the design matrix), it
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may be a bad idea to implement a model which penalizes the intercept.
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Furthermore, in for example Ridge and Lasso regression, the default solutions
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from the library <b>Scikit-Learn</b> (when not shrinking \( \beta_0 \)) for the unknown parameters
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\( \boldsymbol{\beta} \), are derived under the assumption that both \( \boldsymbol{y} \) and
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\( \boldsymbol{X} \) are zero centered, that is we subtract the mean values.
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</p>
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<p>If our predictors represent different scales, then it is important to
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standardize the design matrix \( \boldsymbol{X} \) by subtracting the mean of each
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column from the corresponding column and dividing the column with its
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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,
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the results may differ.
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</p>
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<p>The
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<a href="https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html" target="_self">Standardscaler</a>
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function in <b>Scikit-Learn</b> does this for us. For the data sets we
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have been studying in our various examples, the data are in many cases
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already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a
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survey of your data, with a critical assessment of them in case you need to scale the data.
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</p>
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<p>If you need to scale the data, not doing so will give an <em>unfair</em>
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penalization of the parameters since their magnitude depends on the
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scale of their corresponding predictor.
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</p>
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<p>The <b>Scikit-Learn</b> site <a href="https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section" target="_self"><tt>https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section</tt></a> has a good discussion of different ways of preprocessing data.</p>
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<p>Suppose as an example that you
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you have an input variable given by the heights of different persons.
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Human height might be measured in inches or meters or
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kilometers. If measured in kilometers, a standard linear regression
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model with this predictor would probably give a much bigger
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coefficient term, than if measured in millimeters.
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This can clearly lead to problems in evaluating the cost/loss functions.
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</p>
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<p>Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
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on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as
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</p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="input_area">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">"""</span>
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<span style="color: #BA2121; font-style: italic">#Model training, we compute the mean value of y and X</span>
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|
<span style="color: #BA2121; font-style: italic">y_train_mean = np.mean(y_train)</span>
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<span style="color: #BA2121; font-style: italic">X_train_mean = np.mean(X_train,axis=0)</span>
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<span style="color: #BA2121; font-style: italic">X_train = X_train - X_train_mean</span>
|
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<span style="color: #BA2121; font-style: italic">y_train = y_train - y_train_mean</span>
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|
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<span style="color: #BA2121; font-style: italic"># The we fit our model with the training data</span>
|
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<span style="color: #BA2121; font-style: italic">trained_model = some_model.fit(X_train,y_train)</span>
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|
|
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<span style="color: #BA2121; font-style: italic">#Model prediction, we need also to transform our data set used for the prediction.</span>
|
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<span style="color: #BA2121; font-style: italic">X_test = X_test - X_train_mean #Use mean from training data</span>
|
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<span style="color: #BA2121; font-style: italic">y_pred = trained_model(X_test)</span>
|
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<span style="color: #BA2121; font-style: italic">y_pred = y_pred + y_train_mean</span>
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<span style="color: #BA2121; font-style: italic">"""</span>
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</pre>
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</div>
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</div>
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</div>
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</div>
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<div class="output_wrapper">
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<div class="output">
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<div class="output_area">
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<div class="output_subarea output_stream output_stdout output_text">
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</div>
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</div>
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</div>
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</div>
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</div>
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<p>Let us try to understand what this may imply mathematically when we
|
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subtract the mean values, also known as <em>zero centering</em>. For
|
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simplicity, we will focus on ordinary regression, as done in the above example.
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</p>
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<p>The cost/loss function for regression is</p>
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$$
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C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
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$$
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<p>Recall also that we use the squared value. This expression can lead to an
|
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increased penalty for higher differences between predicted and
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output/target values.
