mhjensen
280 lines
13 KiB
HTML
280 lines
13 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="viewport" content="width=device-width, initial-scale=1.0" />
|
|
<meta name="description" content="Week 36: Resampling techniques and Ordinary Least Square">
|
|
|
|
<title>Week 36: Resampling techniques and Ordinary Least Square</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': [('Plans for week 36', 2, None, '___sec0'),
|
|
('Thursday September 3', 2, None, '___sec1'),
|
|
('Why resampling methods', 2, None, '___sec2'),
|
|
('Resampling methods', 2, None, '___sec3'),
|
|
('Resampling approaches can be computationally expensive',
|
|
2,
|
|
None,
|
|
'___sec4'),
|
|
('Why resampling methods ?', 2, None, '___sec5'),
|
|
('Statistical analysis', 2, None, '___sec6'),
|
|
('Linking the regression analysis with a statistical '
|
|
'interpretation',
|
|
2,
|
|
None,
|
|
'___sec7'),
|
|
('Assumptions made', 2, None, '___sec8'),
|
|
('Expectation value and variance', 2, None, '___sec9'),
|
|
('Expectation value and variance for $\\boldsymbol{\\beta}$',
|
|
2,
|
|
None,
|
|
'___sec10'),
|
|
('Resampling methods', 2, None, '___sec11'),
|
|
('Resampling methods: Jackknife and Bootstrap',
|
|
2,
|
|
None,
|
|
'___sec12'),
|
|
('Resampling methods: Jackknife', 2, None, '___sec13'),
|
|
('Jackknife code example', 2, None, '___sec14'),
|
|
('Resampling methods: Bootstrap', 2, None, '___sec15'),
|
|
('Resampling methods: Bootstrap background', 2, None, '___sec16'),
|
|
('Resampling methods: More Bootstrap background',
|
|
2,
|
|
None,
|
|
'___sec17'),
|
|
('Resampling methods: Bootstrap approach', 2, None, '___sec18'),
|
|
('Resampling methods: Bootstrap steps', 2, None, '___sec19'),
|
|
('Code example for the Bootstrap method', 2, None, '___sec20'),
|
|
('Various steps in cross-validation', 2, None, '___sec21'),
|
|
('How to set up the cross-validation for Ridge and/or Lasso',
|
|
2,
|
|
None,
|
|
'___sec22'),
|
|
('Cross-validation in brief', 2, None, '___sec23'),
|
|
('Code Example for Cross-validation and $k$-fold '
|
|
'Cross-validation',
|
|
2,
|
|
None,
|
|
'___sec24'),
|
|
('The bias-variance tradeoff', 2, None, '___sec25'),
|
|
('Example code for Bias-Variance tradeoff', 2, None, '___sec26'),
|
|
('Understanding what happens', 2, None, '___sec27'),
|
|
('Summing up', 2, None, '___sec28'),
|
|
("Another Example from Scikit-Learn's Repository",
|
|
2,
|
|
None,
|
|
'___sec29'),
|
|
('More examples on bootstrap and cross-validation and errors',
|
|
2,
|
|
None,
|
|
'___sec30'),
|
|
('The same example but now with cross-validation',
|
|
2,
|
|
None,
|
|
'___sec31'),
|
|
('Cross-validation with Ridge', 2, None, '___sec32'),
|
|
('Friday September 4', 2, None, '___sec33')]}
|
|
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="week36-bs.html">Week 36: Resampling techniques and Ordinary Least Square</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="._week36-bs001.html#___sec0" style="font-size: 80%;">Plans for week 36</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs002.html#___sec1" style="font-size: 80%;">Thursday September 3</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs003.html#___sec2" style="font-size: 80%;">Why resampling methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs004.html#___sec3" style="font-size: 80%;">Resampling methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs005.html#___sec4" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs006.html#___sec5" style="font-size: 80%;">Why resampling methods ?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs007.html#___sec6" style="font-size: 80%;">Statistical analysis</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs009.html#___sec8" style="font-size: 80%;">Assumptions made</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs010.html#___sec9" style="font-size: 80%;">Expectation value and variance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs011.html#___sec10" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs012.html#___sec11" style="font-size: 80%;">Resampling methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs013.html#___sec12" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs014.html#___sec13" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs015.html#___sec14" style="font-size: 80%;">Jackknife