added dot files
This commit is contained in:
mhjensen
@@ -0,0 +1,540 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis">
|
||||
|
||||
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<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': [('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec1'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec2'),
|
||||
('Examples', 2, None, '___sec3'),
|
||||
('General linear models', 2, None, '___sec4'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec5'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Optimizing our parameters', 2, None, '___sec9'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec11'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec13'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec16'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec18'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The code', 2, None, '___sec25'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The Boston housing data example', 2, None, '___sec27'),
|
||||
('Housing data, the code', 2, None, '___sec28'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('Preprocessing our data', 2, None, '___sec30'),
|
||||
('More preprocessing', 2, None, '___sec31'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec32'),
|
||||
('The singular value decomposition', 2, None, '___sec33'),
|
||||
('Linear Regression Problems', 2, None, '___sec34'),
|
||||
('Fixing the singularity', 2, None, '___sec35'),
|
||||
('Basic math of the SVD', 2, None, '___sec36'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
|
||||
('Another Example', 2, None, '___sec38'),
|
||||
('Economy-size SVD', 2, None, '___sec39'),
|
||||
('Mathematical Properties', 2, None, '___sec40'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec41'),
|
||||
('More on Ridge Regression', 2, None, '___sec42'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec43'),
|
||||
('More interpretations', 2, None, '___sec44'),
|
||||
('Codes for the SVD', 2, None, '___sec45'),
|
||||
('A better understanding of regularization', 2, None, '___sec46'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec48'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec50'),
|
||||
('Correlation Matrix', 2, None, '___sec51'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec52'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec53'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec54'),
|
||||
('Linking with SVD', 2, None, '___sec55'),
|
||||
('Where are we going?', 2, None, '___sec56'),
|
||||
('Resampling methods', 2, None, '___sec57'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec58'),
|
||||
('Why resampling methods ?', 2, None, '___sec59'),
|
||||
('Statistical analysis', 2, None, '___sec60'),
|
||||
('Statistics', 2, None, '___sec61'),
|
||||
('Statistics, moments', 2, None, '___sec62'),
|
||||
('Statistics, central moments', 2, None, '___sec63'),
|
||||
('Statistics, covariance', 2, None, '___sec64'),
|
||||
('Statistics, more covariance', 2, None, '___sec65'),
|
||||
('Covariance example', 2, None, '___sec66'),
|
||||
('Covariance in numpy', 2, None, '___sec67'),
|
||||
('Statistics, independent variables', 2, None, '___sec68'),
|
||||
('Statistics, more variance', 2, None, '___sec69'),
|
||||
('Statistics and stochastic processes', 2, None, '___sec70'),
|
||||
('Statistics and sample variables', 2, None, '___sec71'),
|
||||
('Statistics, sample variance and covariance',
|
||||
2,
|
||||
None,
|
||||
'___sec72'),
|
||||
('Statistics, law of large numbers', 2, None, '___sec73'),
|
||||
('Statistics, more on sample error', 2, None, '___sec74'),
|
||||
('Statistics', 2, None, '___sec75'),
|
||||
('Statistics, central limit theorem', 2, None, '___sec76'),
|
||||
('Statistics, more technicalities', 2, None, '___sec77'),
|
||||
('Statistics', 2, None, '___sec78'),
|
||||
('Statistics and sample variance', 2, None, '___sec79'),
|
||||
('Statistics, uncorrelated results', 2, None, '___sec80'),
|
||||
('Statistics, computations', 2, None, '___sec81'),
|
||||
('Statistics, more on computations of errors',
|
||||
2,
|
||||
None,
|
||||
'___sec82'),
|
||||
('Statistics, wrapping up 1', 2, None, '___sec83'),
|
||||
('Statistics, final expression', 2, None, '___sec84'),
|
||||
('Statistics, effective number of correlations',
|
||||
2,
|
||||
None,
|
||||
'___sec85'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec86'),
|
||||
('Assumptions made', 2, None, '___sec87'),
|
||||
('Expectation value and variance', 2, None, '___sec88'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec89'),
|
||||
('Resampling methods', 2, None, '___sec90'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec91'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec92'),
|
||||
('Jackknife code example', 2, None, '___sec93'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec94'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec95'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec96'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec97'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec98'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec99'),
|
||||
('Various steps in cross-validation', 2, None, '___sec100'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec101'),
|
||||
('Cross-validation in brief', 2, None, '___sec102'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec103'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec104'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec105'),
|
||||
('Understanding what happens', 2, None, '___sec106'),
|
||||
('Summing up', 2, None, '___sec107'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec108'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec109'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec110'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec111'),
|
||||
('The Ising model', 2, None, '___sec112'),
|
||||
('Reformulating the problem to suit regression',
|
||||
2,
|
||||
None,
|
||||
'___sec113'),
|
||||
('Linear regression', 2, None, '___sec114'),
|
||||
('Singular Value decomposition', 2, None, '___sec115'),
|
||||
('The one-dimensional Ising model', 2, None, '___sec116'),
|
||||
('Ridge regression', 2, None, '___sec117'),
|
||||
('LASSO regression', 2, None, '___sec118'),
|
||||
('Performance as function of the regularization parameter',
|
||||
2,
|
||||
None,
|
||||
'___sec119'),
|
||||
('Finding the optimal value of $\\lambda$',
|
||||
2,
|
||||
None,
|
||||
'___sec120')]}
|
||||
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="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</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="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs114.html#___sec113" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec114" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs116.html#___sec115" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs117.html#___sec116" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs118.html#___sec117" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs119.html#___sec118" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs120.html#___sec119" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs121.html#___sec120" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="part0115"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec114" class="anchor">Linear regression </h2>
|
||||
|
||||
<p>
|
||||
In the ordinary least squares method we choose the cost function
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
|
||||
\tag{25}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
|
||||
This yields the expression for \( \boldsymbol{\beta} \) to be
|
||||
|
||||
$$
|
||||
\boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
|
||||
$$
|
||||
|
||||
<p>
|
||||
which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
|
||||
an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
|
||||
intercept, i.e., a constant term, we must make sure that the
|
||||
first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
|
||||
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
|
||||
axis<span style="color: #666666">=1</span>
|
||||
)
|
||||
X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
|
||||
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
|
||||
axis<span style="color: #666666">=1</span>
|
||||
)
|
||||
</pre></div>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_inv</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-></span> np<span style="color: #666666">.</span>ndarray:
|
||||
<span style="color: #008000; font-weight: bold">return</span> scl<span style="color: #666666">.</span>inv(x<span style="color: #666666">.</span>T <span style="color: #666666">@</span> x) <span style="color: #666666">@</span> (x<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
|
||||
beta <span style="color: #666666">=</span> ols_inv(X_train_own, y_train)
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._Regression-bs114.html">«</a></li>
|
||||
<li><a href="._Regression-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs107.html">108</a></li>
|
||||
<li><a href="._Regression-bs108.html">109</a></li>
|
||||
<li><a href="._Regression-bs109.html">110</a></li>
|
||||
<li><a href="._Regression-bs110.html">111</a></li>
|
||||
<li><a href="._Regression-bs111.html">112</a></li>
|
||||
<li><a href="._Regression-bs112.html">113</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs114.html">115</a></li>
|
||||
<li class="active"><a href="._Regression-bs115.html">116</a></li>
|
||||
<li><a href="._Regression-bs116.html">117</a></li>
|
||||
<li><a href="._Regression-bs117.html">118</a></li>
|
||||
<li><a href="._Regression-bs118.html">119</a></li>
|
||||
<li><a href="._Regression-bs119.html">120</a></li>
|
||||
<li><a href="._Regression-bs120.html">121</a></li>
|
||||
<li><a href="._Regression-bs121.html">122</a></li>
|
||||
<li><a href="._Regression-bs116.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="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,578 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis">
|
||||
|
||||
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<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': [('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec1'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec2'),
|
||||
('Examples', 2, None, '___sec3'),
