306 lines
15 KiB
HTML
306 lines
15 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="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': [('Plans for week 38', 2, None, '___sec0'),
|
|
('Thursday September 17', 2, None, '___sec1'),
|
|
('Ridge and LASSO Regression, reminder', 2, None, '___sec2'),
|
|
('Various steps in cross-validation', 2, None, '___sec3'),
|
|
('How to set up the cross-validation for Ridge and/or Lasso',
|
|
2,
|
|
None,
|
|
'___sec4'),
|
|
('Cross-validation in brief', 2, None, '___sec5'),
|
|
('Code Example for Cross-validation and $k$-fold '
|
|
'Cross-validation',
|
|
2,
|
|
None,
|
|
'___sec6'),
|
|
('Bias-Variance tradeoff with Bootstrap', 2, None, '___sec7'),
|
|
("Another Example from Scikit-Learn's Repository",
|
|
2,
|
|
None,
|
|
'___sec8'),
|
|
('Cross-validation with Ridge', 2, None, '___sec9'),
|
|
('The Ising model', 2, None, '___sec10'),
|
|
('Reformulating the problem to suit regression',
|
|
2,
|
|
None,
|
|
'___sec11'),
|
|
('Linear regression', 2, None, '___sec12'),
|
|
('Singular Value decomposition', 2, None, '___sec13'),
|
|
('The one-dimensional Ising model', 2, None, '___sec14'),
|
|
('Ridge regression', 2, None, '___sec15'),
|
|
('LASSO regression', 2, None, '___sec16'),
|
|
('Performance as function of the regularization parameter',
|
|
2,
|
|
None,
|
|
'___sec17'),
|
|
('Finding the optimal value of $\\lambda$', 2, None, '___sec18'),
|
|
('Friday September 18: Intro to Logistic Regression',
|
|
2,
|
|
None,
|
|
'___sec19'),
|
|
('Logistic Regression', 2, None, '___sec20'),
|
|
('Classification problems', 2, None, '___sec21'),
|
|
('Optimization and Deep learning', 2, None, '___sec22'),
|
|
('Basics', 2, None, '___sec23'),
|
|
('Linear classifier', 2, None, '___sec24'),
|
|
('Some selected properties', 2, None, '___sec25'),
|
|
('Simple example', 2, None, '___sec26'),
|
|
('Plotting the mean value for each group', 2, None, '___sec27'),
|
|
('The logistic function', 2, None, '___sec28'),
|
|
('Examples of likelihood functions used in logistic regression '
|
|
'and nueral networks',
|
|
2,
|
|
None,
|
|
'___sec29'),
|
|
('Two parameters', 2, None, '___sec30'),
|
|
('Maximum likelihood', 2, None, '___sec31'),
|
|
('The cost function rewritten', 2, None, '___sec32'),
|
|
('Minimizing the cross entropy', 2, None, '___sec33'),
|
|
('A more compact expression', 2, None, '___sec34'),
|
|
('Extending to more predictors', 2, None, '___sec35'),
|
|
('Including more classes', 2, None, '___sec36'),
|
|
('More classes', 2, None, '___sec37'),
|
|
('Wisconsin Cancer Data', 2, None, '___sec38'),
|
|
('Using the correlation matrix', 2, None, '___sec39'),
|
|
('Discussing the correlation data', 2, None, '___sec40'),
|
|
('Other measures in classification studies: Cancer Data again',
|
|
2,
|
|
None,
|
|
'___sec41')]}
|
|
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="week38-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="._week38-bs001.html#___sec0" style="font-size: 80%;">Plans for week 38</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs002.html#___sec1" style="font-size: 80%;">Thursday September 17</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">Ridge and LASSO Regression, reminder</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs004.html#___sec3" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs005.html#___sec4" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs006.html#___sec5" style="font-size: 80%;">Cross-validation in brief</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs007.html#___sec6" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs008.html#___sec7" style="font-size: 80%;">Bias-Variance tradeoff with Bootstrap</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs009.html#___sec8" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs010.html#___sec9" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs011.html#___sec10" style="font-size: 80%;">The Ising model</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs012.html#___sec11" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs013.html#___sec12" style="font-size: 80%;">Linear regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs014.html#___sec13" style="font-size: 80%;">Singular Value decomposition</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs015.html#___sec14" style="font-size: 80%;">The one-dimensional Ising model</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs016.html#___sec15" style="font-size: 80%;">Ridge regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs017.html#___sec16" style="font-size: 80%;">LASSO regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs018.html#___sec17" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs019.html#___sec18" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs020.html#___sec19" style="font-size: 