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</p>
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|
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<p>What we have done is to single out the \( \beta_0 \) term in the
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definition of the mean squared error (MSE). The design matrix \( X \)
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does in this case not contain any intercept column. When we take the
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derivative with respect to \( \beta_0 \), we want the derivative to obey
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</p>
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$$
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\frac{\partial C}{\partial \beta_j} = 0,
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$$
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<p>for all \( j \). For \( \beta_0 \) we have</p>
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$$
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\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
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$$
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<p>Multiplying away the constant \( 2/n \), we obtain</p>
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$$
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\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
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$$
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<p>Let us specialize first to the case where we have only two parameters \( \beta_0 \) and \( \beta_1 \).
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Our result for \( \beta_0 \) simplifies then to
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</p>
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$$
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n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
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$$
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<p>We obtain then</p>
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$$
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\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}.
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$$
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<p>If we define</p>
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$$
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\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1},
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$$
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<p>and the mean value of the outputs as</p>
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$$
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\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
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$$
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<p>we have</p>
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$$
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\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}.
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$$
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<p>In the general case with more parameters than \( \beta_0 \) and \( \beta_1 \), we have</p>
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$$
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\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j.
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$$
|
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<p>We can rewrite the latter equation as</p>
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$$
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\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j,
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$$
|
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<p>where we have defined</p>
|
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$$
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\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij},
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$$
|
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<p>the mean value for all elements of the column vector \( \boldsymbol{x}_j \).</p>
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<p>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)</p>
|
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$$
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C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
|
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$$
|
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<p>If we minimize with respect to \( \boldsymbol{\beta} \) we have then</p>
|
|
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|
$$
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\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
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$$
|
|
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<p>where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}} \)
|
|
and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj} \).
|
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</p>
|
|
|
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<p>For Ridge regression we need to add \( \lambda \boldsymbol{\beta}^T\boldsymbol{\beta} \) to the cost function and get then</p>
|
|
$$
|
|
\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
|
|
$$
|
|
|
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<p>What does this mean? And why do we insist on all this? Let us look at some examples.</p>
|
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<p>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 (<em>code example thanks to Øyvind Sigmundson Schøyen</em>). Here our scaling of the data is done by subtracting the mean values only.
|
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Note also that we do not split the data into training and test.
|
|
</p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="inner_cell">
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<div class="input_area">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
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|
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2021</span>)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
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n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">fit_beta</span>(X, y):
|
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
|
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|
|
|
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true_beta <span style="color: #666666">=</span> [<span style="color: #666666">2</span>, <span style="color: #666666">0.5</span>, <span style="color: #666666">3.7</span>]
|
|
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">11</span>)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(
|
|