|
||||
('General linear models', 2, None, '___sec4'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec5'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Optimizing our parameters', 2, None, '___sec9'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec11'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec13'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec16'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec18'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The code', 2, None, '___sec25'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The Boston housing data example', 2, None, '___sec27'),
|
||||
('Housing data, the code', 2, None, '___sec28'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('Preprocessing our data', 2, None, '___sec30'),
|
||||
('More preprocessing', 2, None, '___sec31'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec32'),
|
||||
('The singular value decomposition', 2, None, '___sec33'),
|
||||
('Linear Regression Problems', 2, None, '___sec34'),
|
||||
('Fixing the singularity', 2, None, '___sec35'),
|
||||
('Basic math of the SVD', 2, None, '___sec36'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
|
||||
('Another Example', 2, None, '___sec38'),
|
||||
('Economy-size SVD', 2, None, '___sec39'),
|
||||
('Mathematical Properties', 2, None, '___sec40'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec41'),
|
||||
('More on Ridge Regression', 2, None, '___sec42'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec43'),
|
||||
('More interpretations', 2, None, '___sec44'),
|
||||
('Codes for the SVD', 2, None, '___sec45'),
|
||||
('A better understanding of regularization', 2, None, '___sec46'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec48'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec50'),
|
||||
('Correlation Matrix', 2, None, '___sec51'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec52'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec53'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec54'),
|
||||
('Linking with SVD', 2, None, '___sec55'),
|
||||
('Where are we going?', 2, None, '___sec56'),
|
||||
('Resampling methods', 2, None, '___sec57'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec58'),
|
||||
('Why resampling methods ?', 2, None, '___sec59'),
|
||||
('Statistical analysis', 2, None, '___sec60'),
|
||||
('Statistics', 2, None, '___sec61'),
|
||||
('Statistics, moments', 2, None, '___sec62'),
|
||||
('Statistics, central moments', 2, None, '___sec63'),
|
||||
('Statistics, covariance', 2, None, '___sec64'),
|
||||
('Statistics, more covariance', 2, None, '___sec65'),
|
||||
('Covariance example', 2, None, '___sec66'),
|
||||
('Covariance in numpy', 2, None, '___sec67'),
|
||||
('Statistics, independent variables', 2, None, '___sec68'),
|
||||
('Statistics, more variance', 2, None, '___sec69'),
|
||||
('Statistics and stochastic processes', 2, None, '___sec70'),
|
||||
('Statistics and sample variables', 2, None, '___sec71'),
|
||||
('Statistics, sample variance and covariance',
|
||||
2,
|
||||
None,
|
||||
'___sec72'),
|
||||
('Statistics, law of large numbers', 2, None, '___sec73'),
|
||||
('Statistics, more on sample error', 2, None, '___sec74'),
|
||||
('Statistics', 2, None, '___sec75'),
|
||||
('Statistics, central limit theorem', 2, None, '___sec76'),
|
||||
('Statistics, more technicalities', 2, None, '___sec77'),
|
||||
('Statistics', 2, None, '___sec78'),
|
||||
('Statistics and sample variance', 2, None, '___sec79'),
|
||||
('Statistics, uncorrelated results', 2, None, '___sec80'),
|
||||
('Statistics, computations', 2, None, '___sec81'),
|
||||
('Statistics, more on computations of errors',
|
||||
2,
|
||||
None,
|
||||
'___sec82'),
|
||||
('Statistics, wrapping up 1', 2, None, '___sec83'),
|
||||
('Statistics, final expression', 2, None, '___sec84'),
|
||||
('Statistics, effective number of correlations',
|
||||
2,
|
||||
None,
|
||||
'___sec85'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec86'),
|
||||
('Assumptions made', 2, None, '___sec87'),
|
||||
('Expectation value and variance', 2, None, '___sec88'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec89'),
|
||||
('Resampling methods', 2, None, '___sec90'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec91'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec92'),
|
||||
('Jackknife code example', 2, None, '___sec93'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec94'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec95'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec96'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec97'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec98'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec99'),
|
||||
('Various steps in cross-validation', 2, None, '___sec100'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec101'),
|
||||
('Cross-validation in brief', 2, None, '___sec102'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec103'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec104'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec105'),
|
||||
('Understanding what happens', 2, None, '___sec106'),
|
||||
('Summing up', 2, None, '___sec107'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec108'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec109'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec110'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec111'),
|
||||
('The Ising model', 2, None, '___sec112'),
|
||||
('Reformulating the problem to suit regression',
|
||||
2,
|
||||
None,
|
||||
'___sec113'),
|
||||
('Linear regression', 2, None, '___sec114'),
|
||||
('Singular Value decomposition', 2, None, '___sec115'),
|
||||
('The one-dimensional Ising model', 2, None, '___sec116'),
|
||||
('Ridge regression', 2, None, '___sec117'),
|
||||
('LASSO regression', 2, None, '___sec118'),
|
||||
('Performance as function of the regularization parameter',
|
||||
2,
|
||||
None,
|
||||
'___sec119'),
|
||||
('Finding the optimal value of $\\lambda$',
|
||||
2,
|
||||
None,
|
||||
'___sec120')]}
|
||||
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="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</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="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs114.html#___sec113" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs115.html#___sec114" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec115" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs117.html#___sec116" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs118.html#___sec117" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs119.html#___sec118" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs120.html#___sec119" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs121.html#___sec120" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="part0116"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec115" class="anchor">Singular Value decomposition </h2>
|
||||
|
||||
<p>
|
||||
Doing the inversion directly turns out to be a bad idea since the matrix
|
||||
\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
|
||||
value decomposition</b>. Using the definition of the Moore-Penrose
|
||||
pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
|
||||
|
||||
$$
|
||||
\boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
|
||||
$$
|
||||
|
||||
<p>
|
||||
where the pseudoinverse of \( \boldsymbol{X} \) is given by
|
||||
|
||||
$$
|
||||
\boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
Using singular value decomposition we can decompose the matrix \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
|
||||
where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
|
||||
where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
|
||||
\( \omega \) to
|
||||
$$
|
||||
\begin{align}
|
||||
\boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
|
||||
\tag{26}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
Note that solving this equation by actually doing the pseudoinverse
|
||||
(which is what we will do) is not a good idea as this operation scales
|
||||
as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
|
||||
general matrix. Instead, doing \( QR \)-factorization and solving the
|
||||
linear system as an equation would reduce this down to
|
||||
\( \mathcal{O}(n^2) \) operations.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_svd</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-></span> np<span style="color: #666666">.</span>ndarray:
|
||||
u, s, v <span style="color: #666666">=</span> scl<span style="color: #666666">.</span>svd(x)
|
||||
<span style="color: #008000; font-weight: bold">return</span> v<span style="color: #666666">.</span>T <span style="color: #666666">@</span> scl<span style="color: #666666">.</span>pinv(scl<span style="color: #666666">.</span>diagsvd(s, u<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], v<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])) <span style="color: #666666">@</span> u<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
|
||||
</pre></div>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>beta <span style="color: #666666">=</span> ols_svd(X_train_own,y_train)
|
||||
</pre></div>
|
||||
<p>
|
||||
When extracting the \( J \)-matrix we need to make sure that we remove the intercept, as is done here
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J <span style="color: #666666">=</span> beta[<span style="color: #666666">1</span>:]<span style="color: #666666">.</span>reshape(L, L)
|
||||
</pre></div>
|
||||
<p>
|
||||
A way of looking at the coefficients in \( J \) is to plot the matrices as images.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
|
||||
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J, <span style="color: #666666">**</span>cmap_args)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"OLS"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
|
||||
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
|
||||
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
It is interesting to note that OLS
|
||||
considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
|
||||
valid matrix elements for \( J \).
|
||||
In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
|
||||
this problem can be removed, partly and only with Lasso regression.
