Morten Hjorth-Jensen
442 lines
24 KiB
HTML
442 lines
24 KiB
HTML
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'sections': [('Plans for week 36', 2, None, 'plans-for-week-36'),
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('Thursday September 9', 2, None, 'thursday-september-9'),
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('Summary from last Week and Examples',
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('Simple code for solving the above problem',
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('Another Example, now with a polynomial fit',
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('Linking the regression analysis with a statistical '
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('The bias-variance tradeoff',
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('Understanding what happens',
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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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</div>
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<ul class="nav navbar-nav navbar-right">
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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#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="._week36-bs009.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-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#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.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-bs014.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.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-bs016.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-bs017.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.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-bs023.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.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-bs026.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.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-bs028.html#friday-september-10" style="font-size: 80%;">Friday September 10</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.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-bs030.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.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-bs032.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-bs033.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-bs034.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-bs035.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-bs036.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-bs037.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-bs038.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs039.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs040.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs041.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-bs042.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-bs043.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.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-bs048.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs047.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs049.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#resampling-methods-jackknife-and-bootstrap" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs052.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs053.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs054.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs055.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs056.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs057.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs059.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs060.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs061.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs062.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs063.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs064.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs065.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs066.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs067.html#summing-up" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs068.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs069.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs070.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs071.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
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<!-- ------------------- main content ---------------------- -->
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<div class="jumbotron">
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<center><h1>Week 36: Statistical interpretation of Linear Regression and Resampling techniques</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p>
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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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<p>
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<center><h4>Sep 10, 2021</h4></center> <!-- date -->
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<br>
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<p><a href="._week36-bs001.html" class="btn btn-primary btn-lg">Read »</a></p>
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<li class="active"><a href="._week36-bs000.html">1</a></li>
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<li><a href="._week36-bs002.html">3</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week36-bs071.html">72</a></li>
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<center style="font-size:80%">
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<!-- copyright --> © 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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