Morten Hjorth-Jensen
522 lines
34 KiB
HTML
522 lines
34 KiB
HTML
<!--
|
|
HTML file automatically generated from DocOnce source
|
|
(https://github.com/doconce/doconce/)
|
|
doconce format html week35.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week35-bs --no_mako
|
|
-->
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
|
<meta name="description" content="Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression">
|
|
<title>Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</title>
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<!-- doconce format html week35.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week35-bs --no_mako -->
|
|
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
|
<!-- not necessary
|
|
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
|
-->
|
|
<style type="text/css">
|
|
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
|
.dropdown-menu {
|
|
height: auto;
|
|
max-height: 400px;
|
|
overflow-x: hidden;
|
|
}
|
|
/* Adds an invisible element before each target to offset for the navigation
|
|
bar */
|
|
.anchor::before {
|
|
content:"";
|
|
display:block;
|
|
height:50px; /* fixed header height for style bootstrap */
|
|
margin:-50px 0 0; /* negative fixed header height */
|
|
}
|
|
</style>
|
|
</head>
|
|
|
|
<!-- tocinfo
|
|
{'highest level': 2,
|
|
'sections': [('Plans for week 35', 2, None, 'plans-for-week-35'),
|
|
('Reading recommendations:', 3, None, 'reading-recommendations'),
|
|
('Reminder from last week', 2, None, 'reminder-from-last-week'),
|
|
('The equations for ordinary least squares',
|
|
2,
|
|
None,
|
|
'the-equations-for-ordinary-least-squares'),
|
|
('The cost/loss function', 2, None, 'the-cost-loss-function'),
|
|
('Interpretations and optimizing our parameters',
|
|
2,
|
|
None,
|
|
'interpretations-and-optimizing-our-parameters'),
|
|
('Interpretations and optimizing our parameters',
|
|
2,
|
|
None,
|
|
'interpretations-and-optimizing-our-parameters'),
|
|
('Some useful matrix and vector expressions',
|
|
2,
|
|
None,
|
|
'some-useful-matrix-and-vector-expressions'),
|
|
('The Jacobian', 2, None, 'the-jacobian'),
|
|
('Derivatives, example 1', 2, None, 'derivatives-example-1'),
|
|
('Example 2', 2, None, 'example-2'),
|
|
('Example 3', 2, None, 'example-3'),
|
|
('Example 4', 2, None, 'example-4'),
|
|
('The mean squared error and its derivative',
|
|
2,
|
|
None,
|
|
'the-mean-squared-error-and-its-derivative'),
|
|
('Meet the Hessian Matrix', 2, None, 'meet-the-hessian-matrix'),
|
|
('Interpretations and optimizing our parameters',
|
|
2,
|
|
None,
|
|
'interpretations-and-optimizing-our-parameters'),
|
|
('Example relevant for the exercises',
|
|
2,
|
|
None,
|
|
'example-relevant-for-the-exercises'),
|
|
('Own code for Ordinary Least Squares',
|
|
2,
|
|
None,
|
|
'own-code-for-ordinary-least-squares'),
|
|
('Adding error analysis and training set up',
|
|
2,
|
|
None,
|
|
'adding-error-analysis-and-training-set-up'),
|
|
('Splitting our Data in Training and Test data',
|
|
2,
|
|
None,
|
|
'splitting-our-data-in-training-and-test-data'),
|
|
('The complete code with a simple data set',
|
|
2,
|
|
None,
|
|
'the-complete-code-with-a-simple-data-set'),
|
|
('Making your own test-train splitting',
|
|
2,
|
|
None,
|
|
'making-your-own-test-train-splitting'),
|
|
('Reducing the number of degrees of freedom, overarching view',
|
|
2,
|
|
None,
|
|
'reducing-the-number-of-degrees-of-freedom-overarching-view'),
|
|
('Preprocessing our data', 2, None, 'preprocessing-our-data'),
|
|
('Functionality in Scikit-Learn',
|
|
2,
|
|
None,
|
|
'functionality-in-scikit-learn'),
|
|
('More preprocessing', 2, None, 'more-preprocessing'),
|
|
('Frequently used scaling functions',
|
|
2,
|
|
None,
|
|
'frequently-used-scaling-functions'),
|
|
('Example of own Standard scaling',
|
|
2,
|
|
None,
|
|
'example-of-own-standard-scaling'),
|
|
('Min-Max Scaling', 2, None, 'min-max-scaling'),
|
|
('Testing the Means Squared Error as function of Complexity',
|
|
2,
|
|
None,
|
|
'testing-the-means-squared-error-as-function-of-complexity'),
|
|
('Mathematical Interpretation of Ordinary Least Squares',
|
|
2,
|
|
None,
|
|
'mathematical-interpretation-of-ordinary-least-squares'),
|
|
('Residual Error', 2, None, 'residual-error'),
|
|
('Simple case', 2, None, 'simple-case'),
|
|
('The singular value decomposition',
|
|
2,
|
|
None,
|
|
'the-singular-value-decomposition'),
|