|
||||
|
||||
<p>
|
||||
In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._Regression-bs115.html">«</a></li>
|
||||
<li><a href="._Regression-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs108.html">109</a></li>
|
||||
<li><a href="._Regression-bs109.html">110</a></li>
|
||||
<li><a href="._Regression-bs110.html">111</a></li>
|
||||
<li><a href="._Regression-bs111.html">112</a></li>
|
||||
<li><a href="._Regression-bs112.html">113</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs114.html">115</a></li>
|
||||
<li><a href="._Regression-bs115.html">116</a></li>
|
||||
<li class="active"><a href="._Regression-bs116.html">117</a></li>
|
||||
<li><a href="._Regression-bs117.html">118</a></li>
|
||||
<li><a href="._Regression-bs118.html">119</a></li>
|
||||
<li><a href="._Regression-bs119.html">120</a></li>
|
||||
<li><a href="._Regression-bs120.html">121</a></li>
|
||||
<li><a href="._Regression-bs121.html">122</a></li>
|
||||
<li><a href="._Regression-bs117.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="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,624 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis">
|
||||
|
||||
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<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': [('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec1'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec2'),
|
||||
('Examples', 2, None, '___sec3'),
|
||||
('General linear models', 2, None, '___sec4'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec5'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Optimizing our parameters', 2, None, '___sec9'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec11'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec13'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec16'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec18'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The code', 2, None, '___sec25'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The Boston housing data example', 2, None, '___sec27'),
|
||||
('Housing data, the code', 2, None, '___sec28'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('Preprocessing our data', 2, None, '___sec30'),
|
||||
('More preprocessing', 2, None, '___sec31'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec32'),
|
||||
('The singular value decomposition', 2, None, '___sec33'),
|
||||
('Linear Regression Problems', 2, None, '___sec34'),
|
||||
('Fixing the singularity', 2, None, '___sec35'),
|
||||
('Basic math of the SVD', 2, None, '___sec36'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
|
||||
('Another Example', 2, None, '___sec38'),
|
||||
('Economy-size SVD', 2, None, '___sec39'),
|
||||
('Mathematical Properties', 2, None, '___sec40'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec41'),
|
||||
('More on Ridge Regression', 2, None, '___sec42'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec43'),
|
||||
('More interpretations', 2, None, '___sec44'),
|
||||
('Codes for the SVD', 2, None, '___sec45'),
|
||||
('A better understanding of regularization', 2, None, '___sec46'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec48'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec50'),
|
||||
('Correlation Matrix', 2, None, '___sec51'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec52'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec53'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec54'),
|
||||
('Linking with SVD', 2, None, '___sec55'),
|
||||
('Where are we going?', 2, None, '___sec56'),
|
||||
('Resampling methods', 2, None, '___sec57'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec58'),
|
||||
('Why resampling methods ?', 2, None, '___sec59'),
|
||||
('Statistical analysis', 2, None, '___sec60'),
|
||||
('Statistics', 2, None, '___sec61'),
|
||||
('Statistics, moments', 2, None, '___sec62'),
|
||||
('Statistics, central moments', 2, None, '___sec63'),
|
||||
('Statistics, covariance', 2, None, '___sec64'),
|
||||
('Statistics, more covariance', 2, None, '___sec65'),
|
||||
('Covariance example', 2, None, '___sec66'),
|
||||
('Covariance in numpy', 2, None, '___sec67'),
|
||||
('Statistics, independent variables', 2, None, '___sec68'),
|
||||
('Statistics, more variance', 2, None, '___sec69'),
|
||||
('Statistics and stochastic processes', 2, None, '___sec70'),
|
||||
('Statistics and sample variables', 2, None, '___sec71'),
|
||||
('Statistics, sample variance and covariance',
|
||||
2,
|
||||
None,
|
||||
'___sec72'),
|
||||
('Statistics, law of large numbers', 2, None, '___sec73'),
|
||||
('Statistics, more on sample error', 2, None, '___sec74'),
|
||||
('Statistics', 2, None, '___sec75'),
|
||||
('Statistics, central limit theorem', 2, None, '___sec76'),
|
||||
('Statistics, more technicalities', 2, None, '___sec77'),
|
||||
('Statistics', 2, None, '___sec78'),
|
||||
('Statistics and sample variance', 2, None, '___sec79'),
|
||||
('Statistics, uncorrelated results', 2, None, '___sec80'),
|
||||
('Statistics, computations', 2, None, '___sec81'),
|
||||
('Statistics, more on computations of errors',
|
||||
2,
|
||||
None,
|
||||
'___sec82'),
|
||||
('Statistics, wrapping up 1', 2, None, '___sec83'),
|
||||
('Statistics, final expression', 2, None, '___sec84'),
|
||||
('Statistics, effective number of correlations',
|
||||
2,
|
||||
None,
|
||||
'___sec85'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec86'),
|
||||
('Assumptions made', 2, None, '___sec87'),
|
||||
('Expectation value and variance', 2, None, '___sec88'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec89'),
|
||||
('Resampling methods', 2, None, '___sec90'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec91'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec92'),
|
||||
('Jackknife code example', 2, None, '___sec93'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec94'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec95'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec96'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec97'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec98'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec99'),
|
||||
('Various steps in cross-validation', 2, None, '___sec100'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec101'),
|
||||
('Cross-validation in brief', 2, None, '___sec102'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec103'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec104'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec105'),
|
||||
('Understanding what happens', 2, None, '___sec106'),
|
||||
('Summing up', 2, None, '___sec107'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec108'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec109'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec110'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec111'),
|
||||
('The Ising model', 2, None, '___sec112'),
|
||||
('Reformulating the problem to suit regression',
|
||||
2,
|
||||
None,
|
||||
'___sec113'),
|
||||
('Linear regression', 2, None, '___sec114'),
|
||||
('Singular Value decomposition', 2, None, '___sec115'),
|
||||
('The one-dimensional Ising model', 2, None, '___sec116'),
|
||||
('Ridge regression', 2, None, '___sec117'),
|
||||
('LASSO regression', 2, None, '___sec118'),
|
||||
('Performance as function of the regularization parameter',
|
||||
2,
|
||||
None,
|
||||
'___sec119'),
|
||||
('Finding the optimal value of $\\lambda$',
|
||||
2,
|
||||
None,
|
||||
'___sec120')]}
|
||||
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="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</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="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs114.html#___sec113" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs115.html#___sec114" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs116.html#___sec115" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec116" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs118.html#___sec117" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs119.html#___sec118" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs120.html#___sec119" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs121.html#___sec120" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="part0117"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec116" class="anchor">The one-dimensional Ising model </h2>
|
||||
|
||||
<p>
|
||||
Let us bring back the Ising model again, but now with an additional
|
||||
focus on Ridge and Lasso regression as well. We repeat some of the
|
||||
basic parts of the Ising model and the setup of the training and test
|
||||
data. The one-dimensional Ising model with nearest neighbor
|
||||
interaction, no external field and a constant coupling constant \( J \) is
|
||||
given by
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
H = -J \sum_{k}^L s_k s_{k + 1},
|
||||
\tag{27}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the one-dimensional system there is no phase transition.
|
||||
|
||||
<p>
|
||||
We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><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">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">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</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">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
|
||||
sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
|
||||
cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">'seismic'</span>)
|
||||
|
||||
L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
|
||||
n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
|
||||
|
||||
spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
|
||||
J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
||||
|
||||
energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
|
||||
|
||||
<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>(n):
|
||||
energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
|
||||
</pre></div>
|
||||
<p>
|
||||
A more general form for the one-dimensional Ising model is
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
|
||||
\tag{28}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
Here we allow for interactions beyond the nearest neighbors and a more
|
||||
adaptive coupling matrix. This latter expression can be formulated as
|
||||
a matrix-product on the form
|
||||
$$
|
||||
\begin{align}
|
||||
H = X J,
|
||||
\tag{29}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
|
||||
elements \( -J_{jk} \). This form of writing the energy fits perfectly
|
||||
with the form utilized in linear regression, viz.
