Morten Hjorth-Jensen
329 lines
19 KiB
HTML
329 lines
19 KiB
HTML
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('Thursday September 9', 2, None, 'thursday-september-9'),
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('Interpreting the Ridge results',
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('More interpretations', 2, None, 'more-interpretations'),
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('Writing the Cost Function',
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<a class="navbar-brand" href="week36-bs.html">Week 36: Statistical interpretation of Linear Regression and Resampling techniques</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="._week36-bs002.html#thursday-september-9" style="font-size: 80%;">Thursday September 9</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs003.html#summary-from-last-week-and-examples" style="font-size: 80%;">Summary from last Week and Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.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-bs005.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-bs006.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-bs007.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-bs008.html#inverse-of-rectangular-matrix" style="font-size: 80%;">Inverse of Rectangular Matrix</a></li>
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<!-- navigation toc: --> <li><a href="#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs010.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-bs011.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-bs012.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-bs013.html#comparison-with-ols" style="font-size: 80%;">Comparison with OLS</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.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-bs016.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.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-bs018.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-bs019.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs023.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.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-bs025.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.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-bs028.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.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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<!-- navigation toc: --> <li><a href="._week36-bs030.html#using-cvxopt" style="font-size: 80%;">Using CVXOPT</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#the-simpler-example" style="font-size: 80%;">The simpler Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#friday-september-10" style="font-size: 80%;">Friday September 10</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.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-bs034.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.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-bs036.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-bs037.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-bs038.html#independent-and-identically-distrubuted-iid" style="font-size: 80%;">Independent and Identically Distrubuted (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs039.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-bs040.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-bs041.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-bs042.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs043.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs045.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-bs046.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-bs047.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.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-bs049.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-bs050.html#invoking-bayes-theorem" style="font-size: 80%;">Invoking Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs052.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0009"></a>
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<!-- !split -->
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<h2 id="ridge-and-lasso-regression" class="anchor">Ridge and LASSO Regression </h2>
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<p>Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is
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our optimization problem is
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</p>
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$$
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{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
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$$
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<p>or we can state it as</p>
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$$
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{\displaystyle \min_{\boldsymbol{\beta}\in
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{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
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$$
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<p>where we have used the definition of a norm-2 vector, that is</p>
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$$
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\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
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$$
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<p>
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<li><a href="._week36-bs008.html">«</a></li>
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<li><a href="._week36-bs000.html">1</a></li>
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<li class="active"><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="">...</a></li>
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<a href="https://..."><img width="250" align=right src="https://..."></a>
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</footer>
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-->
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<center style="font-size:80%">
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<!-- copyright only on the titlepage -->
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</center>
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</body>
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</html>
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