lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

added html files

Browse Source
This commit is contained in:
mhjensen
2019-10-24 06:14:09 +02:00
parent 871338800b
commit 3c9acac906
3 changed files with 754 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files
doc/pub/DimRed/html/._DimRed-bs028.html
+256
View File
@@ -0,0 +1,256 @@
<!--
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: Preprocessing and Dimensionality Reduction">
<title>Data Analysis and Machine Learning: Preprocessing and 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'),
('Preprocessing our data', 2, None, '___sec1'),
('More preprocessing', 2, None, '___sec2'),
('Simple preprocessing examples, Franke function and regression',
2,
None,
'___sec3'),
('Simple preprocessing examples, breast cancer data and '
'classification, Support Vector Machines',
2,
None,
'___sec4'),
('More on Cancer Data, now with Logistic Regression',
2,
None,
'___sec5'),
('Why should we think of reducing the dimensionality',
2,
None,
'___sec6'),
('Basic ideas of the Principal Component Analysis (PCA)',
2,
None,
'___sec7'),
('Introducing the Covariance and Correlation functions',
2,
None,
'___sec8'),
('Correlation Function and Design/Feature Matrix',
2,
None,
'___sec9'),
('Covariance Matrix Examples', 2, None, '___sec10'),
('Correlation Matrix', 2, None, '___sec11'),
('Correlation Matrix with Pandas', 2, None, '___sec12'),
('Correlation Matrix with Pandas and the Franke function',
2,
None,
'___sec13'),
('Rewriting the Covariance and/or Correlation Matrix',
2,
None,
'___sec14'),
('Towards the PCA theorem', 2, None, '___sec15'),
('The Algorithm before the Theorem', 2, None, '___sec16'),
('Classical PCA Theorem', 2, None, '___sec17'),
('Proof of the PCA Theorem', 2, None, '___sec18'),
('PCA Proof continued', 2, None, '___sec19'),
('The final step', 2, None, '___sec20'),
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
('Principal Component Analysis', 2, None, '___sec22'),
('PCA and scikit-learn', 2, None, '___sec23'),
('More on the PCA', 2, None, '___sec24'),
('Incremental PCA', 2, None, '___sec25'),
('Randomized PCA', 2, None, '___sec26'),
('Kernel PCA', 2, None, '___sec27'),
('LLE', 2, None, '___sec28'),
('Other techniques', 2, None, '___sec29')]}
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: Preprocessing and 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%;">Preprocessing our data</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">More preprocessing</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">Simple preprocessing examples, breast cancer data and classification, Support Vector Machines</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on Cancer Data, now with Logistic Regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Why should we think of reducing the dimensionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Basic ideas of the Principal Component Analysis (PCA)</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Covariance Matrix Examples</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;">Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
<!-- navigation toc: --> <li><a href="#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div> <!-- end of navigation bar -->
<div class="container">
<p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
<a name="part0028"></a>
<!-- !split -->
<h2 id="___sec27" 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&#8217;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">&quot;rbf&quot;</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-bs027.html">&laquo;</a></li>
<li><a href="._DimRed-bs000.html">1</a></li>
<li><a href="">...</a></li>
<li><a href="._DimRed-bs020.html">21</a></li>
<li><a href="._DimRed-bs021.html">22</a></li>
<li><a href="._DimRed-bs022.html">23</a></li>
<li><a href="._DimRed-bs023.html">24</a></li>
<li><a href="._DimRed-bs024.html">25</a></li>
<li><a href="._DimRed-bs025.html">26</a></li>
<li><a href="._DimRed-bs026.html">27</a></li>
<li><a href="._DimRed-bs027.html">28</a></li>
<li class="active"><a href="._DimRed-bs028.html">29</a></li>
<li><a href="._DimRed-bs029.html">30</a></li>
<li><a href="._DimRed-bs030.html">31</a></li>
<li><a href="._DimRed-bs029.html">&raquo;</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>
doc/pub/DimRed/html/._DimRed-bs029.html
+237
View File
@@ -0,0 +1,237 @@
<!--
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: Preprocessing and Dimensionality Reduction">
<title>Data Analysis and Machine Learning: Preprocessing and 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'),