|
||||
$$
|
||||
\begin{align}
|
||||
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
|
||||
\tag{30}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
We organize the data as we did above
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
|
||||
<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>(n):
|
||||
X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
|
||||
y <span style="color: #666666">=</span> energies
|
||||
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.96</span>)
|
||||
|
||||
X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
|
||||
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
|
||||
axis<span style="color: #666666">=1</span>
|
||||
)
|
||||
|
||||
X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
|
||||
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
|
||||
axis<span style="color: #666666">=1</span>
|
||||
)
|
||||
</pre></div>
|
||||
<p>
|
||||
We will do all fitting with <b>Scikit-Learn</b>,
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X_train, y_train)
|
||||
</pre></div>
|
||||
<p>
|
||||
When extracting the \( J \)-matrix we make sure to remove the intercept
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J_sk <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
|
||||
</pre></div>
|
||||
<p>
|
||||
And then we plot the results
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
|
||||
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J_sk, <span style="color: #666666">**</span>cmap_args)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"LinearRegression from Scikit-learn"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
|
||||
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
|
||||
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
The results perfectly with our previous discussion where we used our own code.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._Regression-bs116.html">«</a></li>
|
||||
<li><a href="._Regression-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs109.html">110</a></li>
|
||||
<li><a href="._Regression-bs110.html">111</a></li>
|
||||
<li><a href="._Regression-bs111.html">112</a></li>
|
||||
<li><a href="._Regression-bs112.html">113</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs114.html">115</a></li>
|
||||
<li><a href="._Regression-bs115.html">116</a></li>
|
||||
<li><a href="._Regression-bs116.html">117</a></li>
|
||||
<li class="active"><a href="._Regression-bs117.html">118</a></li>
|
||||
<li><a href="._Regression-bs118.html">119</a></li>
|
||||
<li><a href="._Regression-bs119.html">120</a></li>
|
||||
<li><a href="._Regression-bs120.html">121</a></li>
|
||||
<li><a href="._Regression-bs121.html">122</a></li>
|
||||
<li><a href="._Regression-bs118.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="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,524 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis">
|
||||
|
||||
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<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': [('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec1'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec2'),
|
||||
('Examples', 2, None, '___sec3'),
|
||||
('General linear models', 2, None, '___sec4'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec5'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Optimizing our parameters', 2, None, '___sec9'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec11'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec13'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec16'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec18'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The code', 2, None, '___sec25'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The Boston housing data example', 2, None, '___sec27'),
|
||||
('Housing data, the code', 2, None, '___sec28'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('Preprocessing our data', 2, None, '___sec30'),
|
||||
('More preprocessing', 2, None, '___sec31'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec32'),
|
||||
('The singular value decomposition', 2, None, '___sec33'),
|
||||
('Linear Regression Problems', 2, None, '___sec34'),
|
||||
('Fixing the singularity', 2, None, '___sec35'),
|
||||
('Basic math of the SVD', 2, None, '___sec36'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
|
||||
('Another Example', 2, None, '___sec38'),
|
||||
('Economy-size SVD', 2, None, '___sec39'),
|
||||
('Mathematical Properties', 2, None, '___sec40'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec41'),
|
||||
('More on Ridge Regression', 2, None, '___sec42'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec43'),
|
||||
('More interpretations', 2, None, '___sec44'),
|
||||
('Codes for the SVD', 2, None, '___sec45'),
|
||||
('A better understanding of regularization', 2, None, '___sec46'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec48'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec50'),
|
||||
('Correlation Matrix', 2, None, '___sec51'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec52'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec53'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec54'),
|
||||
('Linking with SVD', 2, None, '___sec55'),
|
||||
('Where are we going?', 2, None, '___sec56'),
|
||||
('Resampling methods', 2, None, '___sec57'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec58'),
|
||||
('Why resampling methods ?', 2, None, '___sec59'),
|
||||
('Statistical analysis', 2, None, '___sec60'),
|
||||
('Statistics', 2, None, '___sec61'),
|
||||
('Statistics, moments', 2, None, '___sec62'),
|
||||
('Statistics, central moments', 2, None, '___sec63'),
|
||||
('Statistics, covariance', 2, None, '___sec64'),
|
||||
('Statistics, more covariance', 2, None, '___sec65'),
|
||||
('Covariance example', 2, None, '___sec66'),
|
||||
('Covariance in numpy', 2, None, '___sec67'),
|
||||
('Statistics, independent variables', 2, None, '___sec68'),
|
||||
('Statistics, more variance', 2, None, '___sec69'),
|
||||
('Statistics and stochastic processes', 2, None, '___sec70'),
|
||||
('Statistics and sample variables', 2, None, '___sec71'),
|
||||
('Statistics, sample variance and covariance',
|
||||
2,
|
||||
None,
|
||||
'___sec72'),
|
||||
('Statistics, law of large numbers', 2, None, '___sec73'),
|
||||
('Statistics, more on sample error', 2, None, '___sec74'),
|
||||
('Statistics', 2, None, '___sec75'),
|
||||
('Statistics, central limit theorem', 2, None, '___sec76'),
|
||||
('Statistics, more technicalities', 2, None, '___sec77'),
|
||||
('Statistics', 2, None, '___sec78'),
|
||||
('Statistics and sample variance', 2, None, '___sec79'),
|
||||
('Statistics, uncorrelated results', 2, None, '___sec80'),
|
||||
('Statistics, computations', 2, None, '___sec81'),
|
||||
('Statistics, more on computations of errors',
|
||||
2,
|
||||
None,
|
||||
'___sec82'),
|
||||
('Statistics, wrapping up 1', 2, None, '___sec83'),
|
||||
('Statistics, final expression', 2, None, '___sec84'),
|
||||
('Statistics, effective number of correlations',
|
||||
2,
|
||||
None,
|
||||
'___sec85'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec86'),
|
||||
('Assumptions made', 2, None, '___sec87'),
|
||||
('Expectation value and variance', 2, None, '___sec88'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec89'),
|
||||
('Resampling methods', 2, None, '___sec90'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec91'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec92'),
|
||||
('Jackknife code example', 2, None, '___sec93'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec94'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec95'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec96'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec97'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec98'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec99'),
|
||||
('Various steps in cross-validation', 2, None, '___sec100'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec101'),
|
||||
('Cross-validation in brief', 2, None, '___sec102'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec103'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec104'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec105'),
|
||||
('Understanding what happens', 2, None, '___sec106'),
|
||||
('Summing up', 2, None, '___sec107'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec108'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec109'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec110'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec111'),
|
||||
('The Ising model', 2, None, '___sec112'),
|
||||
('Reformulating the problem to suit regression',
|
||||
2,
|
||||
None,
|
||||
'___sec113'),
|
||||
('Linear regression', 2, None, '___sec114'),
|
||||
('Singular Value decomposition', 2, None, '___sec115'),
|
||||
('The one-dimensional Ising model', 2, None, '___sec116'),
|
||||
('Ridge regression', 2, None, '___sec117'),
|
||||
('LASSO regression', 2, None, '___sec118'),
|
||||
('Performance as function of the regularization parameter',
|
||||
2,
|
||||
None,
|
||||
'___sec119'),
|
||||
('Finding the optimal value of $\\lambda$',
|
||||
2,
|
||||
None,
|
||||
'___sec120')]}
|
||||
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="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</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="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs114.html#___sec113" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs115.html#___sec114" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs116.html#___sec115" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs117.html#___sec116" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec117" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs119.html#___sec118" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs120.html#___sec119" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs121.html#___sec120" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="part0118"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec117" class="anchor">Ridge regression </h2>