np<span style="color: #666666">.</span>asarray([x <span style="color: #666666">**</span> p <span style="color: #666666">*</span> b <span style="color: #008000; font-weight: bold">for</span> p, b <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(true_beta)]), axis<span style="color: #666666">=0</span>
|
|
) <span style="color: #666666">+</span> <span style="color: #666666">0.1</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span><span style="color: #008000">len</span>(x))
|
|
|
|
degree <span style="color: #666666">=</span> <span style="color: #666666">3</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x), degree))
|
|
|
|
<span style="color: #408080; font-style: italic"># Include the intercept in the design matrix</span>
|
|
<span style="color: #008000; font-weight: bold">for</span> p <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(degree):
|
|
X[:, p] <span style="color: #666666">=</span> x <span style="color: #666666">**</span> p
|
|
|
|
beta <span style="color: #666666">=</span> fit_beta(X, y)
|
|
|
|
<span style="color: #408080; font-style: italic"># Intercept is included in the design matrix</span>
|
|
skl <span style="color: #666666">=</span> LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)<span style="color: #666666">.</span>fit(X, y)
|
|
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"True beta: </span><span style="color: #BB6688; font-weight: bold">{</span>true_beta<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Fitted beta: </span><span style="color: #BB6688; font-weight: bold">{</span>beta<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Sklearn fitted beta: </span><span style="color: #BB6688; font-weight: bold">{</span>skl<span style="color: #666666">.</span>coef_<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
ypredictOwn <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
|
|
ypredictSKL <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>predict(X)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with intercept column"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y,ypredictOwn))
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with intercept column from SKL"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y,ypredictSKL))
|
|
|
|
|
|
plt<span style="color: #666666">.</span>figure()
|
|
plt<span style="color: #666666">.</span>scatter(x, y, label<span style="color: #666666">=</span><span style="color: #BA2121">"Data"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, X <span style="color: #666666">@</span> beta, label<span style="color: #666666">=</span><span style="color: #BA2121">"Fit"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, skl<span style="color: #666666">.</span>predict(X), label<span style="color: #666666">=</span><span style="color: #BA2121">"Sklearn (fit_intercept=False)"</span>)
|
|
|
|
|
|
<span style="color: #408080; font-style: italic"># Do not include the intercept in the design matrix</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x), degree <span style="color: #666666">-</span> <span style="color: #666666">1</span>))
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> p <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(degree <span style="color: #666666">-</span> <span style="color: #666666">1</span>):
|
|
X[:, p] <span style="color: #666666">=</span> x <span style="color: #666666">**</span> (p <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Intercept is not included in the design matrix</span>
|
|
skl <span style="color: #666666">=</span> LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)<span style="color: #666666">.</span>fit(X, y)
|
|
|
|
<span style="color: #408080; font-style: italic"># Use centered values for X and y when computing coefficients</span>
|
|
y_offset <span style="color: #666666">=</span> np<span style="color: #666666">.</span>average(y, axis<span style="color: #666666">=0</span>)
|
|
X_offset <span style="color: #666666">=</span> np<span style="color: #666666">.</span>average(X, axis<span style="color: #666666">=0</span>)
|
|
|
|
beta <span style="color: #666666">=</span> fit_beta(X <span style="color: #666666">-</span> X_offset, y <span style="color: #666666">-</span> y_offset)
|
|
intercept <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(y_offset <span style="color: #666666">-</span> X_offset <span style="color: #666666">@</span> beta)
|
|
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Manual intercept: </span><span style="color: #BB6688; font-weight: bold">{</span>intercept<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Fitted beta (without intercept): </span><span style="color: #BB6688; font-weight: bold">{</span>beta<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Sklearn intercept: </span><span style="color: #BB6688; font-weight: bold">{</span>skl<span style="color: #666666">.</span>intercept_<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Sklearn fitted beta (without intercept): </span><span style="color: #BB6688; font-weight: bold">{</span>skl<span style="color: #666666">.</span>coef_<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
ypredictOwn <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
|
|
ypredictSKL <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>predict(X)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with Manual intercept"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y,ypredictOwn<span style="color: #666666">+</span>intercept))
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE with Sklearn intercept"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y,ypredictSKL))
|
|
|
|
plt<span style="color: #666666">.</span>plot(x, X <span style="color: #666666">@</span> beta <span style="color: #666666">+</span> intercept, <span style="color: #BA2121">"--"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Fit (manual intercept)"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, skl<span style="color: #666666">.</span>predict(X), <span style="color: #BA2121">"--"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Sklearn (fit_intercept=True)"</span>)
|
|
plt<span style="color: #666666">.</span>grid()
|
|
plt<span style="color: #666666">.</span>legend()
|
|
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div class="output_wrapper">
|
|
<div class="output">
|
|
<div class="output_area">
|
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<div class="output_subarea output_stream output_stdout output_text">
|
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</div>
|
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</div>
|
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</div>
|
|
</div>
|
|
</div>
|
|
|
|
<p>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).
|
|
</p>
|
|
|
|
<p>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.
|
|
</p>
|
|
|
|
<p>To remind the reader, the regularization term, with the intercept in Ridge regression, is given by</p>
|
|
$$
|
|
\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2,
|
|
$$
|
|
|
|
<p>but when we take out the intercept, this equation becomes</p>
|
|
$$
|
|
\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2.
|
|
$$
|
|
|
|
<p>For Lasso regression we have</p>
|
|
$$
|
|
\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert.
|
|
$$
|
|
|
|
<p>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.
