426 lines
24 KiB
HTML
426 lines
24 KiB
HTML
<!--
|
|
HTML file automatically generated from DocOnce source
|
|
(https://github.com/doconce/doconce/)
|
|
doconce format html week47.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week47-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 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods">
|
|
<title>Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods</title>
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<!-- doconce format html week47.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week47-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': [('Plan for week 47', 2, None, 'plan-for-week-47'),
|
|
('Building a tree, regression',
|
|
2,
|
|
None,
|
|
'building-a-tree-regression'),
|
|
('A top-down approach, recursive binary splitting',
|
|
2,
|
|
None,
|
|
'a-top-down-approach-recursive-binary-splitting'),
|
|
('Making a tree', 2, None, 'making-a-tree'),
|
|
('Pruning the tree', 2, None, 'pruning-the-tree'),
|
|
('Cost complexity pruning', 2, None, 'cost-complexity-pruning'),
|
|
('Schematic Regression Procedure',
|
|
2,
|
|
None,
|
|
'schematic-regression-procedure'),
|
|
('A Classification Tree', 2, None, 'a-classification-tree'),
|
|
('Growing a classification tree',
|
|
2,
|
|
None,
|
|
'growing-a-classification-tree'),
|
|
('Classification tree, how to split nodes',
|
|
2,
|
|
None,
|
|
'classification-tree-how-to-split-nodes'),
|
|
('Visualizing the Tree, Classification',
|
|
2,
|
|
None,
|
|
'visualizing-the-tree-classification'),
|
|
('Visualizing the Tree, The Moons',
|
|
2,
|
|
None,
|
|
'visualizing-the-tree-the-moons'),
|
|
('Other ways of visualizing the trees',
|
|
2,
|
|
None,
|
|
'other-ways-of-visualizing-the-trees'),
|
|
('Printing out as text', 2, None, 'printing-out-as-text'),
|
|
('Algorithms for Setting up Decision Trees',
|
|
2,
|
|
None,
|
|
'algorithms-for-setting-up-decision-trees'),
|
|
('The CART algorithm for Classification',
|
|
2,
|
|
None,
|
|
'the-cart-algorithm-for-classification'),
|
|
('The CART algorithm for Regression',
|
|
2,
|
|
None,
|
|
'the-cart-algorithm-for-regression'),
|
|
('Why binary splits?', 2, None, 'why-binary-splits'),
|
|
('Computing a Tree using the Gini Index',
|
|
2,
|
|
None,
|
|
'computing-a-tree-using-the-gini-index'),
|
|
('The Table', 2, None, 'the-table'),
|
|
('Computing the various Gini Indices',
|
|
2,
|
|
None,
|
|
'computing-the-various-gini-indices'),
|
|
('A possible code using Scikit-Learn',
|
|
2,
|
|
None,
|
|
'a-possible-code-using-scikit-learn'),
|
|
('Further example: Computing the Gini index',
|
|
2,
|
|
None,
|
|
'further-example-computing-the-gini-index'),
|
|
('Simple Python Code to read in Data and perform Classification',
|
|
2,
|
|
None,
|
|
'simple-python-code-to-read-in-data-and-perform-classification'),
|
|
('Computing the Gini Factor',
|
|
2,
|
|
None,
|
|
'computing-the-gini-factor'),
|
|
('Regression trees', 2, None, 'regression-trees'),
|
|
('Final regressor code', 2, None, 'final-regressor-code'),
|
|
('Pros and cons of trees, pros',
|
|
2,
|
|
None,
|
|
'pros-and-cons-of-trees-pros'),
|
|
('Disadvantages', 2, None, 'disadvantages'),
|
|
('Ensemble Methods: From a Single Tree to Many Trees and Extreme '
|
|
'Boosting, Meet the Jungle of Methods',
|
|
2,
|
|
None,
|
|
'ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods'),
|
|
('An Overview of Ensemble Methods',
|
|
2,
|
|
None,
|
|
'an-overview-of-ensemble-methods'),
|
|
('Why Voting?', 2, None, 'why-voting'),
|
|
('Tossing coins', 2, None, 'tossing-coins'),
|
|
('Standard imports first', 2, None, 'standard-imports-first'),
|
|
('Simple Voting Example, head or tail',
|
|
2,
|
|
None,
|
|
'simple-voting-example-head-or-tail'),
|
|
('Using the Voting Classifier',
|
|
2,
|
|
None,
|
|
'using-the-voting-classifier'),
|
|
('Voting and Bagging', 2, None, 'voting-and-bagging'),
|
|
('Bagging', 2, None, 'bagging'),
|
|
('More bagging', 2, None, 'more-bagging'),
|
|
('Making your own Bootstrap: Changing the Level of the Decision '
|
|
'Tree',
|
|
2,
|
|
None,
|
|
'making-your-own-bootstrap-changing-the-level-of-the-decision-tree'),
|
|
('Random forests', 2, None, 'random-forests'),
|
|
('Random Forest Algorithm', 2, None, 'random-forest-algorithm'),
|
|
('Random Forests Compared with other Methods on the Cancer Data',
|
|
2,
|
|
None,
|
|