('Preprocessing our data', 2, None, '___sec1'),
('More preprocessing', 2, None, '___sec2'),
('Simple preprocessing examples, Franke function and regression',
2,
None,
'___sec3'),
('Simple preprocessing examples, breast cancer data and '
'classification, Support Vector Machines',
2,
None,
'___sec4'),
('More on Cancer Data, now with Logistic Regression',
2,
None,
'___sec5'),
('Why should we think of reducing the dimensionality',
2,
None,
'___sec6'),
('Basic ideas of the Principal Component Analysis (PCA)',
2,
None,
'___sec7'),
('Introducing the Covariance and Correlation functions',
2,
None,
'___sec8'),
('Correlation Function and Design/Feature Matrix',
2,
None,
'___sec9'),
('Covariance Matrix Examples', 2, None, '___sec10'),
('Correlation Matrix', 2, None, '___sec11'),
('Correlation Matrix with Pandas', 2, None, '___sec12'),
('Correlation Matrix with Pandas and the Franke function',
2,
None,
'___sec13'),
('Rewriting the Covariance and/or Correlation Matrix',
2,
None,
'___sec14'),
('Towards the PCA theorem', 2, None, '___sec15'),
('The Algorithm before the Theorem', 2, None, '___sec16'),
('Classical PCA Theorem', 2, None, '___sec17'),
('Proof of the PCA Theorem', 2, None, '___sec18'),
('PCA Proof continued', 2, None, '___sec19'),
('The final step', 2, None, '___sec20'),
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
('Principal Component Analysis', 2, None, '___sec22'),
('PCA and scikit-learn', 2, None, '___sec23'),
('More on the PCA', 2, None, '___sec24'),
('Incremental PCA', 2, None, '___sec25'),
('Randomized PCA', 2, None, '___sec26'),
('Kernel PCA', 2, None, '___sec27'),
('LLE', 2, None, '___sec28'),
('Other techniques', 2, None, '___sec29')]}
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: Preprocessing and 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%;">Preprocessing our data</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">More preprocessing</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">Simple preprocessing examples, breast cancer data and classification, Support Vector Machines</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on Cancer Data, now with Logistic Regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Why should we think of reducing the dimensionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Basic ideas of the Principal Component Analysis (PCA)</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Covariance Matrix Examples</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;">Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
<!-- navigation toc: --> <li><a href="#___sec28" style="font-size: 80%;">LLE</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div> <!-- end of navigation bar -->
<div class="container">
<p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
<a name="part0029"></a>
<!-- !split -->
<h2 id="___sec28" class="anchor">LLE </h2>
<p>
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
these local relationships are best preserved (more details shortly).
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
<ul class="pagination">
<li><a href="._DimRed-bs028.html">&laquo;</a></li>
<li><a href="._DimRed-bs000.html">1</a></li>
<li><a href="">...</a></li>
<li><a href="._DimRed-bs021.html">22</a></li>
<li><a href="._DimRed-bs022.html">23</a></li>
<li><a href="._DimRed-bs023.html">24</a></li>
<li><a href="._DimRed-bs024.html">25</a></li>
<li><a href="._DimRed-bs025.html">26</a></li>
<li><a href="._DimRed-bs026.html">27</a></li>
<li><a href="._DimRed-bs027.html">28</a></li>
<li><a href="._DimRed-bs028.html">29</a></li>
<li class="active"><a href="._DimRed-bs029.html">30</a></li>
<li><a href="._DimRed-bs030.html">31</a></li>
<li><a href="._DimRed-bs030.html">&raquo;</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>
doc/pub/DimRed/html/._DimRed-bs030.html
+261
View File
@@ -0,0 +1,261 @@
<!--
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: Preprocessing and Dimensionality Reduction">
<title>Data Analysis and Machine Learning: Preprocessing and 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'),
('Preprocessing our data', 2, None, '___sec1'),
('More preprocessing', 2, None, '___sec2'),
('Simple preprocessing examples, Franke function and regression',
2,
None,
'___sec3'),
('Simple preprocessing examples, breast cancer data and '
'classification, Support Vector Machines',
2,
None,
'___sec4'),
('More on Cancer Data, now with Logistic Regression',
2,
None,
'___sec5'),
('Why should we think of reducing the dimensionality',
2,
None,
