Morten Hjorth-Jensen
429 lines
24 KiB
HTML
429 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-august-24-28" style="font-size: 80%;"><b>Plans for week 35, August 24-28</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs002.html#thursday-august-27" style="font-size: 80%;"><b>Thursday August 27</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs003.html#why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week" style="font-size: 80%;"><b>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-bs004.html#regression-analysis-overarching-aims" style="font-size: 80%;"><b>Regression analysis, overarching aims</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs005.html#regression-analysis-overarching-aims-ii" style="font-size: 80%;"><b>Regression analysis, overarching aims II</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#general-linear-models" style="font-size: 80%;"><b>General linear models</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs008.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs009.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs011.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs011.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#optimizing-our-parameters" style="font-size: 80%;"><b>Optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#our-model-for-the-nuclear-binding-energies" style="font-size: 80%;"><b>Our model for the nuclear binding energies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs014.html#optimizing-our-parameters-more-details" style="font-size: 80%;"><b>Optimizing our parameters, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs019.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-bs019.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#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-bs018.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-bs019.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-bs020.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-bs021.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-bs022.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-bs023.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs024.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-bs025.html#the-boston-housing-data-example" style="font-size: 80%;"><b>The Boston housing data example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs026.html#housing-data-the-code" style="font-size: 80%;"><b>Housing data, the code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.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-bs028.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-bs029.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-bs030.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs031.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-bs032.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-bs033.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-bs034.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-bs035.html#more-preprocessing-examples-franke-function-and-regression" style="font-size: 80%;"><b>More preprocessing examples, Franke function and regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.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-bs037.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs038.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs039.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-bs040.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-bs041.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-bs042.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-bs043.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-bs044.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-bs045.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-bs046.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-bs047.html#mathematical-properties" style="font-size: 80%;"><b>Mathematical Properties</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs048.html#friday-september-3" style="font-size: 80%;"><b>Friday September 3</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs049.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-bs050.html#more-on-ridge-regression" style="font-size: 80%;"><b>More on Ridge Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs051.html#interpreting-the-ridge-results" style="font-size: 80%;"><b>Interpreting the Ridge results</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs052.html#more-interpretations" style="font-size: 80%;"><b>More interpretations</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs053.html#a-better-understanding-of-regularization" style="font-size: 80%;"><b>A better understanding of regularization</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs054.html#decomposing-the-ols-and-ridge-expressions" style="font-size: 80%;"><b>Decomposing the OLS and Ridge expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs055.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-bs056.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-bs057.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-bs058.html#correlation-matrix" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs059.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-bs060.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-bs061.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-bs063.html#exercise-1-adding-ridge-and-lasso-regression" style="font-size: 80%;"><b>Exercise 1: Adding Ridge and Lasso Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs063.html#exercise-linear-regression-for-a-two-dimensional-function" style="font-size: 80%;"> Exercise: Linear Regression for a two-dimensional function</a></li>
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<!-- ------------------- main content ---------------------- -->
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<div class="jumbotron">
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<center><h1>Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<b>Morten Hjorth-Jensen</b> [1, 2]
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<center><h4>Sep 2, 2021</h4></center> <!-- date -->
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<br>
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