|
||||
|
||||
<p>
|
||||
Having explored the ordinary least squares we move on to ridge
|
||||
regression. In ridge regression we include a <b>regularizer</b>. This
|
||||
involves a new cost function which leads to a new estimate for the
|
||||
weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
|
||||
cost function is given by
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
|
||||
\tag{31}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>_lambda <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
|
||||
clf_ridge <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X_train, y_train)
|
||||
J_ridge_sk <span style="color: #666666">=</span> clf_ridge<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
|
||||
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J_ridge_sk, <span style="color: #666666">**</span>cmap_args)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"Ridge from Scikit-learn"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
|
||||
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
|
||||
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._Regression-bs117.html">«</a></li>
|
||||
<li><a href="._Regression-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs110.html">111</a></li>
|
||||
<li><a href="._Regression-bs111.html">112</a></li>
|
||||
<li><a href="._Regression-bs112.html">113</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs114.html">115</a></li>
|
||||
<li><a href="._Regression-bs115.html">116</a></li>
|
||||
<li><a href="._Regression-bs116.html">117</a></li>
|
||||
<li><a href="._Regression-bs117.html">118</a></li>
|
||||
<li class="active"><a href="._Regression-bs118.html">119</a></li>
|
||||
<li><a href="._Regression-bs119.html">120</a></li>
|
||||
<li><a href="._Regression-bs120.html">121</a></li>
|
||||
<li><a href="._Regression-bs121.html">122</a></li>
|
||||
<li><a href="._Regression-bs119.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="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,526 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis">
|
||||
|
||||
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<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': [('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec1'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec2'),
|
||||
('Examples', 2, None, '___sec3'),
|
||||
('General linear models', 2, None, '___sec4'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec5'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Optimizing our parameters', 2, None, '___sec9'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec11'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec13'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec16'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec18'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The code', 2, None, '___sec25'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The Boston housing data example', 2, None, '___sec27'),
|
||||
('Housing data, the code', 2, None, '___sec28'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('Preprocessing our data', 2, None, '___sec30'),
|
||||
('More preprocessing', 2, None, '___sec31'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec32'),
|
||||
('The singular value decomposition', 2, None, '___sec33'),
|
||||
('Linear Regression Problems', 2, None, '___sec34'),
|
||||
('Fixing the singularity', 2, None, '___sec35'),
|
||||
('Basic math of the SVD', 2, None, '___sec36'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
|
||||
('Another Example', 2, None, '___sec38'),
|
||||
('Economy-size SVD', 2, None, '___sec39'),
|
||||
('Mathematical Properties', 2, None, '___sec40'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec41'),
|
||||
('More on Ridge Regression', 2, None, '___sec42'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec43'),
|
||||
('More interpretations', 2, None, '___sec44'),
|
||||
('Codes for the SVD', 2, None, '___sec45'),
|
||||
('A better understanding of regularization', 2, None, '___sec46'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec48'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec50'),
|
||||
('Correlation Matrix', 2, None, '___sec51'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec52'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec53'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec54'),
|
||||
('Linking with SVD', 2, None, '___sec55'),
|
||||
('Where are we going?', 2, None, '___sec56'),
|
||||
('Resampling methods', 2, None, '___sec57'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec58'),
|
||||
('Why resampling methods ?', 2, None, '___sec59'),
|
||||
('Statistical analysis', 2, None, '___sec60'),
|
||||
('Statistics', 2, None, '___sec61'),
|
||||
('Statistics, moments', 2, None, '___sec62'),
|
||||
('Statistics, central moments', 2, None, '___sec63'),
|
||||
('Statistics, covariance', 2, None, '___sec64'),
|
||||
('Statistics, more covariance', 2, None, '___sec65'),
|
||||
('Covariance example', 2, None, '___sec66'),
|
||||
('Covariance in numpy', 2, None, '___sec67'),
|
||||
('Statistics, independent variables', 2, None, '___sec68'),
|
||||
('Statistics, more variance', 2, None, '___sec69'),
|
||||
('Statistics and stochastic processes', 2, None, '___sec70'),
|
||||
('Statistics and sample variables', 2, None, '___sec71'),
|
||||
('Statistics, sample variance and covariance',
|
||||
2,
|
||||
None,
|
||||
'___sec72'),
|
||||
('Statistics, law of large numbers', 2, None, '___sec73'),
|
||||
('Statistics, more on sample error', 2, None, '___sec74'),
|
||||
('Statistics', 2, None, '___sec75'),
|
||||
('Statistics, central limit theorem', 2, None, '___sec76'),
|
||||
('Statistics, more technicalities', 2, None, '___sec77'),
|
||||
('Statistics', 2, None, '___sec78'),
|
||||
('Statistics and sample variance', 2, None, '___sec79'),
|
||||
('Statistics, uncorrelated results', 2, None, '___sec80'),
|
||||
('Statistics, computations', 2, None, '___sec81'),
|
||||
('Statistics, more on computations of errors',
|
||||
2,
|
||||
None,
|
||||
'___sec82'),
|
||||
('Statistics, wrapping up 1', 2, None, '___sec83'),
|
||||
('Statistics, final expression', 2, None, '___sec84'),
|
||||
('Statistics, effective number of correlations',
|
||||
2,
|
||||
None,
|
||||
'___sec85'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec86'),
|
||||
('Assumptions made', 2, None, '___sec87'),
|
||||
('Expectation value and variance', 2, None, '___sec88'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec89'),
|
||||
('Resampling methods', 2, None, '___sec90'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec91'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec92'),
|
||||
('Jackknife code example', 2, None, '___sec93'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec94'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec95'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec96'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec97'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec98'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec99'),
|
||||
('Various steps in cross-validation', 2, None, '___sec100'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec101'),
|
||||
('Cross-validation in brief', 2, None, '___sec102'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec103'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec104'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec105'),
|
||||
('Understanding what happens', 2, None, '___sec106'),
|
||||
('Summing up', 2, None, '___sec107'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec108'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec109'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec110'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec111'),
|
||||
('The Ising model', 2, None, '___sec112'),
|
||||
('Reformulating the problem to suit regression',
|
||||
2,
|
||||
None,
|
||||
'___sec113'),
|
||||
('Linear regression', 2, None, '___sec114'),
|
||||
('Singular Value decomposition', 2, None, '___sec115'),
|
||||
('The one-dimensional Ising model', 2, None, '___sec116'),
|
||||
('Ridge regression', 2, None, '___sec117'),
|
||||
('LASSO regression', 2, None, '___sec118'),
|
||||
('Performance as function of the regularization parameter',
|
||||
2,
|
||||
None,
|
||||
'___sec119'),
|
||||
('Finding the optimal value of $\\lambda$',
|
||||
2,
|
||||
None,
|
||||
'___sec120')]}
|
||||
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="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</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="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs114.html#___sec113" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs115.html#___sec114" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs116.html#___sec115" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs117.html#___sec116" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs118.html#___sec117" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec118" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs120.html#___sec119" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs121.html#___sec120" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="part0119"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec118" class="anchor">LASSO regression </h2>
|
||||
|
||||
<p>
|
||||
In the <b>Least Absolute Shrinkage and Selection Operator</b> (LASSO)-method we get a third cost function.
