mhjensen
183 lines
7.1 KiB
HTML
183 lines
7.1 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: Dimensionality Reduction">
|
|
|
|
<title>Data Analysis and Machine Learning: Dimensionality Reduction</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': [('Reducing the number of degrees of freedom, overarching view',
|
|
2,
|
|
None,
|
|
'___sec0'),
|
|
('Principal Component Analysis', 2, None, '___sec1'),
|
|
('PCA and scikit-learn', 2, None, '___sec2'),
|
|
('More on the PCA', 2, None, '___sec3'),
|
|
('Incremental PCA', 2, None, '___sec4'),
|
|
('Randomized PCA', 2, None, '___sec5'),
|
|
('Kernel PCA', 2, None, '___sec6'),
|
|
('LLE', 2, None, '___sec7'),
|
|
('Other techniques', 2, None, '___sec8')]}
|
|
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="DimRed-bs.html">Data Analysis and Machine Learning: Dimensionality Reduction</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="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Principal Component Analysis</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">More on the PCA</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">Incremental PCA</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Randomized PCA</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">Kernel PCA</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">LLE</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Other techniques</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="part0007"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec6" class="anchor">Kernel PCA </h2>
|
|
<div class="panel panel-default">
|
|
<div class="panel-body">
|
|
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
|
|
|
<p>
|
|
The kernel trick is a mathematical technique that implicitly maps instances into a
|
|
very high-dimensional space (called the feature space), enabling nonlinear classification and regression
|
|
with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
|
|
space corresponds to a complex nonlinear decision boundary in the original space.
|
|
It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
|
|
projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
|
|
preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
|
|
twisted manifold.
|
|
For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
|
|
<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">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> KernelPCA
|
|
rbf_pca <span style="color: #666666">=</span> KernelPCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>, kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=0.04</span>)
|
|
X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #666666">.</span>fit_transform(X)
|
|
</pre></div>
|
|
<p>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._DimRed-bs006.html">«</a></li>
|
|
<li><a href="._DimRed-bs000.html">1</a></li>
|
|
<li><a href="._DimRed-bs001.html">2</a></li>
|
|
<li><a href="._DimRed-bs002.html">3</a></li>
|
|
<li><a href="._DimRed-bs003.html">4</a></li>
|
|
<li><a href="._DimRed-bs004.html">5</a></li>
|
|
<li><a href="._DimRed-bs005.html">6</a></li>
|
|
<li><a href="._DimRed-bs006.html">7</a></li>
|
|
<li class="active"><a href="._DimRed-bs007.html">8</a></li>
|
|
<li><a href="._DimRed-bs008.html">9</a></li>
|
|
<li><a href="._DimRed-bs009.html">10</a></li>
|
|
<li><a href="._DimRed-bs008.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>
|
|
|
|
|