mhjensen
248 lines
13 KiB
HTML
248 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="description" content="Data Analysis and Machine Learning: Logistic Regression">
|
|
|
|
<title>Data Analysis and Machine Learning: Logistic Regression</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': [('Logistic Regression', 2, None, '___sec0'),
|
|
('Optimization and Deep learning', 2, None, '___sec1'),
|
|
('Basics', 2, None, '___sec2'),
|
|
('Linear classifier', 2, None, '___sec3'),
|
|
('Some selected properties', 2, None, '___sec4'),
|
|
('The logistic function', 2, None, '___sec5'),
|
|
('Two parameters', 2, None, '___sec6'),
|
|
('Maximum likelihood', 2, None, '___sec7'),
|
|
('The cost function rewritten', 2, None, '___sec8'),
|
|
('Minimizing the cross entropy', 2, None, '___sec9'),
|
|
('A more compact expression', 2, None, '___sec10'),
|
|
('Extending to more predictors', 2, None, '___sec11'),
|
|
('Including more classes', 2, None, '___sec12'),
|
|
('The Softmax function', 2, None, '___sec13'),
|
|
('A _scikit-learn_ example', 2, None, '___sec14'),
|
|
('A simple classification problem', 2, None, '___sec15'),
|
|
('The two-dimensional Ising model, Predicting phase transition '
|
|
'of the two-dimensional Ising model',
|
|
2,
|
|
None,
|
|
'___sec16'),
|
|
('Reading in the data', 2, None, '___sec17'),
|
|
('Logistic regression', 2, None, '___sec18'),
|
|
('Exploring the logistic regression', 2, None, '___sec19'),
|
|
('Accuracy of a classification model', 2, None, '___sec20'),
|
|
('Analyzing the results', 2, None, '___sec21')]}
|
|
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="LogReg-bs.html">Data Analysis and Machine Learning: Logistic Regression</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="._LogReg-bs001.html#___sec0" style="font-size: 80%;">Logistic Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs002.html#___sec1" style="font-size: 80%;">Optimization and Deep learning</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs003.html#___sec2" style="font-size: 80%;">Basics</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs004.html#___sec3" style="font-size: 80%;">Linear classifier</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs005.html#___sec4" style="font-size: 80%;">Some selected properties</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs006.html#___sec5" style="font-size: 80%;">The logistic function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs007.html#___sec6" style="font-size: 80%;">Two parameters</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs008.html#___sec7" style="font-size: 80%;">Maximum likelihood</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs009.html#___sec8" style="font-size: 80%;">The cost function rewritten</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs010.html#___sec9" style="font-size: 80%;">Minimizing the cross entropy</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs011.html#___sec10" style="font-size: 80%;">A more compact expression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs012.html#___sec11" style="font-size: 80%;">Extending to more predictors</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A <b>scikit-learn</b> example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">A simple classification problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;">The two-dimensional Ising model, Predicting phase transition of the two-dimensional Ising model</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs018.html#___sec17" style="font-size: 80%;">Reading in the data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs019.html#___sec18" style="font-size: 80%;">Logistic regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs020.html#___sec19" style="font-size: 80%;">Exploring the logistic regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs021.html#___sec20" style="font-size: 80%;">Accuracy of a classification model</a></li>
|
|
<!-- navigation toc: --> <li><a href="._LogReg-bs022.html#___sec21" style="font-size: 80%;">Analyzing the results</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="part0017"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec16" class="anchor">The two-dimensional Ising model, Predicting phase transition of the two-dimensional Ising model </h2>
|
|
|
|
<p>
|
|
The Hamiltonian of the two-dimensional Ising model without an external field for a constant coupling constant \( J \) is given by
|
|
$$
|
|
\begin{align}
|
|
H = -J \sum_{\langle ij\rangle} S_i S_j,
|
|
\tag{2}
|
|
\end{align}
|
|
$$
|
|
|
|
where \( S_i \in \{-1, 1\} \) and \( \langle ij \rangle \) signifies that we only iterate over the nearest neighbors in the lattice. We will be looking at a system of \( L = 40 \) spins in each dimension, i.e., \( L^2 = 1600 \) spins in total. Opposed to the one-dimensional Ising model we will get a phase transition from an <b>ordered</b> phase to a <b>disordered</b> phase at the critical temperature
|
|
|
|
$$
|
|
\begin{align}
|
|
\frac{T_c}{J} = \frac{2}{\log\left(1 + \sqrt{2}\right)} \approx 2.26,
|
|
\tag{3}
|
|
\end{align}
|
|
$$
|
|
|
|
as shown by Lars Onsager.
|
|
|
|
<p>
|
|
Here we use <b>logistic regression</b> to predict when a phase transition
|
|
occurs. The data we will look at is a set of spin configurations,
|
|
i.e., individual lattices with spins, labeled <b>ordered</b> <code>1</code> or
|
|
<b>disordered</b> <code>0</code>. Our job is to build a model which will take in a
|
|
spin configuration and predict whether or not the spin configuration
|
|
constitutes an ordered or a disordered phase. To achieve this we will
|
|
represent the lattices as flattened arrays with \( 1600 \) elements
|
|
instead of a matrix of \( 40 \times 40 \) elements. As an extra test of
|
|
the performance of the algorithms we will divide the dataset into
|
|
three pieces. We will do a conventional train-test-split on a
|
|
combination of totally ordered and totally disordered phases. The
|
|
remaining "critical-like" states will be used as test data which we
|
|
hope the model will be able to make good extrapolated predictions on.
|
|
|
|
<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">pickle</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">glob</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">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skms</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skm</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">copy</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">time</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
|
|
|
|
<span style="color: #666666">%</span>matplotlib inline
|
|
|
|
sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000">True</span>)
|
|
</pre></div>
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._LogReg-bs016.html">«</a></li>
|
|
<li><a href="._LogReg-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._LogReg-bs009.html">10</a></li>
|
|
<li><a href="._LogReg-bs010.html">11</a></li>
|
|
<li><a href="._LogReg-bs011.html">12</a></li>
|
|
<li><a href="._LogReg-bs012.html">13</a></li>
|
|
<li><a href="._LogReg-bs013.html">14</a></li>
|
|
<li><a href="._LogReg-bs014.html">15</a></li>
|
|
<li><a href="._LogReg-bs015.html">16</a></li>
|
|
<li><a href="._LogReg-bs016.html">17</a></li>
|
|
<li class="active"><a href="._LogReg-bs017.html">18</a></li>
|
|
<li><a href="._LogReg-bs018.html">19</a></li>
|
|
<li><a href="._LogReg-bs019.html">20</a></li>
|
|
<li><a href="._LogReg-bs020.html">21</a></li>
|
|
<li><a href="._LogReg-bs021.html">22</a></li>
|
|
<li><a href="._LogReg-bs022.html">23</a></li>
|
|
<li><a href="._LogReg-bs018.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>
|
|
|
|
|