Morten Hjorth-Jensen
422 lines
26 KiB
HTML
422 lines
26 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#for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week" style="font-size: 80%;"><b>For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat 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-bs016.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#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#other-useful-relations" style="font-size: 80%;"><b>Other useful relations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs015.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-bs016.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-bs017.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-bs018.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-bs019.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-bs020.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-bs021.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-bs022.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-bs023.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-bs024.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-bs025.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-bs026.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.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-bs028.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-bs029.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-bs030.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-bs031.html#more-preprocessing-examples-two-dimensional-example-the-franke-function" style="font-size: 80%;"><b>More preprocessing examples, two-dimensional example, the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs032.html#to-think-about-first-part" style="font-size: 80%;"><b>To think about, first part</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs033.html#more-thinking" style="font-size: 80%;"><b>More thinking</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs034.html#still-thinking" style="font-size: 80%;"><b>Still thinking</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs035.html#what-does-centering-subtracting-the-mean-values-mean-mathematically" style="font-size: 80%;"><b>What does centering (subtracting the mean values) mean mathematically?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.html#further-manipulations" style="font-size: 80%;"><b>Further Manipulations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs037.html#wrapping-it-up" style="font-size: 80%;"><b>Wrapping it up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs038.html#linear-regression-code-intercept-handling-first" style="font-size: 80%;"><b>Linear Regression code, Intercept handling first</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs039.html#material-for-lecture-monday-august-26" style="font-size: 80%;"><b>Material for lecture Monday, August 26</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs040.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-bs041.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs042.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs043.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-bs044.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-bs045.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-bs046.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-bs047.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-bs048.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-bs049.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-bs050.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-bs051.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-bs052.html#example-matrix" style="font-size: 80%;"><b>Example Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs053.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-bs054.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-bs055.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-bs056.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="._week35-bs057.html#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-bs058.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-bs059.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-bs060.html#correlation-matrix" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs061.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-bs062.html#correlation-matrix-with-pandas-and-the-franke-function" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs063.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-bs064.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-bs065.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-bs066.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-bs067.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-bs068.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-bs069.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-bs070.html#more-interpretations" style="font-size: 80%;"><b>More interpretations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs071.html#deriving-the-lasso-regression-equations" style="font-size: 80%;"><b>Deriving the Lasso Regression Equations</b></a></li>
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<h1>Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</h1>
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<!-- author(s): Morten Hjorth-Jensen -->
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<b>Morten Hjorth-Jensen</b>
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<b>Department of Physics, University of Oslo</b>
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<h4>August 26-30, 2024</h4>
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