'___sec6'),
('Basic ideas of the Principal Component Analysis (PCA)',
2,
None,
'___sec7'),
('Introducing the Covariance and Correlation functions',
2,
None,
'___sec8'),
('Correlation Function and Design/Feature Matrix',
2,
None,
'___sec9'),
('Covariance Matrix Examples', 2, None, '___sec10'),
('Correlation Matrix', 2, None, '___sec11'),
('Correlation Matrix with Pandas', 2, None, '___sec12'),
('Correlation Matrix with Pandas and the Franke function',
2,
None,
'___sec13'),
('Rewriting the Covariance and/or Correlation Matrix',
2,
None,
'___sec14'),
('Towards the PCA theorem', 2, None, '___sec15'),
('The Algorithm before the Theorem', 2, None, '___sec16'),
('Classical PCA Theorem', 2, None, '___sec17'),
('Proof of the PCA Theorem', 2, None, '___sec18'),
('PCA Proof continued', 2, None, '___sec19'),
('The final step', 2, None, '___sec20'),
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
('Principal Component Analysis', 2, None, '___sec22'),
('PCA and scikit-learn', 2, None, '___sec23'),
('More on the PCA', 2, None, '___sec24'),
('Incremental PCA', 2, None, '___sec25'),
('Randomized PCA', 2, None, '___sec26'),
('Kernel PCA', 2, None, '___sec27'),
('LLE', 2, None, '___sec28'),
('Other techniques', 2, None, '___sec29')]}
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: Preprocessing and 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%;">Preprocessing our data</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">More preprocessing</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">Simple preprocessing examples, breast cancer data and classification, Support Vector Machines</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on Cancer Data, now with Logistic Regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Why should we think of reducing the dimensionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Basic ideas of the Principal Component Analysis (PCA)</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Covariance Matrix Examples</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;">Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
<!-- navigation toc: --> <li><a href="#___sec29" style="font-size: 80%;">Other techniques</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div> <!-- end of navigation bar -->
<div class="container">
<p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
<a name="part0030"></a>
<!-- !split -->
<h2 id="___sec29" class="anchor">Other techniques </h2>
<p>
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
<p>
Here are some of the most popular:
<ul>
<li> <b>Multidimensional Scaling (MDS)</b> reduces dimensionality while trying to preserve the distances between the instances.</li>
<li> <b>Isomap</b> creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.</li>
<li> <b>t-Distributed Stochastic Neighbor Embedding</b> (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).</li>
<li> Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.</li>
</ul>
Here are other examples where we use the <b>DataFrame</b> functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
of dimensionality \( 10\times 5 \) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other 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">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">100</span>)
<span style="color: #408080; font-style: italic"># setting up a 10 x 5 matrix</span>
rows <span style="color: #666666">=</span> <span style="color: #666666">10</span>
cols <span style="color: #666666">=</span> <span style="color: #666666">5</span>
a <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(rows,cols)
df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(a)
display(df)
<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>mean())
<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>std())
display(df<span style="color: #666666">**2</span>)
</pre></div>
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
<ul class="pagination">
<li><a href="._DimRed-bs029.html">&laquo;</a></li>
<li><a href="._DimRed-bs000.html">1</a></li>
<li><a href="">...</a></li>
<li><a href="._DimRed-bs022.html">23</a></li>
<li><a href="._DimRed-bs023.html">24</a></li>
<li><a href="._DimRed-bs024.html">25</a></li>
<li><a href="._DimRed-bs025.html">26</a></li>
<li><a href="._DimRed-bs026.html">27</a></li>
<li><a href="._DimRed-bs027.html">28</a></li>
<li><a href="._DimRed-bs028.html">29</a></li>
<li><a href="._DimRed-bs029.html">30</a></li>
<li class="active"><a href="._DimRed-bs030.html">31</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>