'random-forests-compared-with-other-methods-on-the-cancer-data'),
|
|
('Compare Bagging on Trees with Random Forests',
|
|
2,
|
|
None,
|
|
'compare-bagging-on-trees-with-random-forests'),
|
|
("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'),
|
|
('Making an ADAboost code yourself',
|
|
2,
|
|
None,
|
|
'making-an-adaboost-code-yourself'),
|
|
('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'),
|
|
('Gradient boosting, making our own code for a regression case',
|
|
2,
|
|
None,
|
|
'gradient-boosting-making-our-own-code-for-a-regression-case')]}
|
|
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="week47-bs.html">Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting 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="._week47-bs001.html#plan-for-week-47" style="font-size: 80%;">Plan for week 47</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs002.html#building-a-tree-regression" style="font-size: 80%;">Building a tree, regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs003.html#a-top-down-approach-recursive-binary-splitting" style="font-size: 80%;">A top-down approach, recursive binary splitting</a></li>
|
|
<!-- navigation toc: --> <li><a href="#making-a-tree" style="font-size: 80%;">Making a tree</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs005.html#pruning-the-tree" style="font-size: 80%;">Pruning the tree</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs006.html#cost-complexity-pruning" style="font-size: 80%;">Cost complexity pruning</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs007.html#schematic-regression-procedure" style="font-size: 80%;">Schematic Regression Procedure</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs008.html#a-classification-tree" style="font-size: 80%;">A Classification Tree</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs009.html#growing-a-classification-tree" style="font-size: 80%;">Growing a classification tree</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs010.html#classification-tree-how-to-split-nodes" style="font-size: 80%;">Classification tree, how to split nodes</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs011.html#visualizing-the-tree-classification" style="font-size: 80%;">Visualizing the Tree, Classification</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs012.html#visualizing-the-tree-the-moons" style="font-size: 80%;">Visualizing the Tree, The Moons</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs013.html#other-ways-of-visualizing-the-trees" style="font-size: 80%;">Other ways of visualizing the trees</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs014.html#printing-out-as-text" style="font-size: 80%;">Printing out as text</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs015.html#algorithms-for-setting-up-decision-trees" style="font-size: 80%;">Algorithms for Setting up Decision Trees</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs016.html#the-cart-algorithm-for-classification" style="font-size: 80%;">The CART algorithm for Classification</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs017.html#the-cart-algorithm-for-regression" style="font-size: 80%;">The CART algorithm for Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs018.html#why-binary-splits" style="font-size: 80%;">Why binary splits?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs019.html#computing-a-tree-using-the-gini-index" style="font-size: 80%;">Computing a Tree using the Gini Index</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs020.html#the-table" style="font-size: 80%;">The Table</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs021.html#computing-the-various-gini-indices" style="font-size: 80%;">Computing the various Gini Indices</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs022.html#a-possible-code-using-scikit-learn" style="font-size: 80%;">A possible code using Scikit-Learn</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs023.html#further-example-computing-the-gini-index" style="font-size: 80%;">Further example: Computing the Gini index</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs024.html#simple-python-code-to-read-in-data-and-perform-classification" style="font-size: 80%;">Simple Python Code to read in Data and perform Classification</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs025.html#computing-the-gini-factor" style="font-size: 