Morten Hjorth-Jensen
403 lines
24 KiB
HTML
403 lines
24 KiB
HTML
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<a class="navbar-brand" href="week35-bs.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</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="._week35-bs001.html#plans-for-week-35" style="font-size: 80%;"><b>Plans for week 35</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#reading-recommendations" style="font-size: 80%;"> Reading recommendations:</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs002.html#reminder-from-last-week" style="font-size: 80%;"><b>Reminder from last week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs003.html#the-equations-for-ordinary-least-squares" style="font-size: 80%;"><b>The equations for ordinary least squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs004.html#the-cost-loss-function" style="font-size: 80%;"><b>The cost/loss function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs015.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs015.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#some-useful-matrix-and-vector-expressions" style="font-size: 80%;"><b>Some useful matrix and vector expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs008.html#the-jacobian" style="font-size: 80%;"><b>The Jacobian</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs009.html#derivatives-example-1" style="font-size: 80%;"><b>Derivatives, example 1</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs010.html#example-2" style="font-size: 80%;"><b>Example 2</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs011.html#example-3" style="font-size: 80%;"><b>Example 3</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#example-4" style="font-size: 80%;"><b>Example 4</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#the-mean-squared-error-and-its-derivative" style="font-size: 80%;"><b>The mean squared error and its derivative</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs014.html#meet-the-hessian-matrix" style="font-size: 80%;"><b>Meet the Hessian Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs015.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#example-relevant-for-the-exercises" style="font-size: 80%;"><b>Example relevant for the exercises</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs017.html#own-code-for-ordinary-least-squares" style="font-size: 80%;"><b>Own code for Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#adding-error-analysis-and-training-set-up" style="font-size: 80%;"><b>Adding error analysis and training set up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs019.html#splitting-our-data-in-training-and-test-data" style="font-size: 80%;"><b>Splitting our Data in Training and Test data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs020.html#the-complete-code-with-a-simple-data-set" style="font-size: 80%;"><b>The complete code with a simple data set</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs021.html#making-your-own-test-train-splitting" style="font-size: 80%;"><b>Making your own test-train splitting</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs022.html#reducing-the-number-of-degrees-of-freedom-overarching-view" style="font-size: 80%;"><b>Reducing the number of degrees of freedom, overarching view</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#preprocessing-our-data" style="font-size: 80%;"><b>Preprocessing our data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs024.html#functionality-in-scikit-learn" style="font-size: 80%;"><b>Functionality in Scikit-Learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs025.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs026.html#frequently-used-scaling-functions" style="font-size: 80%;"><b>Frequently used scaling functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.html#example-of-own-standard-scaling" style="font-size: 80%;"><b>Example of own Standard scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs028.html#min-max-scaling" style="font-size: 80%;"><b>Min-Max Scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs029.html#testing-the-means-squared-error-as-function-of-complexity" style="font-size: 80%;"><b>Testing the Means Squared Error as function of Complexity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs030.html#mathematical-interpretation-of-ordinary-least-squares" style="font-size: 80%;"><b>Mathematical Interpretation of Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs031.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs032.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs033.html#the-singular-value-decomposition" style="font-size: 80%;"><b>The singular value decomposition</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs034.html#linear-regression-problems" style="font-size: 80%;"><b>Linear Regression Problems</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs035.html#fixing-the-singularity" style="font-size: 80%;"><b>Fixing the singularity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.html#ridge-and-lasso-regression" style="font-size: 80%;"><b>Ridge and LASSO Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs037.html#deriving-the-ridge-regression-equations" style="font-size: 80%;"><b>Deriving the Ridge Regression Equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs038.html#basic-math-of-the-svd" style="font-size: 80%;"><b>Basic math of the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs039.html#the-svd-a-fantastic-algorithm" style="font-size: 80%;"><b>The SVD, a Fantastic Algorithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs040.html#economy-size-svd" style="font-size: 80%;"><b>Economy-size SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs041.html#codes-for-the-svd" style="font-size: 80%;"><b>Codes for the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs042.html#note-about-svd-calculations" style="font-size: 80%;"><b>Note about SVD Calculations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs043.html#mathematics-of-the-svd-and-implications" style="font-size: 80%;"><b>Mathematics of the SVD and implications</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs044.html#example-matrix" style="font-size: 80%;"><b>Example Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs045.html#setting-up-the-matrix-to-be-inverted" style="font-size: 80%;"><b>Setting up the Matrix to be inverted</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs046.html#further-properties-important-for-our-analyses-later" style="font-size: 80%;"><b>Further properties (important for our analyses later)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs047.html#meet-the-covariance-matrix" style="font-size: 80%;"><b>Meet the Covariance Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs048.html#introducing-the-covariance-and-correlation-functions" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="#covariance-and-correlation-matrix" style="font-size: 80%;"><b>Covariance and Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs050.html#correlation-function-and-design-feature-matrix" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs051.html#covariance-matrix-examples" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs052.html#correlation-matrix" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs053.html#correlation-matrix-with-pandas" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs054.html#rewriting-the-covariance-and-or-correlation-matrix" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs055.html#linking-with-the-svd" style="font-size: 80%;"><b>Linking with the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs056.html#what-does-it-mean" style="font-size: 80%;"><b>What does it mean?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs057.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;"><b>And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs058.html#back-to-ridge-and-lasso-regression" style="font-size: 80%;"><b>Back to Ridge and LASSO Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs059.html#interpreting-the-ridge-results" style="font-size: 80%;"><b>Interpreting the Ridge results</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs060.html#more-interpretations" style="font-size: 80%;"><b>More interpretations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs061.html#deriving-the-lasso-regression-equations" style="font-size: 80%;"><b>Deriving the Lasso Regression Equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs062.html#material-for-exercises-week-35" style="font-size: 80%;"><b>Material for exercises week 35</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs062.html#important-technicalities-more-on-rescaling-data" style="font-size: 80%;"><b>Important technicalities: More on Rescaling data</b></a></li>
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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="part0049"></a>
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<!-- !split -->
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<h2 id="covariance-and-correlation-matrix" class="anchor">Covariance and Correlation Matrix </h2>
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<p>The covariance takes values between zero and infinity and may thus
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lead to problems with loss of numerical precision for particularly
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large values. It is common to scale the covariance matrix by
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introducing instead the correlation matrix defined via the so-called
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correlation function
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</p>
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$$
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\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
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$$
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<p>The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
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\in [-1,1] \). This avoids eventual problems with too large values. We
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can then define the correlation matrix for the two vectors \( \boldsymbol{x} \)
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and \( \boldsymbol{y} \) as
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</p>
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$$
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\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
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\mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
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\end{bmatrix},
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$$
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<p>In the above example this is the function we constructed using <b>pandas</b>.</p>
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<p>
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