|
('Linear Regression Problems',
|
|
2,
|
|
None,
|
|
'linear-regression-problems'),
|
|
('Fixing the singularity', 2, None, 'fixing-the-singularity'),
|
|
('Ridge and LASSO Regression',
|
|
2,
|
|
None,
|
|
'ridge-and-lasso-regression'),
|
|
('Deriving the Ridge Regression Equations',
|
|
2,
|
|
None,
|
|
'deriving-the-ridge-regression-equations'),
|
|
('Basic math of the SVD', 2, None, 'basic-math-of-the-svd'),
|
|
('The SVD, a Fantastic Algorithm',
|
|
2,
|
|
None,
|
|
'the-svd-a-fantastic-algorithm'),
|
|
('Economy-size SVD', 2, None, 'economy-size-svd'),
|
|
('Codes for the SVD', 2, None, 'codes-for-the-svd'),
|
|
('Note about SVD Calculations',
|
|
2,
|
|
None,
|
|
'note-about-svd-calculations'),
|
|
('Mathematics of the SVD and implications',
|
|
2,
|
|
None,
|
|
'mathematics-of-the-svd-and-implications'),
|
|
('Example Matrix', 2, None, 'example-matrix'),
|
|
('Setting up the Matrix to be inverted',
|
|
2,
|
|
None,
|
|
'setting-up-the-matrix-to-be-inverted'),
|
|
('Further properties (important for our analyses later)',
|
|
2,
|
|
None,
|
|
'further-properties-important-for-our-analyses-later'),
|
|
('Meet the Covariance Matrix',
|
|
2,
|
|
None,
|
|
'meet-the-covariance-matrix'),
|
|
('Introducing the Covariance and Correlation functions',
|
|
2,
|
|
None,
|
|
'introducing-the-covariance-and-correlation-functions'),
|
|
('Covariance and Correlation Matrix',
|
|
2,
|
|
None,
|
|
'covariance-and-correlation-matrix'),
|
|
('Correlation Function and Design/Feature Matrix',
|
|
2,
|
|
None,
|
|
'correlation-function-and-design-feature-matrix'),
|
|
('Covariance Matrix Examples',
|
|
2,
|
|
None,
|
|
'covariance-matrix-examples'),
|
|
('Correlation Matrix', 2, None, 'correlation-matrix'),
|
|
('Correlation Matrix with Pandas',
|
|
2,
|
|
None,
|
|
'correlation-matrix-with-pandas'),
|
|
('Rewriting the Covariance and/or Correlation Matrix',
|
|
2,
|
|
None,
|
|
'rewriting-the-covariance-and-or-correlation-matrix'),
|
|
('Linking with the SVD', 2, None, 'linking-with-the-svd'),
|
|
('What does it mean?', 2, None, 'what-does-it-mean'),
|
|
('And finally $\\boldsymbol{X}\\boldsymbol{X}^T$',
|
|
2,
|
|
None,
|
|
'and-finally-boldsymbol-x-boldsymbol-x-t'),
|
|
('Back to Ridge and LASSO Regression',
|
|
2,
|
|
None,
|
|
'back-to-ridge-and-lasso-regression'),
|
|
('Interpreting the Ridge results',
|
|
2,
|
|
None,
|
|
'interpreting-the-ridge-results'),
|
|
('More interpretations', 2, None, 'more-interpretations'),
|
|
('Deriving the Lasso Regression Equations',
|
|
2,
|
|
None,
|
|
'deriving-the-lasso-regression-equations'),
|
|
('Material for exercises week 35',
|
|
2,
|
|
None,
|
|
'material-for-exercises-week-35'),
|
|
('Important technicalities: More on Rescaling data',
|
|
2,
|
|
None,
|
|
'important-technicalities-more-on-rescaling-data')]}
|
|
end of tocinfo -->
|
|
|
|
<body>
|
|
|
|
|
|
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
TeX: {
|
|
equationNumbers: { autoNumber: "none" },
|
|
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript" async
|
|
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
|
</script>
|
|
|
|
|
|
<!-- Bootstrap navigation bar -->
|
|
<div class="navbar navbar-default navbar-fixed-top">
|
|
<div class="navbar-header">
|
|
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
</button>
|
|
<a class="navbar-brand" href="week35-bs.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a>
|
|
</div>
|
|
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
|
<ul class="nav navbar-nav navbar-right">
|
|
<li class="dropdown">
|
|
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
|
<ul class="dropdown-menu">
|
|
<!-- navigation toc: --> <li><a href="._week35-bs001.html#plans-for-week-35" style="font-size: 80%;"><b>Plans for week 35</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs001.html#reading-recommendations" style="font-size: 80%;"> Reading recommendations:</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs002.html#reminder-from-last-week" style="font-size: 80%;"><b>Reminder from last week</b></a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs004.html#the-cost-loss-function" style="font-size: 80%;"><b>The cost/loss function</b></a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs008.html#the-jacobian" style="font-size: 80%;"><b>The Jacobian</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs009.html#derivatives-example-1" style="font-size: 80%;"><b>Derivatives, example 