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
|
||||
\tag{32}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from <b>Scikit-Learn</b>.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>clf_lasso <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Lasso(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X_train, y_train)
|
||||
J_lasso_sk <span style="color: #666666">=</span> clf_lasso<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
|
||||
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J_lasso_sk, <span style="color: #666666">**</span>cmap_args)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"Lasso from Scikit-learn"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
|
||||
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
|
||||
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
It is quite striking how LASSO breaks the symmetry of the coupling
|
||||
constant as opposed to ridge and OLS. We get a sparse solution with
|
||||
\( J_{j, j + 1} = -1 \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._Regression-bs118.html">«</a></li>
|
||||
<li><a href="._Regression-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs111.html">112</a></li>
|
||||
<li><a href="._Regression-bs112.html">113</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs114.html">115</a></li>
|
||||
<li><a href="._Regression-bs115.html">116</a></li>
|
||||
<li><a href="._Regression-bs116.html">117</a></li>
|
||||
<li><a href="._Regression-bs117.html">118</a></li>
|
||||
<li><a href="._Regression-bs118.html">119</a></li>
|
||||
<li class="active"><a href="._Regression-bs119.html">120</a></li>
|
||||
<li><a href="._Regression-bs120.html">121</a></li>
|
||||
<li><a href="._Regression-bs121.html">122</a></li>
|
||||
<li><a href="._Regression-bs120.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="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,541 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis">
|
||||
|
||||
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<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': [('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec1'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec2'),
|
||||
('Examples', 2, None, '___sec3'),
|
||||
('General linear models', 2, None, '___sec4'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec5'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Optimizing our parameters', 2, None, '___sec9'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec11'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec13'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec16'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec18'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The code', 2, None, '___sec25'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The Boston housing data example', 2, None, '___sec27'),
|
||||
('Housing data, the code', 2, None, '___sec28'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('Preprocessing our data', 2, None, '___sec30'),
|
||||
('More preprocessing', 2, None, '___sec31'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec32'),
|
||||
('The singular value decomposition', 2, None, '___sec33'),
|
||||
('Linear Regression Problems', 2, None, '___sec34'),
|
||||
('Fixing the singularity', 2, None, '___sec35'),
|
||||
('Basic math of the SVD', 2, None, '___sec36'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
|
||||
('Another Example', 2, None, '___sec38'),
|
||||
('Economy-size SVD', 2, None, '___sec39'),
|
||||
('Mathematical Properties', 2, None, '___sec40'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec41'),
|
||||
('More on Ridge Regression', 2, None, '___sec42'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec43'),
|
||||
('More interpretations', 2, None, '___sec44'),
|
||||
('Codes for the SVD', 2, None, '___sec45'),
|
||||
('A better understanding of regularization', 2, None, '___sec46'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec48'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec50'),
|
||||
('Correlation Matrix', 2, None, '___sec51'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec52'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec53'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec54'),
|
||||
('Linking with SVD', 2, None, '___sec55'),
|
||||
('Where are we going?', 2, None, '___sec56'),
|
||||
('Resampling methods', 2, None, '___sec57'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec58'),
|
||||
('Why resampling methods ?', 2, None, '___sec59'),
|
||||
('Statistical analysis', 2, None, '___sec60'),
|
||||
('Statistics', 2, None, '___sec61'),
|
||||
('Statistics, moments', 2, None, '___sec62'),
|
||||
('Statistics, central moments', 2, None, '___sec63'),
|
||||
('Statistics, covariance', 2, None, '___sec64'),
|
||||
('Statistics, more covariance', 2, None, '___sec65'),
|
||||
('Covariance example', 2, None, '___sec66'),
|
||||
('Covariance in numpy', 2, None, '___sec67'),
|
||||
('Statistics, independent variables', 2, None, '___sec68'),
|
||||
('Statistics, more variance', 2, None, '___sec69'),
|
||||
('Statistics and stochastic processes', 2, None, '___sec70'),
|
||||
('Statistics and sample variables', 2, None, '___sec71'),
|
||||
('Statistics, sample variance and covariance',
|
||||
2,
|
||||
None,
|
||||
'___sec72'),
|
||||
('Statistics, law of large numbers', 2, None, '___sec73'),
|
||||
('Statistics, more on sample error', 2, None, '___sec74'),
|
||||
('Statistics', 2, None, '___sec75'),
|
||||
('Statistics, central limit theorem', 2, None, '___sec76'),
|
||||
('Statistics, more technicalities', 2, None, '___sec77'),
|
||||
('Statistics', 2, None, '___sec78'),
|
||||
('Statistics and sample variance', 2, None, '___sec79'),
|
||||
('Statistics, uncorrelated results', 2, None, '___sec80'),
|
||||
('Statistics, computations', 2, None, '___sec81'),
|
||||
('Statistics, more on computations of errors',
|
||||
2,
|
||||
None,
|
||||
'___sec82'),
|
||||
('Statistics, wrapping up 1', 2, None, '___sec83'),
|
||||
('Statistics, final expression', 2, None, '___sec84'),
|
||||
('Statistics, effective number of correlations',
|
||||
2,
|
||||
None,
|
||||
'___sec85'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec86'),
|
||||
('Assumptions made', 2, None, '___sec87'),
|
||||
('Expectation value and variance', 2, None, '___sec88'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec89'),
|
||||
('Resampling methods', 2, None, '___sec90'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec91'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec92'),
|
||||
('Jackknife code example', 2, None, '___sec93'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec94'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec95'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec96'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec97'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec98'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec99'),
|
||||
('Various steps in cross-validation', 2, None, '___sec100'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec101'),
|
||||
('Cross-validation in brief', 2, None, '___sec102'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec103'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec104'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec105'),
|
||||
('Understanding what happens', 2, None, '___sec106'),
|
||||
('Summing up', 2, None, '___sec107'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec108'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec109'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec110'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec111'),
|
||||
('The Ising model', 2, None, '___sec112'),
|
||||
('Reformulating the problem to suit regression',
|
||||
2,
|
||||
None,
|
||||
'___sec113'),
|
||||
('Linear regression', 2, None, '___sec114'),
|
||||
('Singular Value decomposition', 2, None, '___sec115'),
|
||||
('The one-dimensional Ising model', 2, None, '___sec116'),
|
||||
('Ridge regression', 2, None, '___sec117'),
|
||||
('LASSO regression', 2, None, '___sec118'),
|
||||
('Performance as function of the regularization parameter',
|
||||
2,
|
||||
None,
|
||||
'___sec119'),
|
||||
('Finding the optimal value of $\\lambda$',
|
||||
2,
|
||||
None,
|
||||
'___sec120')]}
|
||||
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="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</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="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs114.html#___sec113" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs115.html#___sec114" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs116.html#___sec115" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs117.html#___sec116" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs118.html#___sec117" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs119.html#___sec118" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec119" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs121.html#___sec120" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="part0120"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec119" class="anchor">Performance as function of the regularization parameter </h2>
|
||||
|
||||
<p>
|
||||
We see how the different models perform for a different set of values for \( \lambda \).
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">5</span>, <span style="color: #666666">10</span>)
|
||||
|
||||
train_errors <span style="color: #666666">=</span> {
|
||||
<span style="color: #BA2121">"ols_sk"</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
|
||||
<span style="color: #BA2121">"ridge_sk"</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
|
||||
<span style="color: #BA2121">"lasso_sk"</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size)
|
||||
}
|
||||
|
||||
test_errors <span style="color: #666666">=</span> {
|
||||
<span style="color: #BA2121">"ols_sk"</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
|
||||
<span style="color: #BA2121">"ridge_sk"</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
|
||||
<span style="color: #BA2121">"lasso_sk"</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size)
|
||||
}
|
||||
|
||||
plot_counter <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">32</span>, <span style="color: #666666">54</span>))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> i, _lambda <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(tqdm<span style="color: #666666">.</span>tqdm(lambdas)):
|
||||
<span style="color: #008000; font-weight: bold">for</span> key, method <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">zip</span>(
|
||||
[<span style="color: #BA2121">"ols_sk"</span>, <span style="color: #BA2121">"ridge_sk"</span>, <span style="color: #BA2121">"lasso_sk"</span>],
|
||||
[skl<span style="color: #666666">.</span>LinearRegression(), skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda), skl<span style="color: #666666">.</span>Lasso(alpha<span style="color: #666666">=</span>_lambda)]
|
