Morten Hjorth-Jensen
304 lines
16 KiB
HTML
304 lines
16 KiB
HTML
<!--
|
|
HTML file automatically generated from DocOnce source
|
|
(https://github.com/doconce/doconce/)
|
|
doconce format html week45.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week45-bs --no_mako
|
|
-->
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
|
<meta name="description" content="Week 45: Decisions Trees, Random Forests, Bagging and Boosting">
|
|
<title>Week 45: Decisions Trees, Random Forests, Bagging and Boosting</title>
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<!-- doconce format html week45.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week45-bs --no_mako -->
|
|
<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': [('Overview of week 45', 2, None, 'overview-of-week-45'),
|
|
('Brief code reminder from last wekk',
|
|
2,
|
|
None,
|
|
'brief-code-reminder-from-last-wekk'),
|
|
("Boosting, a Bird's Eye View",
|
|
2,
|
|
None,
|
|
'boosting-a-bird-s-eye-view'),
|
|
('What is boosting? Additive Modelling/Iterative Fitting',
|
|
2,
|
|
None,
|
|
'what-is-boosting-additive-modelling-iterative-fitting'),
|
|
('Iterative Fitting, Regression and Squared-error Cost Function',
|
|
2,
|
|
None,
|
|
'iterative-fitting-regression-and-squared-error-cost-function'),
|
|
('Squared-Error Example and Iterative Fitting',
|
|
2,
|
|
None,
|
|
'squared-error-example-and-iterative-fitting'),
|
|
('Iterative Fitting, Classification and AdaBoost',
|
|
2,
|
|
None,
|
|
'iterative-fitting-classification-and-adaboost'),
|
|
('Adaptive Boosting, AdaBoost',
|
|
2,
|
|
None,
|
|
'adaptive-boosting-adaboost'),
|
|
('Building up AdaBoost', 2, None, 'building-up-adaboost'),
|
|
('Adaptive boosting: AdaBoost, Basic Algorithm',
|
|
2,
|
|
None,
|
|
'adaptive-boosting-adaboost-basic-algorithm'),
|
|
('Basic Steps of AdaBoost', 2, None, 'basic-steps-of-adaboost'),
|
|
('AdaBoost Examples', 2, None, 'adaboost-examples'),
|
|
('Gradient boosting: Basics with Steepest Descent/Functional '
|
|
'Gradient Descent',
|
|
2,
|
|
None,
|
|
'gradient-boosting-basics-with-steepest-descent-functional-gradient-descent'),
|
|
('The Squared-Error again! Steepest Descent',
|
|
2,
|
|
None,
|
|
'the-squared-error-again-steepest-descent'),
|
|
('Steepest Descent Example', 2, None, 'steepest-descent-example'),
|
|
('Gradient Boosting, algorithm',
|
|
2,
|
|
None,
|
|
'gradient-boosting-algorithm'),
|
|
('Gradient Boosting, Examples of Regression',
|
|
2,
|
|
None,
|
|
'gradient-boosting-examples-of-regression'),
|
|
('Gradient Boosting, Classification Example',
|
|
2,
|
|
None,
|
|
'gradient-boosting-classification-example'),
|
|
('XGBoost: Extreme Gradient Boosting',
|
|
2,
|
|
None,
|
|
'xgboost-extreme-gradient-boosting'),
|
|
('Regression Case', 2, None, 'regression-case'),
|
|
('Xgboost on the Cancer Data',
|
|
2,
|
|
None,
|
|
'xgboost-on-the-cancer-data'),
|
|
('Support Vector Machines, overarching aims',
|
|
2,
|
|
None,
|
|
'support-vector-machines-overarching-aims'),
|
|
('Hyperplanes and all that', 2, None, 'hyperplanes-and-all-that'),
|
|
('What is a hyperplane?', 2, None, 'what-is-a-hyperplane'),
|
|
('A $p$-dimensional space of features',
|
|
2,
|
|
None,
|
|
'a-p-dimensional-space-of-features'),
|
|
('The two-dimensional case', 2, None, 'the-two-dimensional-case'),
|
|
('Getting into the details', 2, None, 'getting-into-the-details'),
|
|
('First attempt at a minimization approach',
|
|
2,
|
|
None,
|
|
'first-attempt-at-a-minimization-approach'),
|
|
('Solving the equations', 2, None, 'solving-the-equations'),
|
|
('Code Example', 2, None, 'code-example'),
|
|
('Problems with the Simpler Approach',
|
|
2,
|
|
None,
|
|
'problems-with-the-simpler-approach'),
|
|
('A better approach', 2, None, 'a-better-approach'),
|
|
('A quick Reminder on Lagrangian Multipliers',
|
|
2,
|
|
None,
|
|
'a-quick-reminder-on-lagrangian-multipliers'),
|
|
('Adding the Multiplier', 2, None, 'adding-the-multiplier'),
|
|
('Setting up the Problem', 2, None, 'setting-up-the-problem'),
|
|
('The problem to solve', 2, None, 'the-problem-to-solve'),
|
|
('The last steps', 2, None, 'the-last-steps'),
|
|
('A soft classifier', 2, None, 'a-soft-classifier'),
|
|
('Soft optmization problem',
|
|
2,
|
|
None,
|
|
'soft-optmization-problem')]}
|
|
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="week45-bs.html">Week 45: Decisions Trees, Random Forests, Bagging and Boosting</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="._week45-bs001.html#overview-of-week-45" style="font-size: 80%;">Overview of week 45</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs002.html#brief-code-reminder-from-last-wekk" style="font-size: 80%;">Brief code reminder from last wekk</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs003.html#boosting-a-bird-s-eye-view" style="font-size: 80%;">Boosting, a Bird's Eye View</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs004.html#what-is-boosting-additive-modelling-iterative-fitting" style="font-size: 80%;">What is boosting? Additive Modelling/Iterative Fitting</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs005.html#iterative-fitting-regression-and-squared-error-cost-function" style="font-size: 80%;">Iterative Fitting, Regression and Squared-error Cost Function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs006.html#squared-error-example-and-iterative-fitting" style="font-size: 80%;">Squared-Error Example and Iterative Fitting</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs007.html#iterative-fitting-classification-and-adaboost" style="font-size: 80%;">Iterative Fitting, Classification and AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs008.html#adaptive-boosting-adaboost" style="font-size: 