1</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs010.html#example-2" style="font-size: 80%;"><b>Example 2</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs011.html#example-3" style="font-size: 80%;"><b>Example 3</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs012.html#example-4" style="font-size: 80%;"><b>Example 4</b></a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs014.html#meet-the-hessian-matrix" style="font-size: 80%;"><b>Meet the Hessian Matrix</b></a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs023.html#preprocessing-our-data" style="font-size: 80%;"><b>Preprocessing our data</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs024.html#functionality-in-scikit-learn" style="font-size: 80%;"><b>Functionality in Scikit-Learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs025.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs026.html#frequently-used-scaling-functions" style="font-size: 80%;"><b>Frequently used scaling functions</b></a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs028.html#min-max-scaling" style="font-size: 80%;"><b>Min-Max Scaling</b></a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs031.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs032.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs033.html#the-singular-value-decomposition" style="font-size: 80%;"><b>The singular value decomposition</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs034.html#linear-regression-problems" style="font-size: 80%;"><b>Linear Regression Problems</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs035.html#fixing-the-singularity" style="font-size: 80%;"><b>Fixing the singularity</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs036.html#ridge-and-lasso-regression" style="font-size: 80%;"><b>Ridge and LASSO Regression</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="#deriving-the-ridge-regression-equations" style="font-size: 80%;"><b>Deriving the Ridge Regression Equations</b></a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs040.html#economy-size-svd" style="font-size: 80%;"><b>Economy-size SVD</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs041.html#codes-for-the-svd" style="font-size: 80%;"><b>Codes for the SVD</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs042.html#note-about-svd-calculations" style="font-size: 80%;"><b>Note about SVD Calculations</b></a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs044.html#example-matrix" style="font-size: 80%;"><b>Example Matrix</b></a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs047.html#meet-the-covariance-matrix" style="font-size: 80%;"><b>Meet the Covariance Matrix</b></a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs049.html#covariance-and-correlation-matrix" style="font-size: 80%;"><b>Covariance and Correlation Matrix</b></a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs051.html#covariance-matrix-examples" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs052.html#correlation-matrix" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs053.html#correlation-matrix-with-pandas" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs055.html#linking-with-the-svd" style="font-size: 80%;"><b>Linking with the SVD</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs056.html#what-does-it-mean" style="font-size: 80%;"><b>What does it mean?</b></a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs059.html#interpreting-the-ridge-results" style="font-size: 80%;"><b>Interpreting the Ridge results</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs060.html#more-interpretations" style="font-size: 80%;"><b>More interpretations</b></a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div> <!-- end of navigation bar -->
|
|
<div class="container">
|
|
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
|
<a name="part0037"></a>
|
|
<!-- !split -->
|
|
<h2 id="deriving-the-ridge-regression-equations" class="anchor">Deriving the Ridge Regression Equations </h2>
|
|
|
|
<p>Using the matrix-vector expression for Ridge regression and dropping the parameter \( 1/n \) in front of the standard means squared error equation, we have</p>
|
|
|
|
$$
|
|
C(\boldsymbol{X},\boldsymbol{\theta})=\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta})\right\}+\lambda\boldsymbol{\theta}^T\boldsymbol{\theta},
|
|
$$
|
|
|
|
<p>and
|
|