||||
):
|
||||
method <span style="color: #666666">=</span> method<span style="color: #666666">.</span>fit(X_train, y_train)
|
||||
|
||||
train_errors[key][i] <span style="color: #666666">=</span> method<span style="color: #666666">.</span>score(X_train, y_train)
|
||||
test_errors[key][i] <span style="color: #666666">=</span> method<span style="color: #666666">.</span>score(X_test, y_test)
|
||||
|
||||
omega <span style="color: #666666">=</span> method<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">10</span>, <span style="color: #666666">5</span>, plot_counter)
|
||||
plt<span style="color: #666666">.</span>imshow(omega, <span style="color: #666666">**</span>cmap_args)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"</span><span style="color: #BB6688; font-weight: bold">%s</span><span style="color: #BA2121">, $\lambda = </span><span style="color: #BB6688; font-weight: bold">%.4f</span><span style="color: #BA2121">$"</span> <span style="color: #666666">%</span> (key, _lambda))
|
||||
plot_counter <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
We see that LASSO reaches a good solution for low
|
||||
values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
|
||||
much. Ridge is more stable over a larger range of values for
|
||||
\( \lambda \), but eventually also fades away.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._Regression-bs119.html">«</a></li>
|
||||
<li><a href="._Regression-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs112.html">113</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs114.html">115</a></li>
|
||||
<li><a href="._Regression-bs115.html">116</a></li>
|
||||
<li><a href="._Regression-bs116.html">117</a></li>
|
||||
<li><a href="._Regression-bs117.html">118</a></li>
|
||||
<li><a href="._Regression-bs118.html">119</a></li>
|
||||
<li><a href="._Regression-bs119.html">120</a></li>
|
||||
<li class="active"><a href="._Regression-bs120.html">121</a></li>
|
||||
<li><a href="._Regression-bs121.html">122</a></li>
|
||||
<li><a href="._Regression-bs121.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="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -0,0 +1,538 @@
|
||||
<!--
|
||||
Automatically generated HTML file from DocOnce source
|
||||
(https://github.com/hplgit/doconce/)
|
||||
-->
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis">
|
||||
|
||||
<title>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<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': [('Why Linear Regression (aka Ordinary Least Squares and family)',
|
||||
2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Regression analysis, overarching aims', 2, None, '___sec1'),
|
||||
('Regression analysis, overarching aims II', 2, None, '___sec2'),
|
||||
('Examples', 2, None, '___sec3'),
|
||||
('General linear models', 2, None, '___sec4'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec5'),
|
||||
('Rewriting the fitting procedure as a linear algebra problem, '
|
||||
'more details',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec7'),
|
||||
('Generalizing the fitting procedure as a linear algebra problem',
|
||||
2,
|
||||
None,
|
||||
'___sec8'),
|
||||
('Optimizing our parameters', 2, None, '___sec9'),
|
||||
('Our model for the nuclear binding energies',
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('Optimizing our parameters, more details', 2, None, '___sec11'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec12'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec13'),
|
||||
('Some useful matrix and vector expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec14'),
|
||||
('Interpretations and optimizing our parameters',
|
||||
2,
|
||||
None,
|
||||
'___sec15'),
|
||||
('Own code for Ordinary Least Squares', 2, None, '___sec16'),
|
||||
('Adding error analysis and training set up',
|
||||
2,
|
||||
None,
|
||||
'___sec17'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec18'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec19'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec20'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec21'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec22'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec23'),
|
||||
('Fitting an Equation of State for Dense Nuclear Matter',
|
||||
2,
|
||||
None,
|
||||
'___sec24'),
|
||||
('The code', 2, None, '___sec25'),
|
||||
('Splitting our Data in Training and Test data',
|
||||
2,
|
||||
None,
|
||||
'___sec26'),
|
||||
('The Boston housing data example', 2, None, '___sec27'),
|
||||
('Housing data, the code', 2, None, '___sec28'),
|
||||
('Reducing the number of degrees of freedom, overarching view',
|
||||
2,
|
||||
None,
|
||||
'___sec29'),
|
||||
('Preprocessing our data', 2, None, '___sec30'),
|
||||
('More preprocessing', 2, None, '___sec31'),
|
||||
('Simple preprocessing examples, Franke function and regression',
|
||||
2,
|
||||
None,
|
||||
'___sec32'),
|
||||
('The singular value decomposition', 2, None, '___sec33'),
|
||||
('Linear Regression Problems', 2, None, '___sec34'),
|
||||
('Fixing the singularity', 2, None, '___sec35'),
|
||||
('Basic math of the SVD', 2, None, '___sec36'),
|
||||
('The SVD, a Fantastic Algorithm', 2, None, '___sec37'),
|
||||
('Another Example', 2, None, '___sec38'),
|
||||
('Economy-size SVD', 2, None, '___sec39'),
|
||||
('Mathematical Properties', 2, None, '___sec40'),
|
||||
('Ridge and LASSO Regression', 2, None, '___sec41'),
|
||||
('More on Ridge Regression', 2, None, '___sec42'),
|
||||
('Interpreting the Ridge results', 2, None, '___sec43'),
|
||||
('More interpretations', 2, None, '___sec44'),
|
||||
('Codes for the SVD', 2, None, '___sec45'),
|
||||
('A better understanding of regularization', 2, None, '___sec46'),
|
||||
('Decomposing the OLS and Ridge expressions',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec48'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec50'),
|
||||
('Correlation Matrix', 2, None, '___sec51'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec52'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec53'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec54'),
|
||||
('Linking with SVD', 2, None, '___sec55'),
|
||||
('Where are we going?', 2, None, '___sec56'),
|
||||
('Resampling methods', 2, None, '___sec57'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
2,
|
||||
None,
|
||||
'___sec58'),
|
||||
('Why resampling methods ?', 2, None, '___sec59'),
|
||||
('Statistical analysis', 2, None, '___sec60'),
|
||||
('Statistics', 2, None, '___sec61'),
|
||||
('Statistics, moments', 2, None, '___sec62'),
|
||||
('Statistics, central moments', 2, None, '___sec63'),
|
||||
('Statistics, covariance', 2, None, '___sec64'),
|
||||
('Statistics, more covariance', 2, None, '___sec65'),
|
||||
('Covariance example', 2, None, '___sec66'),
|
||||
('Covariance in numpy', 2, None, '___sec67'),
|
||||
('Statistics, independent variables', 2, None, '___sec68'),
|
||||
('Statistics, more variance', 2, None, '___sec69'),
|
||||
('Statistics and stochastic processes', 2, None, '___sec70'),
|
||||
('Statistics and sample variables', 2, None, '___sec71'),
|
||||
('Statistics, sample variance and covariance',
|
||||
2,
|
||||
None,
|
||||
'___sec72'),
|
||||
('Statistics, law of large numbers', 2, None, '___sec73'),
|
||||
('Statistics, more on sample error', 2, None, '___sec74'),
|
||||
('Statistics', 2, None, '___sec75'),
|
||||
('Statistics, central limit theorem', 2, None, '___sec76'),
|
||||
('Statistics, more technicalities', 2, None, '___sec77'),
|
||||
('Statistics', 2, None, '___sec78'),
|
||||
('Statistics and sample variance', 2, None, '___sec79'),
|
||||
('Statistics, uncorrelated results', 2, None, '___sec80'),
|
||||
('Statistics, computations', 2, None, '___sec81'),
|
||||
('Statistics, more on computations of errors',
|
||||
2,
|
||||
None,
|
||||
'___sec82'),
|
||||
('Statistics, wrapping up 1', 2, None, '___sec83'),
|
||||
('Statistics, final expression', 2, None, '___sec84'),
|
||||
('Statistics, effective number of correlations',
|
||||
2,
|
||||
None,
|
||||
'___sec85'),
|
||||
('Linking the regression analysis with a statistical '
|
||||
'interpretation',
|
||||
2,
|
||||
None,
|
||||
'___sec86'),
|
||||
('Assumptions made', 2, None, '___sec87'),
|
||||
('Expectation value and variance', 2, None, '___sec88'),
|
||||
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
||||
2,
|
||||
None,
|
||||
'___sec89'),
|
||||
('Resampling methods', 2, None, '___sec90'),
|
||||
('Resampling methods: Jackknife and Bootstrap',
|
||||
2,
|
||||
None,
|
||||
'___sec91'),
|
||||
('Resampling methods: Jackknife', 2, None, '___sec92'),
|
||||
('Jackknife code example', 2, None, '___sec93'),
|
||||
('Resampling methods: Bootstrap', 2, None, '___sec94'),
|
||||
('Resampling methods: Bootstrap background', 2, None, '___sec95'),
|
||||
('Resampling methods: More Bootstrap background',
|
||||
2,
|
||||
None,
|
||||
'___sec96'),
|
||||
('Resampling methods: Bootstrap approach', 2, None, '___sec97'),
|
||||
('Resampling methods: Bootstrap steps', 2, None, '___sec98'),
|
||||
('Code example for the Bootstrap method', 2, None, '___sec99'),
|
||||
('Various steps in cross-validation', 2, None, '___sec100'),
|
||||
('How to set up the cross-validation for Ridge and/or Lasso',
|
||||
2,
|
||||
None,
|
||||
'___sec101'),
|
||||
('Cross-validation in brief', 2, None, '___sec102'),
|
||||
('Code Example for Cross-validation and $k$-fold '
|
||||
'Cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec103'),
|
||||
('The bias-variance tradeoff', 2, None, '___sec104'),
|
||||
('Example code for Bias-Variance tradeoff', 2, None, '___sec105'),
|
||||
('Understanding what happens', 2, None, '___sec106'),
|
||||
('Summing up', 2, None, '___sec107'),
|
||||
("Another Example from Scikit-Learn's Repository",
|
||||
2,
|
||||
None,
|
||||
'___sec108'),
|
||||
('More examples on bootstrap and cross-validation and errors',
|
||||
2,
|
||||
None,
|
||||
'___sec109'),
|
||||
('The same example but now with cross-validation',
|
||||
2,
|
||||
None,
|
||||
'___sec110'),
|
||||
('Cross-validation with Ridge', 2, None, '___sec111'),
|
||||
('The Ising model', 2, None, '___sec112'),
|
||||
('Reformulating the problem to suit regression',
|
||||
2,
|
||||
None,
|
||||
'___sec113'),
|
||||
('Linear regression', 2, None, '___sec114'),
|
||||
('Singular Value decomposition', 2, None, '___sec115'),
|
||||
('The one-dimensional Ising model', 2, None, '___sec116'),
|
||||
('Ridge regression', 2, None, '___sec117'),
|
||||
('LASSO regression', 2, None, '___sec118'),
|
||||
('Performance as function of the regularization parameter',
|
||||
2,
|
||||
None,
|
||||
'___sec119'),
|
||||
('Finding the optimal value of $\\lambda$',
|
||||
2,
|
||||
None,
|
||||
'___sec120')]}
|
||||