80%;">Adaptive Boosting, AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs009.html#building-up-adaboost" style="font-size: 80%;">Building up AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs010.html#adaptive-boosting-adaboost-basic-algorithm" style="font-size: 80%;">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs011.html#basic-steps-of-adaboost" style="font-size: 80%;">Basic Steps of AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs012.html#adaboost-examples" style="font-size: 80%;">AdaBoost Examples</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs013.html#gradient-boosting-basics-with-steepest-descent-functional-gradient-descent" style="font-size: 80%;">Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs014.html#the-squared-error-again-steepest-descent" style="font-size: 80%;">The Squared-Error again! Steepest Descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs015.html#steepest-descent-example" style="font-size: 80%;">Steepest Descent Example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs016.html#gradient-boosting-algorithm" style="font-size: 80%;">Gradient Boosting, algorithm</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs017.html#gradient-boosting-examples-of-regression" style="font-size: 80%;">Gradient Boosting, Examples of Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs018.html#gradient-boosting-classification-example" style="font-size: 80%;">Gradient Boosting, Classification Example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs019.html#xgboost-extreme-gradient-boosting" style="font-size: 80%;">XGBoost: Extreme Gradient Boosting</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs020.html#regression-case" style="font-size: 80%;">Regression Case</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs021.html#xgboost-on-the-cancer-data" style="font-size: 80%;">Xgboost on the Cancer Data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs022.html#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs023.html#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs024.html#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs025.html#a-p-dimensional-space-of-features" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs026.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs027.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs028.html#first-attempt-at-a-minimization-approach" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs029.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs030.html#code-example" style="font-size: 80%;">Code Example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs031.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs032.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs033.html#a-quick-reminder-on-lagrangian-multipliers" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
|
<!-- navigation toc: --> <li><a href="#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs035.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs036.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs037.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs038.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week45-bs039.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</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="part0034"></a>
|
|
<!-- !split -->
|
|
<h2 id="adding-the-multiplier" class="anchor">Adding the Multiplier </h2>
|
|
|
|
<p>However, we can add to</p>
|
|
$$
|
|
df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz,
|
|
$$
|
|
|
|
<p>a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in</p>
|
|
$$
|
|
df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda
|
|
\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+
|
|
(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0.
|
|
$$
|
|
|
|
<p>Our multiplier is chosen so that</p>
|
|
$$
|
|
\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0.
|
|
$$
|
|
|
|
<p>We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have</p>
|
|
$$
|
|
\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0,
|
|
$$
|
|
|
|
<p>and</p>
|
|
$$
|
|
\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0.
|
|
$$
|
|
|
|
<p>When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and
|
|
\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined,
|
|
it is therefore often called
|
|
Lagrange's undetermined multiplier.
|
|
If we have a set of constraints \( \phi_k \) we have the equations
|
|
</p>
|
|
$$
|
|
\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0.
|
|
$$
|
|
|
|
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week45-bs033.html">«</a></li>
|
|
<li><a href="._week45-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week45-bs026.html">27</a></li>
|
|
<li><a href="._week45-bs027.html">28</a></li>
|
|
<li><a href="._week45-bs028.html">29</a></li>
|
|
<li><a href="._week45-bs029.html">30</a></li>
|
|
<li><a href="._week45-bs030.html">31</a></li>
|
|
<li><a href="._week45-bs031.html">32</a></li>
|
|
<li><a href="._week45-bs032.html">33</a></li>
|
|
<li><a href="._week45-bs033.html">34</a></li>
|
|
<li class="active"><a href="._week45-bs034.html">35</a></li>
|
|
<li><a href="._week45-bs035.html">36</a></li>
|
|
<li><a href="._week45-bs036.html">37</a></li>
|
|
<li><a href="._week45-bs037.html">38</a></li>
|
|
<li><a href="._week45-bs038.html">39</a></li>
|
|
<li><a href="._week45-bs039.html">40</a></li>
|
|
<li><a href="._week45-bs035.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="https://..."><img width="250" align=right src="https://..."></a>
|
|
</footer>
|
|
-->
|
|
<center style="font-size:80%">
|
|
<!-- copyright only on the titlepage -->
|
|
</center>
|
|
</body>
|
|
</html>
|
|
|