lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
Files
5c6c05efbe058f033f3f35def695f64a7c70b9ce
FYS-STK4155/doc/pub/week43/html/._week43-bs009.html
T
mhjensen 427bbed498 updating week 43
2020-10-17 22:09:23 +02:00

280 lines
16 KiB
HTML
Raw Blame History

<!--
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 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods">
<title>Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods</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': [('Recurrent Neural Networks', 2, None, '___sec0'),
('Solving ODEs with Deep Learning', 2, None, '___sec1'),
('Why should we think of reducing the dimensionality',
2,
None,
'___sec2'),
('Basic ideas of the Principal Component Analysis (PCA)',
2,
None,
'___sec3'),
('Introducing the Covariance and Correlation functions',
2,
None,
'___sec4'),
('Correlation Function and Design/Feature Matrix',
2,
None,
'___sec5'),
('Covariance Matrix Examples', 2, None, '___sec6'),
('Correlation Matrix', 2, None, '___sec7'),
('Correlation Matrix with Pandas', 2, None, '___sec8'),
('Correlation Matrix with Pandas and the Franke function',
2,
None,
'___sec9'),
('Rewriting the Covariance and/or Correlation Matrix',
2,
None,
'___sec10'),
('Towards the PCA theorem', 2, None, '___sec11'),
('The Algorithm before the Theorem', 2, None, '___sec12'),
('Writing our own PCA code', 2, None, '___sec13'),
('Compute the sample mean and center the data',
3,
None,
'___sec14'),
('Compute the sample covariance', 3, None, '___sec15'),
('Diagonalize the sample covariance matrix to obtain the '
'principal components',
3,
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'),
('Geometric Interpretation and link with Singular Value '
'Decomposition',
2,
None,
'___sec21'),
('Principal Component Analysis', 2, None, '___sec22'),
('PCA and scikit-learn', 2, None, '___sec23'),
('Back to the Cancer Data', 2, None, '___sec24'),
('More on the PCA', 2, None, '___sec25'),
('Incremental PCA', 2, None, '___sec26'),
('Randomized PCA', 2, None, '___sec27'),
('Kernel PCA', 2, None, '___sec28'),
('LLE', 2, None, '___sec29'),
('Other techniques', 2, None, '___sec30')]}
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="week43-bs.html">Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods</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="._week43-bs002.html#___sec0" style="font-size: 80%;"><b>Recurrent Neural Networks</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs003.html#___sec1" style="font-size: 80%;"><b>Solving ODEs with Deep Learning</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs004.html#___sec2" style="font-size: 80%;"><b>Why should we think of reducing the dimensionality</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs005.html#___sec3" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs006.html#___sec4" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs007.html#___sec5" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs008.html#___sec6" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs010.html#___sec8" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs011.html#___sec9" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs012.html#___sec10" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs013.html#___sec11" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs014.html#___sec12" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec13" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec14" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample mean and center the data</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec15" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample covariance</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec16" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Diagonalize the sample covariance matrix to obtain the principal components</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs016.html#___sec17" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs017.html#___sec18" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec19" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec20" style="font-size: 80%;"><b>The final step</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec21" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec22" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec23" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec24" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec25" style="font-size: 80%;"><b>More on the PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs025.html#___sec26" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs026.html#___sec27" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec28" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec29" style="font-size: 80%;"><b>LLE</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec30" style="font-size: 80%;"><b>Other techniques</b></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="part0009"></a>
<!-- !split -->
<h2 id="___sec7" class="anchor">Correlation Matrix </h2>
<p>
The previous example can be converted into the correlation matrix by
simply scaling the matrix elements with the variances. We should also
subtract the mean values for each column. This leads to the following
code which sets up the correlations matrix for the previous example in
a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors).
<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>
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
<span style="color: #408080; font-style: italic"># define two vectors </span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>random(size<span style="color: #666666">=</span>n)
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
<span style="color: #408080; font-style: italic">#scaling the x and y vectors </span>
x <span style="color: #666666">=</span> x <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(x)
y <span style="color: #666666">=</span> y <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y)
variance_x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #AA22FF">@x</span>)<span style="color: #666666">/</span>n
variance_y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(y<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>n
<span style="color: #008000">print</span>(variance_x)
<span style="color: #008000">print</span>(variance_y)
cov_xy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>n
cov_xx <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #AA22FF">@x</span>)<span style="color: #666666">/</span>n
cov_yy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(y<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>n
C <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #666666">2</span>,<span style="color: #666666">2</span>))
C[<span style="color: #666666">0</span>,<span style="color: #666666">0</span>]<span style="color: #666666">=</span> cov_xx<span style="color: #666666">/</span>variance_x
C[<span style="color: #666666">1</span>,<span style="color: #666666">1</span>]<span style="color: #666666">=</span> cov_yy<span style="color: #666666">/</span>variance_y
C[<span style="color: #666666">0</span>,<span style="color: #666666">1</span>]<span style="color: #666666">=</span> cov_xy<span style="color: #666666">/</span>np<span style="color: #666666">.</span>sqrt(variance_y<span style="color: #666666">*</span>variance_x)
C[<span style="color: #666666">1</span>,<span style="color: #666666">0</span>]<span style="color: #666666">=</span> C[<span style="color: #666666">0</span>,<span style="color: #666666">1</span>]
<span style="color: #008000">print</span>(C)
</pre></div>
<p>
We see that the matrix elements along the diagonal are one as they
should be and that the matrix is symmetric. Furthermore, diagonalizing
this matrix we easily see that it is a positive definite matrix.
<p>
The above procedure with <b>numpy</b> can be made more compact if we use <b>pandas</b>.
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
<ul class="pagination">
<li><a href="._week43-bs008.html">&laquo;</a></li>
<li><a href="._week43-bs000.html">1</a></li>
<li><a href="._week43-bs001.html">2</a></li>
<li><a href="._week43-bs002.html">3</a></li>
<li><a href="._week43-bs003.html">4</a></li>
<li><a href="._week43-bs004.html">5</a></li>
<li><a href="._week43-bs005.html">6</a></li>
<li><a href="._week43-bs006.html">7</a></li>
<li><a href="._week43-bs007.html">8</a></li>
<li><a href="._week43-bs008.html">9</a></li>
<li class="active"><a href="._week43-bs009.html">10</a></li>
<li><a href="._week43-bs010.html">11</a></li>
<li><a href="._week43-bs011.html">12</a></li>
<li><a href="._week43-bs012.html">13</a></li>
<li><a href="._week43-bs013.html">14</a></li>
<li><a href="._week43-bs014.html">15</a></li>
<li><a href="._week43-bs015.html">16</a></li>
<li><a href="._week43-bs016.html">17</a></li>
<li><a href="._week43-bs017.html">18</a></li>
<li><a href="._week43-bs018.html">19</a></li>
<li><a href="">...</a></li>
<li><a href="._week43-bs029.html">30</a></li>
<li><a href="._week43-bs010.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>
Reference in New Issue View Git Blame Copy Permalink