80%;">Computing the Gini Factor</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs026.html#regression-trees" style="font-size: 80%;">Regression trees</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs027.html#final-regressor-code" style="font-size: 80%;">Final regressor code</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs028.html#pros-and-cons-of-trees-pros" style="font-size: 80%;">Pros and cons of trees, pros</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs029.html#disadvantages" style="font-size: 80%;">Disadvantages</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs030.html#ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods" style="font-size: 80%;">Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs031.html#an-overview-of-ensemble-methods" style="font-size: 80%;">An Overview of Ensemble Methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs032.html#why-voting" style="font-size: 80%;">Why Voting?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs033.html#tossing-coins" style="font-size: 80%;">Tossing coins</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs034.html#standard-imports-first" style="font-size: 80%;">Standard imports first</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs035.html#simple-voting-example-head-or-tail" style="font-size: 80%;">Simple Voting Example, head or tail</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs036.html#using-the-voting-classifier" style="font-size: 80%;">Using the Voting Classifier</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs037.html#voting-and-bagging" style="font-size: 80%;">Voting and Bagging</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs038.html#bagging" style="font-size: 80%;">Bagging</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs039.html#more-bagging" style="font-size: 80%;">More bagging</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs040.html#making-your-own-bootstrap-changing-the-level-of-the-decision-tree" style="font-size: 80%;">Making your own Bootstrap: Changing the Level of the Decision Tree</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs041.html#random-forests" style="font-size: 80%;">Random forests</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs042.html#random-forest-algorithm" style="font-size: 80%;">Random Forest Algorithm</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs043.html#random-forests-compared-with-other-methods-on-the-cancer-data" style="font-size: 80%;">Random Forests Compared with other Methods on the Cancer Data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs044.html#compare-bagging-on-trees-with-random-forests" style="font-size: 80%;">Compare Bagging on Trees with Random Forests</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs045.html#boosting-a-bird-s-eye-view" style="font-size: 80%;">Boosting, a Bird's Eye View</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs046.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="._week47-bs047.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="._week47-bs048.html#squared-error-example-and-iterative-fitting" style="font-size: 80%;">Squared-Error Example and Iterative Fitting</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs049.html#iterative-fitting-classification-and-adaboost" style="font-size: 80%;">Iterative Fitting, Classification and AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs050.html#adaptive-boosting-adaboost" style="font-size: 80%;">Adaptive Boosting, AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs051.html#building-up-adaboost" style="font-size: 80%;">Building up AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs052.html#adaptive-boosting-adaboost-basic-algorithm" style="font-size: 80%;">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs053.html#basic-steps-of-adaboost" style="font-size: 80%;">Basic Steps of AdaBoost</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs054.html#adaboost-examples" style="font-size: 80%;">AdaBoost Examples</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs055.html#making-an-adaboost-code-yourself" style="font-size: 80%;">Making an ADAboost code yourself</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs056.