code example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs016.html#___sec15" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs017.html#___sec16" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs018.html#___sec17" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs019.html#___sec18" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs020.html#___sec19" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs021.html#___sec20" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs022.html#___sec21" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs023.html#___sec22" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs024.html#___sec23" style="font-size: 80%;">Cross-validation in brief</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs025.html#___sec24" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs026.html#___sec25" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs027.html#___sec26" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs028.html#___sec27" style="font-size: 80%;">Understanding what happens</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs029.html#___sec28" style="font-size: 80%;">Summing up</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs030.html#___sec29" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs031.html#___sec30" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs032.html#___sec31" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs033.html#___sec32" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs034.html#___sec33" style="font-size: 80%;">Friday September 4</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="part0008"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec7" class="anchor">Linking the regression analysis with a statistical interpretation </h2>
|
|
|
|
<p>
|
|
The
|
|
advantage of doing linear regression is that we actually end up with
|
|
analytical expressions for several statistical quantities.
|
|
Standard least squares and Ridge regression allow us to
|
|
derive quantities like the variance and other expectation values in a
|
|
rather straightforward way.
|
|
|
|
<p>
|
|
It is assumed that \( \varepsilon_i
|
|
\sim \mathcal{N}(0, \sigma^2) \) and the \( \varepsilon_{i} \) are
|
|
independent, i.e.:
|
|
$$
|
|
\begin{align*}
|
|
\mbox{Cov}(\varepsilon_{i_1},
|
|
\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
|
|
& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right.
|
|
\end{align*}
|
|
$$
|
|
|
|
The randomness of \( \varepsilon_i \) implies that
|
|
\( \mathbf{y}_i \) is also a random variable. In particular,
|
|
\( \mathbf{y}_i \) is normally distributed, because \( \varepsilon_i \sim
|
|
\mathcal{N}(0, \sigma^2) \) and \( \mathbf{X}_{i,\ast} \, \boldsymbol{\beta} \) is a
|
|
non-random scalar. To specify the parameters of the distribution of
|
|
\( \mathbf{y}_i \) we need to calculate its first two moments.
|
|
|
|
<p>
|
|
Recall that \( \boldsymbol{X} \) is a matrix of dimensionality \( n\times p \). The
|
|
notation above \( \mathbf{X}_{i,\ast} \) means that we are looking at the
|
|
row number \( i \) and perform a sum over all values \( p \).
|
|
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week36-bs007.html">«</a></li>
|
|
<li><a href="._week36-bs000.html">1</a></li>
|
|
<li><a href="._week36-bs001.html">2</a></li>
|
|
<li><a href="._week36-bs002.html">3</a></li>
|
|
<li><a href="._week36-bs003.html">4</a></li>
|
|
<li><a href="._week36-bs004.html">5</a></li>
|
|
<li><a href="._week36-bs005.html">6</a></li>
|
|
<li><a href="._week36-bs006.html">7</a></li>
|
|
<li><a href="._week36-bs007.html">8</a></li>
|
|
<li class="active"><a href="._week36-bs008.html">9</a></li>
|
|
<li><a href="._week36-bs009.html">10</a></li>
|
|
<li><a href="._week36-bs010.html">11</a></li>
|
|
<li><a href="._week36-bs011.html">12</a></li>
|
|
<li><a href="._week36-bs012.html">13</a></li>
|
|
<li><a href="._week36-bs013.html">14</a></li>
|
|
<li><a href="._week36-bs014.html">15</a></li>
|
|
<li><a href="._week36-bs015.html">16</a></li>
|
|
<li><a href="._week36-bs016.html">17</a></li>
|
|
<li><a href="._week36-bs017.html">18</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week36-bs034.html">35</a></li>
|
|
<li><a href="._week36-bs009.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>
|
|
|
|
|