80%;">Friday September 18: Intro to Logistic Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs021.html#___sec20" style="font-size: 80%;">Logistic Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs022.html#___sec21" style="font-size: 80%;">Classification problems</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs023.html#___sec22" style="font-size: 80%;">Optimization and Deep learning</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs024.html#___sec23" style="font-size: 80%;">Basics</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs025.html#___sec24" style="font-size: 80%;">Linear classifier</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs026.html#___sec25" style="font-size: 80%;">Some selected properties</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs027.html#___sec26" style="font-size: 80%;">Simple example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs028.html#___sec27" style="font-size: 80%;">Plotting the mean value for each group</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs029.html#___sec28" style="font-size: 80%;">The logistic function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs030.html#___sec29" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs031.html#___sec30" style="font-size: 80%;">Two parameters</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs032.html#___sec31" style="font-size: 80%;">Maximum likelihood</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs033.html#___sec32" style="font-size: 80%;">The cost function rewritten</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs034.html#___sec33" style="font-size: 80%;">Minimizing the cross entropy</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs035.html#___sec34" style="font-size: 80%;">A more compact expression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Using the correlation matrix</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs041.html#___sec40" style="font-size: 80%;">Discussing the correlation data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week38-bs042.html#___sec41" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</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="part0003"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec2" class="anchor">Ridge and LASSO Regression, reminder </h2>
|
|
|
|
<p>
|
|
The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is
|
|
our optimization problem is
|
|
$$
|
|
{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
|
|
$$
|
|
|
|
or we can state it as
|
|
$$
|
|
{\displaystyle \min_{\boldsymbol{\beta}\in
|
|
{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
|
|
$$
|
|
|
|
where we have used the definition of a norm-2 vector, that is
|
|
$$
|
|
\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
|
|
$$
|
|
|
|
<p>
|
|
By minimizing the above equation with respect to the parameters
|
|
\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the
|
|
parameters \( \boldsymbol{\beta} \). We can add a regularization parameter \( \lambda \) by
|
|
defining a new cost function to be optimized, that is
|
|
|
|
$$
|
|
{\displaystyle \min_{\boldsymbol{\beta}\in
|
|
{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
|
|
$$
|
|
|
|
<p>
|
|
which leads to the Ridge regression minimization problem where we
|
|
require that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is
|
|
a finite number larger than zero. By defining
|
|
|
|
$$
|
|
C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,
|
|
$$
|
|
|
|
<p>
|
|
we have a new optimization equation
|
|
$$
|
|
{\displaystyle \min_{\boldsymbol{\beta}\in
|
|
{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1
|
|
$$
|
|
|
|
which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.
|
|
|
|
<p>
|
|
Here we have defined the norm-1 as
|
|
$$
|
|
\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert.
|
|
$$
|
|
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week38-bs002.html">«</a></li>
|
|
<li><a href="._week38-bs000.html">1</a></li>
|
|
<li><a href="._week38-bs001.html">2</a></li>
|
|
<li><a href="._week38-bs002.html">3</a></li>
|
|
<li class="active"><a href="._week38-bs003.html">4</a></li>
|
|
<li><a href="._week38-bs004.html">5</a></li>
|
|
<li><a href="._week38-bs005.html">6</a></li>
|
|
<li><a href="._week38-bs006.html">7</a></li>
|
|
<li><a href="._week38-bs007.html">8</a></li>
|
|
<li><a href="._week38-bs008.html">9</a></li>
|
|
<li><a href="._week38-bs009.html">10</a></li>
|
|
<li><a href="._week38-bs010.html">11</a></li>
|
|
<li><a href="._week38-bs011.html">12</a></li>
|
|
<li><a href="._week38-bs012.html">13</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week38-bs042.html">43</a></li>
|
|
<li><a href="._week38-bs004.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>
|
|
|
|
|