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="Regression-bs.html">Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</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="._Regression-bs001.html#___sec0" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">General linear models</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">Adding error analysis and training set up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">The code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">The Boston housing data example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Housing data, the code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">More preprocessing</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">The singular value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">Linear Regression Problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">Fixing the singularity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Basic math of the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Another Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">Economy-size SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Mathematical Properties</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">More on Ridge Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">More interpretations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Codes for the SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">A better understanding of regularization</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs048.html#___sec47" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs054.html#___sec53" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs055.html#___sec54" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs056.html#___sec55" style="font-size: 80%;">Linking with SVD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs057.html#___sec56" style="font-size: 80%;">Where are we going?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs058.html#___sec57" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs059.html#___sec58" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs060.html#___sec59" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs061.html#___sec60" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs062.html#___sec61" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs063.html#___sec62" style="font-size: 80%;">Statistics, moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs064.html#___sec63" style="font-size: 80%;">Statistics, central moments</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs065.html#___sec64" style="font-size: 80%;">Statistics, covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs066.html#___sec65" style="font-size: 80%;">Statistics, more covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs067.html#___sec66" style="font-size: 80%;">Covariance example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs068.html#___sec67" style="font-size: 80%;">Covariance in numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs069.html#___sec68" style="font-size: 80%;">Statistics, independent variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs070.html#___sec69" style="font-size: 80%;">Statistics, more variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs071.html#___sec70" style="font-size: 80%;">Statistics and stochastic processes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs072.html#___sec71" style="font-size: 80%;">Statistics and sample variables</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs073.html#___sec72" style="font-size: 80%;">Statistics, sample variance and covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs074.html#___sec73" style="font-size: 80%;">Statistics, law of large numbers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs075.html#___sec74" style="font-size: 80%;">Statistics, more on sample error</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs076.html#___sec75" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs077.html#___sec76" style="font-size: 80%;">Statistics, central limit theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs078.html#___sec77" style="font-size: 80%;">Statistics, more technicalities</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs079.html#___sec78" style="font-size: 80%;">Statistics</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs080.html#___sec79" style="font-size: 80%;">Statistics and sample variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs081.html#___sec80" style="font-size: 80%;">Statistics, uncorrelated results</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs082.html#___sec81" style="font-size: 80%;">Statistics, computations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs083.html#___sec82" style="font-size: 80%;">Statistics, more on computations of errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs084.html#___sec83" style="font-size: 80%;">Statistics, wrapping up 1</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs085.html#___sec84" style="font-size: 80%;">Statistics, final expression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs086.html#___sec85" style="font-size: 80%;">Statistics, effective number of correlations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs087.html#___sec86" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs088.html#___sec87" style="font-size: 80%;">Assumptions made</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs089.html#___sec88" style="font-size: 80%;">Expectation value and variance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs090.html#___sec89" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs091.html#___sec90" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs092.html#___sec91" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs093.html#___sec92" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs094.html#___sec93" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs095.html#___sec94" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs096.html#___sec95" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs097.html#___sec96" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs098.html#___sec97" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs099.html#___sec98" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs100.html#___sec99" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs101.html#___sec100" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs102.html#___sec101" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs103.html#___sec102" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs104.html#___sec103" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs105.html#___sec104" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs106.html#___sec105" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs107.html#___sec106" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs108.html#___sec107" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs109.html#___sec108" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs110.html#___sec109" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs111.html#___sec110" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs112.html#___sec111" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs113.html#___sec112" style="font-size: 80%;">The Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs114.html#___sec113" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs115.html#___sec114" style="font-size: 80%;">Linear regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs116.html#___sec115" style="font-size: 80%;">Singular Value decomposition</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs117.html#___sec116" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs118.html#___sec117" style="font-size: 80%;">Ridge regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs119.html#___sec118" style="font-size: 80%;">LASSO regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs120.html#___sec119" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec120" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</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="part0121"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec120" class="anchor">Finding the optimal value of \( \lambda \) </h2>
|
||||
|
||||
<p>
|
||||
To determine which value of \( \lambda \) is best we plot the accuracy of
|
||||
the models when predicting the training and the testing set. We expect
|
||||
the accuracy of the training set to be quite good, but if the accuracy
|
||||
of the testing set is much lower this tells us that we might be
|
||||
subject to an overfit model. The ideal scenario is an accuracy on the
|
||||
testing set that is close to the accuracy of the training set.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
|
||||
|
||||
colors <span style="color: #666666">=</span> {
|
||||
<span style="color: #BA2121">"ols_sk"</span>: <span style="color: #BA2121">"r"</span>,
|
||||
<span style="color: #BA2121">"ridge_sk"</span>: <span style="color: #BA2121">"y"</span>,
|
||||
<span style="color: #BA2121">"lasso_sk"</span>: <span style="color: #BA2121">"c"</span>
|
||||
}
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> train_errors:
|
||||
plt<span style="color: #666666">.</span>semilogx(
|
||||
lambdas,
|
||||
train_errors[key],
|
||||
colors[key],
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">"Train </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(key),
|
||||
linewidth<span style="color: #666666">=4.0</span>
|
||||
)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> test_errors:
|
||||
plt<span style="color: #666666">.</span>semilogx(
|
||||
lambdas,
|
||||
test_errors[key],
|
||||
colors[key] <span style="color: #666666">+</span> <span style="color: #BA2121">"--"</span>,
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">"Test </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(key),
|
||||
linewidth<span style="color: #666666">=4.0</span>
|
||||
)
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"best"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$\lambda$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$R^2$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>tick_params(labelsize<span style="color: #666666">=18</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
|
||||
achieves a very good accuracy on the test set. This by far surpasses the
|
||||
other models for all values of \( \lambda \).
|
||||
|
||||
<p>
|
||||
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
<li><a href="._Regression-bs120.html">«</a></li>
|
||||
<li><a href="._Regression-bs000.html">1</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs113.html">114</a></li>
|
||||
<li><a href="._Regression-bs114.html">115</a></li>
|
||||
<li><a href="._Regression-bs115.html">116</a></li>
|
||||
<li><a href="._Regression-bs116.html">117</a></li>
|
||||
<li><a href="._Regression-bs117.html">118</a></li>
|
||||
<li><a href="._Regression-bs118.html">119</a></li>
|
||||
<li><a href="._Regression-bs119.html">120</a></li>
|
||||
<li><a href="._Regression-bs120.html">121</a></li>
|
||||
<li class="active"><a href="._Regression-bs121.html">122</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="http://..."><img width="250" align=right src="http://..."></a>
|
||||
</footer>
|
||||
-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
Reference in New Issue
Block a user