taking the derivatives with respect to \( \boldsymbol{\theta} \) we obtain then
|
|
a slightly modified matrix inversion problem which for finite values
|
|
of \( \lambda \) does not suffer from singularity problems. We obtain
|
|
the optimal parameters
|
|
</p>
|
|
$$
|
|
\hat{\boldsymbol{\theta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
|
|
$$
|
|
|
|
<p>with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that</p>
|
|
|
|
$$
|
|
\sum_{i=0}^{p-1} \theta_i^2 \leq t,
|
|
$$
|
|
|
|
<p>with \( t \) a finite positive number. </p>
|
|
|
|
<p>If we keep the \( 1/n \) factor, the equation for the optimal \( \theta \) changes to</p>
|
|
$$
|
|
\hat{\boldsymbol{\theta}}_{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+n\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
|
|
$$
|
|
|
|
<p>In many textbooks the \( 1/n \) term is often omitted. Note that a library like <b>Scikit-Learn</b> does not include the \( 1/n \) factor in the setup of the cost function.</p>
|
|
|
|
<p>When we compare this with the ordinary least squares result we have</p>
|
|
$$
|
|
\hat{\boldsymbol{\theta}}_{\mathrm{OLS}} = \left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
|
|
$$
|
|
|
|
<p>which can lead to singular matrices. However, with the SVD, we can always compute the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \).</p>
|
|
|
|
<p>We see that Ridge regression is nothing but the standard OLS with a
|
|
modified diagonal term added to \( \boldsymbol{X}^T\boldsymbol{X} \). The consequences, in
|
|
particular for our discussion of the bias-variance tradeoff are rather
|
|
interesting. We will see that for specific values of \( \lambda \), we may
|
|
even reduce the variance of the optimal parameters \( \boldsymbol{\theta} \). These topics and other related ones, will be discussed after the more linear algebra oriented analysis here.
|
|
</p>
|
|
|
|
<p>When we have discussed the singular value decomposition of the design
|
|
matrix \( \boldsymbol{X} \), we will in turn perform a more rigorous mathematical
|
|
discussion of Ridge regression.
|
|
</p>
|
|
|
|
<p>The code here is a simple demonstration of how to implement Ridge regression with our own code and compare this with scikit-learn.</p>
|
|
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="cell border-box-sizing code_cell rendered">
|
|
<div class="input">
|
|
<div class="inner_cell">
|
|
<div class="input_area">
|
|
<div class="highlight" style="background: #f8f8f8">
|
|
<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> linear_model
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
|
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
|
|
|
|
|
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
|
|
<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
|
|
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)
|
|
|
|
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">20</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n,Maxpolydegree))
|
|
<span style="color: #408080; font-style: italic">#We include explicitely the intercept column</span>
|
|
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Maxpolydegree):
|
|
X[:,degree] <span style="color: #666666">=</span> x<span style="color: #666666">**</span>degree
|
|
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
|
|
|
|
p <span style="color: #666666">=</span> Maxpolydegree
|
|
I <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(p,p)
|
|
<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
|
|
nlambdas <span style="color: #666666">=</span> <span style="color: #666666">6</span>
|
|
MSEOwnRidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
|
|
MSERidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
|
|
lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">2</span>, nlambdas)
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(nlambdas):
|
|
lmb <span style="color: #666666">=</span> lambdas[i]
|
|
OwnRidgeTheta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X_train<span style="color: #666666">+</span>lmb<span style="color: #666666">*</span>I) <span style="color: #666666">@</span> X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y_train
|
|
<span style="color: #408080; font-style: italic"># Note: we include the intercept column and no scaling</span>
|
|
RegRidge <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>Ridge(lmb,fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
|
|
