Morten Hjorth-Jensen
386 lines
23 KiB
HTML
386 lines
23 KiB
HTML
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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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<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="._week36-bs002.html#summary-from-last-week-and-discussion-of-svd-ridge-and-lasso-regression-with-examples" style="font-size: 80%;">Summary from last Week and discussion of SVD, Ridge and Lasso regression with examples</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs003.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-bs004.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-bs005.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-bs006.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-bs007.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-bs008.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-bs009.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-bs010.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-bs011.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-bs012.html#comparison-with-ols" style="font-size: 80%;">Comparison with OLS</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.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-bs015.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.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-bs017.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-bs018.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs023.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-bs024.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.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-bs027.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.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-bs029.html#to-think-about-first-part" style="font-size: 80%;">To think about, first part</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs030.html#more-thinking" style="font-size: 80%;">More thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#still-thinking" style="font-size: 80%;">Still thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#what-does-centering-subtracting-the-mean-values-mean-mathematically" style="font-size: 80%;">What does centering (subtracting the mean values) mean mathematically?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.html#further-manipulations" style="font-size: 80%;">Further Manipulations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.html#linear-regression-code-intercept-handling-first" style="font-size: 80%;">Linear Regression code, Intercept handling first</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs036.html#code-examples" style="font-size: 80%;">Code Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs037.html#taking-out-the-mean" style="font-size: 80%;">Taking out the mean</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs038.html#friday-september-9" style="font-size: 80%;">Friday September 9</a></li>
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<!-- navigation toc: --> <li><a href="#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-bs040.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs041.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-bs042.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-bs043.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-bs044.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-bs045.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-bs046.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-bs047.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-bs048.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs049.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.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-bs052.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-bs053.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs054.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-bs055.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-bs056.html#invoking-bayes-theorem" style="font-size: 80%;">Invoking Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs057.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#exercise-1-mean-values-and-variances-in-linear-regression" style="font-size: 80%;">Exercise 1: mean values and variances in linear regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#exercise-2-adding-ridge-and-lasso-regression" style="font-size: 80%;">Exercise 2: Adding Ridge and Lasso Regression</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="part0039"></a>
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<!-- !split -->
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<h2 id="linking-the-regression-analysis-with-a-statistical-interpretation" class="anchor">Linking the regression analysis with a statistical interpretation </h2>
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<p>We will now couple the discussions of ordinary least squares, Ridge
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and Lasso regression with a statistical interpretation, that is we
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move from a linear algebra analysis to a statistical analysis. In
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particular, we will focus on what the regularization terms can result
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in. We will amongst other things show that the regularization
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parameter can reduce considerably the variance of the parameters
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\( \beta \).
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</p>
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<p>The
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advantage of doing linear regression is that we actually end up with
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analytical expressions for several statistical quantities.
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Standard least squares and Ridge regression allow us to
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derive quantities like the variance and other expectation values in a
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rather straightforward way.
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</p>
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<p>It is assumed that \( \varepsilon_i
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\sim \mathcal{N}(0, \sigma^2) \) and the \( \varepsilon_{i} \) are
|
|
independent, i.e.:
|
|
</p>
|
|
$$
|
|
\begin{align*}
|
|
\mbox{Cov}(\varepsilon_{i_1},
|
|
\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
|
|
& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right.
|
|
\end{align*}
|
|
$$
|
|
|
|
<p>The randomness of \( \varepsilon_i \) implies that
|
|
\( \mathbf{y}_i \) is also a random variable. In particular,
|
|
\( \mathbf{y}_i \) is normally distributed, because \( \varepsilon_i \sim
|
|
\mathcal{N}(0, \sigma^2) \) and \( \mathbf{X}_{i,\ast} \, \boldsymbol{\beta} \) is a
|
|
non-random scalar. To specify the parameters of the distribution of
|
|
\( \mathbf{y}_i \) we need to calculate its first two moments.
|
|
</p>
|
|
|
|
<p>Recall that \( \boldsymbol{X} \) is a matrix of dimensionality \( n\times p \). The
|
|
notation above \( \mathbf{X}_{i,\ast} \) means that we are looking at the
|
|
row number \( i \) and perform a sum over all values \( p \).
|
|
</p>
|
|
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|
<p>
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