|
|
</p>
|
|
|
|
<p>Armed with this wisdom, we attempt first to simply set the intercept equal to <b>False</b> in our implementation of Ridge regression for our well-known vanilla data set.</p>
|
|
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="inner_cell">
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> linear_model
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
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n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
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<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
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<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
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n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)
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Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">20</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n,Maxpolydegree))
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<span style="color: #408080; font-style: italic">#We include explicitely the intercept column</span>
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<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Maxpolydegree):
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X[:,degree] <span style="color: #666666">=</span> x<span style="color: #666666">**</span>degree
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<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
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X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
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p <span style="color: #666666">=</span> Maxpolydegree
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I <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(p,p)
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<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
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nlambdas <span style="color: #666666">=</span> <span style="color: #666666">6</span>
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MSEOwnRidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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MSERidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">2</span>, nlambdas)
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(nlambdas):
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lmb <span style="color: #666666">=</span> lambdas[i]
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OwnRidgeBeta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X_train<span style="color: #666666">+</span>lmb<span style="color: #666666">*</span>I) <span style="color: #666666">@</span> X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y_train
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<span style="color: #408080; font-style: italic"># Note: we include the intercept column and no scaling</span>
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RegRidge <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>Ridge(lmb,fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
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RegRidge<span style="color: #666666">.</span>fit(X_train,y_train)
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<span style="color: #408080; font-style: italic"># and then make the prediction</span>
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ytildeOwnRidge <span style="color: #666666">=</span> X_train <span style="color: #666666">@</span> OwnRidgeBeta
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ypredictOwnRidge <span style="color: #666666">=</span> X_test <span style="color: #666666">@</span> OwnRidgeBeta
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ytildeRidge <span style="color: #666666">=</span> RegRidge<span style="color: #666666">.</span>predict(X_train)
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ypredictRidge <span style="color: #666666">=</span> RegRidge<span style="color: #666666">.</span>predict(X_test)
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MSEOwnRidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictOwnRidge)
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MSERidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictRidge)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Beta values for own Ridge implementation"</span>)
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<span style="color: #008000">print</span>(OwnRidgeBeta)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Beta values for Scikit-Learn Ridge implementation"</span>)
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<span style="color: #008000">print</span>(RegRidge<span style="color: #666666">.</span>coef_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for own Ridge implementation"</span>)
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<span style="color: #008000">print</span>(MSEOwnRidgePredict[i])
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for Scikit-Learn Ridge implementation"</span>)
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<span style="color: #008000">print</span>(MSERidgePredict[i])
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<span style="color: #408080; font-style: italic"># Now plot the results</span>
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plt<span style="color: #666666">.</span>figure()
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSEOwnRidgePredict, <span style="color: #BA2121">'r'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE own Ridge Test'</span>)
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSERidgePredict, <span style="color: #BA2121">'g'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE Ridge Test'</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MSE'</span>)
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plt<span style="color: #666666">.</span>legend()
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plt<span style="color: #666666">.</span>show()
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</pre>
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<p>The results here agree when we force <b>Scikit-Learn</b>'s Ridge function to include the first column in our design matrix.
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We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
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What happens if we do not include the intercept in our fit?
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Let us see how we can change this code by zero centering.