RegRidge<span style="color: #666666">.</span>fit(X_train,y_train)
|
|
<span style="color: #408080; font-style: italic"># and then make the prediction</span>
|
|
ytildeOwnRidge <span style="color: #666666">=</span> X_train <span style="color: #666666">@</span> OwnRidgeTheta
|
|
ypredictOwnRidge <span style="color: #666666">=</span> X_test <span style="color: #666666">@</span> OwnRidgeTheta
|
|
ytildeRidge <span style="color: #666666">=</span> RegRidge<span style="color: #666666">.</span>predict(X_train)
|
|
ypredictRidge <span style="color: #666666">=</span> RegRidge<span style="color: #666666">.</span>predict(X_test)
|
|
MSEOwnRidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictOwnRidge)
|
|
MSERidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictRidge)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Theta values for own Ridge implementation"</span>)
|
|
<span style="color: #008000">print</span>(OwnRidgeTheta)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Theta values for Scikit-Learn Ridge implementation"</span>)
|
|
<span style="color: #008000">print</span>(RegRidge<span style="color: #666666">.</span>coef_)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for own Ridge implementation"</span>)
|
|
<span style="color: #008000">print</span>(MSEOwnRidgePredict[i])
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"MSE values for Scikit-Learn Ridge implementation"</span>)
|
|
<span style="color: #008000">print</span>(MSERidgePredict[i])
|
|
|
|
<span style="color: #408080; font-style: italic"># Now plot the results</span>
|
|
plt<span style="color: #666666">.</span>figure()
|
|
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSEOwnRidgePredict, <span style="color: #BA2121">'r'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE own Ridge Test'</span>)
|
|
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSERidgePredict, <span style="color: #BA2121">'g'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE Ridge Test'</span>)
|
|
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MSE'</span>)
|
|
plt<span style="color: #666666">.</span>legend()
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div class="output_wrapper">
|
|
<div class="output">
|
|
<div class="output_area">
|
|
<div class="output_subarea output_stream output_stdout output_text">
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<p>The results here agree when we force <b>Scikit-Learn</b>'s Ridge function to include the first column in our design matrix.
|
|
We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
|
|
What happens if we do not include the intercept in our fit? We will discuss this in more detail next week.
|
|
</p>
|
|
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week35-bs036.html">«</a></li>
|
|
<li><a href="._week35-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week35-bs029.html">30</a></li>
|
|
<li><a href="._week35-bs030.html">31</a></li>
|
|
<li><a href="._week35-bs031.html">32</a></li>
|
|
<li><a href="._week35-bs032.html">33</a></li>
|
|
<li><a href="._week35-bs033.html">34</a></li>
|
|
<li><a href="._week35-bs034.html">35</a></li>
|
|
<li><a href="._week35-bs035.html">36</a></li>
|
|
<li><a href="._week35-bs036.html">37</a></li>
|
|
<li class="active"><a href="._week35-bs037.html">38</a></li>
|
|
<li><a href="._week35-bs038.html">39</a></li>
|
|
<li><a href="._week35-bs039.html">40</a></li>
|
|
<li><a href="._week35-bs040.html">41</a></li>
|
|
<li><a href="._week35-bs041.html">42</a></li>
|
|
<li><a href="._week35-bs042.html">43</a></li>
|
|
<li><a href="._week35-bs043.html">44</a></li>
|
|
<li><a href="._week35-bs044.html">45</a></li>
|
|
<li><a href="._week35-bs045.html">46</a></li>
|
|
<li><a href="._week35-bs046.html">47</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week35-bs062.html">63</a></li>
|
|
<li><a href="._week35-bs038.html">»</a></li>
|
|
</ul>
|
|
<!-- ------------------- end of main content --------------- -->
|
|
</div> <!-- end container -->
|
|
<!-- include javascript, jQuery *first* -->
|
|
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
|
|
<script src="https://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
|
|
<!-- Bootstrap footer
|
|
<footer>
|
|
<a href="https://..."><img width="250" align=right src="https://..."></a>
|
|
</footer>
|
|
-->
|
|
<center style="font-size:80%">
|
|
<!-- copyright only on the titlepage -->
|
|
</center>
|
|
</body>
|
|
</html>
|
|
|