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="._week47-bs057.html#the-squared-error-again-steepest-descent" style="font-size: 80%;">The Squared-Error again! Steepest Descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs058.html#steepest-descent-example" style="font-size: 80%;">Steepest Descent Example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs059.html#gradient-boosting-algorithm" style="font-size: 80%;">Gradient Boosting, algorithm</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs060.html#gradient-boosting-examples-of-regression" style="font-size: 80%;">Gradient Boosting, Examples of Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs061.html#gradient-boosting-classification-example" style="font-size: 80%;">Gradient Boosting, Classification Example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs062.html#xgboost-extreme-gradient-boosting" style="font-size: 80%;">XGBoost: Extreme Gradient Boosting</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs063.html#regression-case" style="font-size: 80%;">Regression Case</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs064.html#xgboost-on-the-cancer-data" style="font-size: 80%;">Xgboost on the Cancer Data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week47-bs065.html#gradient-boosting-making-our-own-code-for-a-regression-case" style="font-size: 80%;">Gradient boosting, making our own code for a regression case</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="part0004"></a>
|
|
<!-- !split -->
|
|
<h2 id="making-a-tree" class="anchor">Making a tree </h2>
|
|
|
|
<p>In order to implement the recursive binary splitting we start by selecting
|
|
the predictor \( x_j \) and a cutpoint \( s \) that splits the predictor space into two regions \( R_1 \) and \( R_2 \)
|
|
</p>
|
|
$$
|
|
\left\{X\vert x_j < s\right\},
|
|
$$
|
|
|
|
<p>and</p>
|
|
$$
|
|
\left\{X\vert x_j \geq s\right\},
|
|
$$
|
|
|
|
<p>so that we obtain the lowest MSE, that is</p>
|
|
$$
|
|
\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2,
|
|
$$
|
|
|
|
<p>which we want to minimize by considering all predictors
|
|
\( x_1,x_2,\dots,x_p \). We consider also all possible values of \( s \) for
|
|
each predictor. These values could be determined by randomly assigned
|
|
numbers or by starting at the midpoint and then proceed till we find
|
|
an optimal value.
|
|
</p>
|
|
|
|
<p>For any \( j \) and \( s \), we define the pair of half-planes where
|
|
\( \overline{y}_{R_1} \) is the mean response for the training
|
|
observations in \( R_1(j,s) \), and \( \overline{y}_{R_2} \) is the mean
|
|
response for the training observations in \( R_2(j,s) \).
|
|
</p>
|
|
|
|
<p>Finding the values of \( j \) and \( s \) that minimize the above equation can be
|
|
done quite quickly, especially when the number of features \( p \) is not
|
|
too large.
|
|
</p>
|
|
|
|
<p>Next, we repeat the process, looking
|
|
for the best predictor and best cutpoint in order to split the data
|
|
further so as to minimize the MSE within each of the resulting
|
|
regions. However, this time, instead of splitting the entire predictor
|
|
space, we split one of the two previously identified regions. We now
|
|
have three regions. Again, we look to split one of these three regions
|
|
further, so as to minimize the MSE. The process continues until a
|
|
stopping criterion is reached; for instance, we may continue until no
|
|
region contains more than five observations.
|
|
</p>
|
|
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week47-bs003.html">«</a></li>
|
|
<li><a href="._week47-bs000.html">1</a></li>
|
|
<li><a href="._week47-bs001.html">2</a></li>
|
|
<li><a href="._week47-bs002.html">3</a></li>
|
|
<li><a href="._week47-bs003.html">4</a></li>
|
|
<li class="active"><a href="._week47-bs004.html">5</a></li>
|
|
<li><a href="._week47-bs005.html">6</a></li>
|
|
<li><a href="._week47-bs006.html">7</a></li>
|
|
<li><a href="._week47-bs007.html">8</a></li>
|
|
<li><a href="._week47-bs008.html">9</a></li>
|
|
<li><a href="._week47-bs009.html">10</a></li>
|
|
<li><a href="._week47-bs010.html">11</a></li>
|
|
<li><a href="._week47-bs011.html">12</a></li>
|
|
<li><a href="._week47-bs012.html">13</a></li>
|
|
<li><a href="._week47-bs013.html">14</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week47-bs065.html">66</a></li>
|
|
<li><a href="._week47-bs005.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>
|
|
|