mhjensen
275 lines
12 KiB
HTML
275 lines
12 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="Week 37: Ridge and Lasso Regression">
|
|
|
|
<title>Week 37: Ridge and Lasso 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 37', 2, None, '___sec0'),
|
|
('Thursday September 10', 2, None, '___sec1'),
|
|
('A Bayesian approach to develop intuition about skrinkage '
|
|
'methods',
|
|
2,
|
|
None,
|
|
'___sec2'),
|
|
('The singular value decomposition', 2, None, '___sec3'),
|
|
('Linear Regression Problems', 2, None, '___sec4'),
|
|
('Fixing the singularity', 2, None, '___sec5'),
|
|
('Basic math of the SVD', 2, None, '___sec6'),
|
|
('The SVD, a Fantastic Algorithm', 2, None, '___sec7'),
|
|
('Economy-size SVD', 2, None, '___sec8'),
|
|
('Codes for the SVD', 2, None, '___sec9'),
|
|
('Mathematical Properties', 2, None, '___sec10'),
|
|
('Friday September 12', 2, None, '___sec11'),
|
|
('Ridge and LASSO Regression', 2, None, '___sec12'),
|
|
('More on Ridge Regression', 2, None, '___sec13'),
|
|
('Interpreting the Ridge results', 2, None, '___sec14'),
|
|
('More interpretations', 2, None, '___sec15'),
|
|
('A better understanding of regularization', 2, None, '___sec16'),
|
|
('Decomposing the OLS and Ridge expressions',
|
|
2,
|
|
None,
|
|
'___sec17'),
|
|
('Introducing the Covariance and Correlation functions',
|
|
2,
|
|
None,
|
|
'___sec18'),
|
|
('Correlation Function and Design/Feature Matrix',
|
|
2,
|
|
None,
|
|
'___sec19'),
|
|
('Covariance Matrix Examples', 2, None, '___sec20'),
|
|
('Correlation Matrix', 2, None, '___sec21'),
|
|
('Correlation Matrix with Pandas', 2, None, '___sec22'),
|
|
('Correlation Matrix with Pandas and the Franke function',
|
|
2,
|
|
None,
|
|
'___sec23'),
|
|
('Rewriting the Covariance and/or Correlation Matrix',
|
|
2,
|
|
None,
|
|
'___sec24'),
|
|
('Linking with SVD', 2, None, '___sec25')]}
|
|
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="week37-bs.html">Week 37: Ridge and Lasso 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="._week37-bs001.html#___sec0" style="font-size: 80%;">Plans for week 37</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs002.html#___sec1" style="font-size: 80%;">Thursday September 10</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs003.html#___sec2" style="font-size: 80%;">A Bayesian approach to develop intuition about skrinkage methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs004.html#___sec3" style="font-size: 80%;">The singular value decomposition</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs005.html#___sec4" style="font-size: 80%;">Linear Regression Problems</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs006.html#___sec5" style="font-size: 80%;">Fixing the singularity</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs007.html#___sec6" style="font-size: 80%;">Basic math of the SVD</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs008.html#___sec7" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs009.html#___sec8" style="font-size: 80%;">Economy-size SVD</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs010.html#___sec9" style="font-size: 80%;">Codes for the SVD</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs011.html#___sec10" style="font-size: 80%;">Mathematical Properties</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs012.html#___sec11" style="font-size: 80%;">Friday September 12</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs013.html#___sec12" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec13" style="font-size: 80%;">More on Ridge Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs015.html#___sec14" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs016.html#___sec15" style="font-size: 80%;">More interpretations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs017.html#___sec16" style="font-size: 80%;">A better understanding of regularization</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs018.html#___sec17" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs019.html#___sec18" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs020.html#___sec19" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs021.html#___sec20" style="font-size: 80%;">Covariance Matrix Examples</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs022.html#___sec21" style="font-size: 80%;">Correlation Matrix</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs023.html#___sec22" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs024.html#___sec23" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs025.html#___sec24" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs026.html#___sec25" style="font-size: 80%;">Linking with SVD</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="part0014"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec13" class="anchor">More on Ridge Regression </h2>
|
|
|
|
<p>
|
|
Using the matrix-vector expression for Ridge regression,
|
|
|
|
$$
|
|
C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta},
|
|
$$
|
|
|
|
<p>
|
|
by taking the derivatives with respect to \( \boldsymbol{\beta} \) we obtain then
|
|
a slightly modified matrix inversion problem which for finite values
|
|
of \( \lambda \) does not suffer from singularity problems. We obtain
|
|
|
|
$$
|
|
\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
|
|
$$
|
|
|
|
<p>
|
|
with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that
|
|
|
|
$$
|
|
\sum_{i=0}^{p-1} \beta_i^2 \leq t,
|
|
$$
|
|
|
|
<p>
|
|
with \( t \) a finite positive number.
|
|
|
|
<p>
|
|
We see that Ridge regression is nothing but the standard
|
|
OLS with a modified diagonal term added to \( \boldsymbol{X}^T\boldsymbol{X} \). The
|
|
consequences, in particular for our discussion of the bias-variance tradeoff
|
|
are rather interesting.
|
|
|
|
<p>
|
|
Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had
|
|
$$
|
|
(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}.
|
|
$$
|
|
|
|
<p>
|
|
We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \( \boldsymbol{U} \) as
|
|
$$
|
|
\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}
|
|
$$
|
|
|
|
<p>
|
|
For Ridge regression this becomes
|
|
|
|
$$
|
|
\boldsymbol{X}\boldsymbol{\beta}^{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
|
|
$$
|
|
|
|
<p>
|
|
with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \).
|
|
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week37-bs013.html">«</a></li>
|
|
<li><a href="._week37-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week37-bs006.html">7</a></li>
|
|
<li><a href="._week37-bs007.html">8</a></li>
|
|
<li><a href="._week37-bs008.html">9</a></li>
|
|
<li><a href="._week37-bs009.html">10</a></li>
|
|
<li><a href="._week37-bs010.html">11</a></li>
|
|
<li><a href="._week37-bs011.html">12</a></li>
|
|
<li><a href="._week37-bs012.html">13</a></li>
|
|
<li><a href="._week37-bs013.html">14</a></li>
|
|
<li class="active"><a href="._week37-bs014.html">15</a></li>
|
|
<li><a href="._week37-bs015.html">16</a></li>
|
|
<li><a href="._week37-bs016.html">17</a></li>
|
|
<li><a href="._week37-bs017.html">18</a></li>
|
|
<li><a href="._week37-bs018.html">19</a></li>
|
|
<li><a href="._week37-bs019.html">20</a></li>
|
|
<li><a href="._week37-bs020.html">21</a></li>
|
|
<li><a href="._week37-bs021.html">22</a></li>
|
|
<li><a href="._week37-bs022.html">23</a></li>
|
|
<li><a href="._week37-bs023.html">24</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week37-bs026.html">27</a></li>
|
|
<li><a href="._week37-bs015.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>
|
|
|
|
|