337 lines
18 KiB
HTML
337 lines
18 KiB
HTML
<!--
|
|
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="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': [('Regression analysis, overarching aims', 2, None, '___sec0'),
|
|
('Regression analysis, overarching aims II', 2, None, '___sec1'),
|
|
('General linear models', 2, None, '___sec2'),
|
|
('Rewriting the fitting procedure as a linear algebra problem',
|
|
2,
|
|
None,
|
|
'___sec3'),
|
|
('Rewriting the fitting procedure as a linear algebra problem, '
|
|
'follows',
|
|
2,
|
|
None,
|
|
'___sec4'),
|
|
('Generalizing the fitting procedure as a linear algebra problem',
|
|
2,
|
|
None,
|
|
'___sec5'),
|
|
('Generalizing the fitting procedure as a linear algebra problem',
|
|
2,
|
|
None,
|
|
'___sec6'),
|
|
('Optimizing our parameters', 2, None, '___sec7'),
|
|
('Optimizing our parameters, more details', 2, None, '___sec8'),
|
|
('Interpretations and optimizing our parameters',
|
|
2,
|
|
None,
|
|
'___sec9'),
|
|
('Interpretations and optimizing our parameters',
|
|
2,
|
|
None,
|
|
'___sec10'),
|
|
('Interpretations and optimizing our parameters',
|
|
2,
|
|
None,
|
|
'___sec11'),
|
|
('The $\\chi^2$ function', 2, None, '___sec12'),
|
|
('The $\\chi^2$ function', 2, None, '___sec13'),
|
|
('The $\\chi^2$ function', 2, None, '___sec14'),
|
|
('The $\\chi^2$ function', 2, None, '___sec15'),
|
|
('The $\\chi^2$ function', 2, None, '___sec16'),
|
|
('The $\\chi^2$ function', 2, None, '___sec17'),
|
|
('Simple regression model', 2, None, '___sec18'),
|
|
('Simple regression model, now using _scikit-learn_',
|
|
2,
|
|
None,
|
|
'___sec19'),
|
|
('Simple linear regression model using _scikit-learn_',
|
|
2,
|
|
None,
|
|
'___sec20'),
|
|
('Simple linear regression model', 2, None, '___sec21'),
|
|
('Less noise', 2, None, '___sec22'),
|
|
('How to study our fits', 2, None, '___sec23'),
|
|
('Minimizing the cost function', 2, None, '___sec24'),
|
|
('Relative error', 2, None, '___sec25'),
|
|
('The richness of _scikit-learn_', 2, None, '___sec26'),
|
|
('Functions in _scikit-learn_', 2, None, '___sec27'),
|
|
('Other functions in _scikit-learn_', 2, None, '___sec28'),
|
|
('The mean absolute error and other functions in _scikit-learn_',
|
|
2,
|
|
None,
|
|
'___sec29'),
|
|
('Cubic polynomial in _scikit-learn_', 2, None, '___sec30'),
|
|
('Polynomial Regression', 2, None, '___sec31'),
|
|
('Linking the regression analysis with a statistical '
|
|
'interpretation',
|
|
2,
|
|
None,
|
|
'___sec32'),
|
|
('Expectation value and variance', 2, None, '___sec33'),
|
|
('The singular value decompostion', 2, None, '___sec34'),
|
|
('From standard regression to Ridge regressions',
|
|
2,
|
|
None,
|
|
'___sec35'),
|
|
('Fixing the singularity', 2, None, '___sec36'),
|
|
('Fitting vs. predicting when data is in the model class',
|
|
2,
|
|
None,
|
|
'___sec37'),
|
|
('Fitting versus predicting when data is not in the model class',
|
|
2,
|
|
None,
|
|
'___sec38'),
|
|
('An example code without the model assessment part',
|
|
2,
|
|
None,
|
|
'___sec39'),
|
|
('Generating test data', 2, None, '___sec40'),
|
|
('How can we effectively evaluate the various models?',
|
|
2,
|
|
None,
|
|
'___sec41'),
|
|
('Code examples for Ridge and Lasso Regression',
|
|
2,
|
|
None,
|
|
'___sec42'),
|
|
('A second-order polynomial with Ridge and Lasso',
|
|
2,
|
|
None,
|
|
'___sec43'),
|
|
('Resampling methods', 2, None, '___sec44'),
|
|
('Resampling approaches can be computationally expensive',
|
|
2,
|
|
None,
|
|
'___sec45'),
|
|
('Log-likelihood', 2, None, '___sec46'),
|
|
('Cross-validation', 2, None, '___sec47'),
|
|
('Computationally expensive', 2, None, '___sec48'),
|
|
('Various steps in cross-validation', 2, None, '___sec49'),
|
|
('How to set up the cross-validation for Ridge and/or Lasso',
|
|
2,
|
|
None,
|
|
'___sec50'),
|
|
('Predicted Residual Error Sum of Squares', 2, None, '___sec51'),
|
|
('Bootstrap', 2, None, '___sec52')]}
|
|
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%;">Regression analysis, overarching aims</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs002.html#___sec1" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs003.html#___sec2" style="font-size: 80%;">General linear models</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs004.html#___sec3" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs005.html#___sec4" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, follows</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs006.html#___sec5" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs007.html#___sec6" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs008.html#___sec7" style="font-size: 80%;">Optimizing our parameters</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs019.html#___sec18" style="font-size: 80%;">Simple regression model</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs020.html#___sec19" style="font-size: 80%;">Simple regression model, now using <b>scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs021.html#___sec20" style="font-size: 80%;">Simple linear regression model using <b>scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs022.html#___sec21" style="font-size: 80%;">Simple linear regression model</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs023.html#___sec22" style="font-size: 80%;">Less noise</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs024.html#___sec23" style="font-size: 80%;">How to study our fits</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs025.html#___sec24" style="font-size: 80%;">Minimizing the cost function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs026.html#___sec25" style="font-size: 80%;">Relative error</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs027.html#___sec26" style="font-size: 80%;">The richness of <b>scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs028.html#___sec27" style="font-size: 80%;">Functions