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</p>
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> linear_model
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
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n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
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<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
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<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">315</span>)
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n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)
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Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">20</span>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n,Maxpolydegree<span style="color: #666666">-1</span>))
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<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,Maxpolydegree): <span style="color: #408080; font-style: italic">#No intercept column</span>
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X[:,degree<span style="color: #666666">-1</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**</span>(degree)
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<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
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X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
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<span style="color: #408080; font-style: italic">#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable</span>
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X_train_mean <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(X_train,axis<span style="color: #666666">=0</span>)
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<span style="color: #408080; font-style: italic">#Center by removing mean from each feature</span>
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X_train_scaled <span style="color: #666666">=</span> X_train <span style="color: #666666">-</span> X_train_mean
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X_test_scaled <span style="color: #666666">=</span> X_test <span style="color: #666666">-</span> X_train_mean
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<span style="color: #408080; font-style: italic">#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)</span>
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<span style="color: #408080; font-style: italic">#Remove the intercept from the training data.</span>
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y_scaler <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(y_train)
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y_train_scaled <span style="color: #666666">=</span> y_train <span style="color: #666666">-</span> y_scaler
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p <span style="color: #666666">=</span> Maxpolydegree<span style="color: #666666">-1</span>
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I <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(p,p)
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<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
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nlambdas <span style="color: #666666">=</span> <span style="color: #666666">6</span>
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MSEOwnRidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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MSERidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
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lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">2</span>, nlambdas)
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(nlambdas):
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lmb <span style="color: #666666">=</span> lambdas[i]
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OwnRidgeBeta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X_train_scaled<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X_train_scaled<span style="color: #666666">+</span>lmb<span style="color: #666666">*</span>I) <span style="color: #666666">@</span> X_train_scaled<span style="color: #666666">.</span>T <span style="color: #666666">@</span> (y_train_scaled)
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intercept_ <span style="color: #666666">=</span> y_scaler <span style="color: #666666">-</span> X_train_mean<span style="color: #AA22FF">@OwnRidgeBeta</span> <span style="color: #408080; font-style: italic">#The intercept can be shifted so the model can predict on uncentered data</span>
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<span style="color: #408080; font-style: italic">#Add intercept to prediction</span>
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ypredictOwnRidge <span style="color: #666666">=</span> X_test_scaled <span style="color: #666666">@</span> OwnRidgeBeta <span style="color: #666666">+</span> y_scaler
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RegRidge <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>Ridge(lmb)
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RegRidge<span style="color: #666666">.</span>fit(X_train,y_train)
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ypredictRidge <span style="color: #666666">=</span> RegRidge<span style="color: #666666">.</span>predict(X_test)
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MSEOwnRidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictOwnRidge)
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MSERidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictRidge)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Beta values for own Ridge implementation"</span>)
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<span style="color: #008000">print</span>(OwnRidgeBeta) <span style="color: #408080; font-style: italic">#Intercept is given by mean of target variable</span>
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"Beta values for Scikit-Learn Ridge implementation"</span>)
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<span style="color: #008000">print</span>(RegRidge<span style="color: #666666">.</span>coef_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Intercept from own implementation:'</span>)
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<span style="color: #008000">print</span>(intercept_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Intercept from Scikit-Learn Ridge implementation'</span>)
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<span style="color: #008000">print</span>(RegRidge<span style="color: #666666">.</span>intercept_)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for own Ridge implementation"</span>)
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<span style="color: #008000">print</span>(MSEOwnRidgePredict[i])
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for Scikit-Learn Ridge implementation"</span>)
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<span style="color: #008000">print</span>(MSERidgePredict[i])
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<span style="color: #408080; font-style: italic"># Now plot the results</span>
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plt<span style="color: #666666">.</span>figure()
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSEOwnRidgePredict, <span style="color: #BA2121">'b--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE own Ridge Test'</span>)
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plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSERidgePredict, <span style="color: #BA2121">'g--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE SL Ridge Test'</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MSE'</span>)
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plt<span style="color: #666666">.</span>legend()
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plt<span style="color: #666666">.</span>show()
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</pre>
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<p>We see here, when compared to the code which includes explicitely the
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intercept column, that our MSE value is actually smaller. This is
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because the regularization term does not include the intercept value
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\( \beta_0 \) in the fitting. This applies to Lasso regularization as
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well. It means that our optimization is now done only with the
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centered matrix and/or vector that enter the fitting procedure.
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</p>
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<p>
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<li><a href="._week36-bs007.html">8</a></li>
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<li><a href="._week36-bs008.html">9</a></li>
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<li><a href="._week36-bs009.html">10</a></li>
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<li><a href="._week36-bs010.html">11</a></li>
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<li><a href="._week36-bs011.html">12</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week36-bs048.html">49</a></li>
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