in <b>scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs029.html#___sec28" style="font-size: 80%;">Other functions in <b>scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs030.html#___sec29" style="font-size: 80%;">The mean absolute error and other functions in <b>scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs031.html#___sec30" style="font-size: 80%;">Cubic polynomial in <b>scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs032.html#___sec31" style="font-size: 80%;">Polynomial Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs033.html#___sec32" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs034.html#___sec33" style="font-size: 80%;">Expectation value and variance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs035.html#___sec34" style="font-size: 80%;">The singular value decompostion</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs036.html#___sec35" style="font-size: 80%;">From standard regression to Ridge regressions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs037.html#___sec36" style="font-size: 80%;">Fixing the singularity</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs038.html#___sec37" style="font-size: 80%;">Fitting vs. predicting when data is in the model class</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs039.html#___sec38" style="font-size: 80%;">Fitting versus predicting when data is not in the model class</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs040.html#___sec39" style="font-size: 80%;">An example code without the model assessment part</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs041.html#___sec40" style="font-size: 80%;">Generating test data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs042.html#___sec41" style="font-size: 80%;">How can we effectively evaluate the various models?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs043.html#___sec42" style="font-size: 80%;">Code examples for Ridge and Lasso Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs044.html#___sec43" style="font-size: 80%;">A second-order polynomial with Ridge and Lasso</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs045.html#___sec44" style="font-size: 80%;">Resampling methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs046.html#___sec45" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs047.html#___sec46" style="font-size: 80%;">Log-likelihood</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec47" style="font-size: 80%;">Cross-validation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs049.html#___sec48" style="font-size: 80%;">Computationally expensive</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs050.html#___sec49" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs051.html#___sec50" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs052.html#___sec51" style="font-size: 80%;">Predicted Residual Error Sum of Squares</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Regression-bs053.html#___sec52" style="font-size: 80%;">Bootstrap</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="part0048"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec47" class="anchor">Cross-validation </h2>
|
|
|
|
<p>
|
|
Instead of choosing the penalty parameter to balance model fit with
|
|
model complexity, cross-validation requires it (i.e. the penalty
|
|
parameter) to yield a model with good prediction
|
|
performance. Commonly, this performance is evaluated on novel
|
|
data. Novel data need not be easy to come by and one has to make do
|
|
with the data at hand. The setting of `original' and novel data is
|
|
then mimicked by sample splitting: the data set is divided into two
|
|
(groups of samples). One of these two data sets, called the <em>training
|
|
set</em>, plays the role of `original' data on which the model is
|
|
built. The second of these data sets, called the <em>test set</em>, plays the
|
|
role of the `novel' data and is used to evaluate the prediction
|
|
performance (often operationalized as the log-likelihood or the
|
|
prediction error or its square or the R2 score) of the model built on the training data set. This
|
|
procedure (model building and prediction evaluation on training and
|
|
test set, respectively) is done for a collection of possible penalty
|
|
parameter choices. The penalty parameter that yields the model with
|
|
the best prediction performance is to be preferred. The thus obtained
|
|
performance evaluation depends on the actual split of the data set. To
|
|
remove this dependence the data set is split many times into a
|
|
training and test set. For each split the model parameters are
|
|
estimated for all choices of \( \lambda \) using the training data and
|
|
estimated parameters are evaluated on the corresponding test set. The
|
|
penalty parameter that on average over the test sets performs best (in
|
|
some sense) is then selected.
|
|
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._Regression-bs047.html">«</a></li>
|
|
<li><a href="._Regression-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._Regression-bs040.html">41</a></li>
|
|
<li><a href="._Regression-bs041.html">42</a></li>
|
|
<li><a href="._Regression-bs042.html">43</a></li>
|
|
<li><a href="._Regression-bs043.html">44</a></li>
|
|
<li><a href="._Regression-bs044.html">45</a></li>
|
|
<li><a href="._Regression-bs045.html">46</a></li>
|
|
<li><a href="._Regression-bs046.html">47</a></li>
|
|
<li><a href="._Regression-bs047.html">48</a></li>
|
|
<li class="active"><a href="._Regression-bs048.html">49</a></li>
|
|
<li><a href="._Regression-bs049.html">50</a></li>
|
|
<li><a href="._Regression-bs050.html">51</a></li>
|
|
<li><a href="._Regression-bs051.html">52</a></li>
|
|
<li><a href="._Regression-bs052.html">53</a></li>
|
|
<li><a href="._Regression-bs053.html">54</a></li>
|
|
<li><a href="._Regression-bs049.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>
|
|
|
|
|