mhjensen
329 lines
19 KiB
HTML
329 lines
19 KiB
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('An Overview of Ensemble Methods', 2, None, '___sec32'),
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('Simple Voting Example, head or tail', 2, None, '___sec35'),
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('Gradient boosting: Basics', 2, None, '___sec50'),
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('Gradient Boosting, algorithm', 2, None, '___sec51'),
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('Gradient Boosting, Examples', 2, None, '___sec52'),
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('Gradient Boots with Early Stopping', 2, None, '___sec53'),
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('XGBoost: Extreme Gradient Boosting', 2, None, '___sec54'),
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs001.html#___sec0" style="font-size: 80%;">Decision trees, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs002.html#___sec1" style="font-size: 80%;">A typical Decision Tree with its pertinent Jargon, Classification Problem</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs003.html#___sec2" style="font-size: 80%;">General Features</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs004.html#___sec3" style="font-size: 80%;">How do we set it up?</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs005.html#___sec4" style="font-size: 80%;">Decision trees and Regression</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs006.html#___sec5" style="font-size: 80%;">Building a tree, regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DecisionTrees-bs007.html#___sec6" style="font-size: 80%;">A top-down approach, recursive binary splitting</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs008.html#___sec7" style="font-size: 80%;">Making a tree</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs009.html#___sec8" style="font-size: 80%;">Pruning the tree</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs010.html#___sec9" style="font-size: 80%;">Cost complexity pruning</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs011.html#___sec10" style="font-size: 80%;">Schematic Regression Procedure</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs012.html#___sec11" style="font-size: 80%;">A Classification Tree</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs013.html#___sec12" style="font-size: 80%;">Growing a classification tree</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs014.html#___sec13" style="font-size: 80%;">Classification tree, how to split nodes</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs015.html#___sec14" style="font-size: 80%;">Visualizing the Tree, Classification</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs016.html#___sec15" style="font-size: 80%;">Visualizing the Tree, The Moons</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs017.html#___sec16" style="font-size: 80%;">Algorithms for Setting up Decision Trees</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs018.html#___sec17" style="font-size: 80%;">The CART algorithm for Classification</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs019.html#___sec18" style="font-size: 80%;">The CART algorithm for Regression</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs020.html#___sec19" style="font-size: 80%;">Computing the Gini index</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs021.html#___sec20" style="font-size: 80%;">Simple Python Code to read in Data and perform Classification</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs022.html#___sec21" style="font-size: 80%;">Computing the Gini Factor</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs023.html#___sec22" style="font-size: 80%;">Entropy and the ID3 algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs024.html#___sec23" style="font-size: 80%;">Implementing the ID3 Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs025.html#___sec24" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs026.html#___sec25" style="font-size: 80%;">Another example, the moons again</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs027.html#___sec26" style="font-size: 80%;">Playing around with regions</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs028.html#___sec27" style="font-size: 80%;">Regression trees</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs029.html#___sec28" style="font-size: 80%;">Final regressor code</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs030.html#___sec29" style="font-size: 80%;">Pros and cons of trees, pros</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs031.html#___sec30" style="font-size: 80%;">Disadvantages</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs032.html#___sec31" style="font-size: 80%;">Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs033.html#___sec32" style="font-size: 80%;">An Overview of Ensemble Methods</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs034.html#___sec33" style="font-size: 80%;">Bagging</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs035.html#___sec34" style="font-size: 80%;">More bagging</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs036.html#___sec35" style="font-size: 80%;">Simple Voting Example, head or tail</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs037.html#___sec36" style="font-size: 80%;">Using the Voting Classifier</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs038.html#___sec37" style="font-size: 80%;">Please, not the moons again! Voting and Bagging</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs039.html#___sec38" style="font-size: 80%;">Now Bagging</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs040.html#___sec39" style="font-size: 80%;">Making our own Bagging with Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs041.html#___sec40" style="font-size: 80%;">Changing the Level of the Decision Tree</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs042.html#___sec41" style="font-size: 80%;">Random forests</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs043.html#___sec42" style="font-size: 80%;">Random Forest Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs044.html#___sec43" style="font-size: 80%;">Bootstrap with Random Forests Instead of a Single Tree, own Bagging</a></li>
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|
<!-- navigation toc: --> <li><a href="._DecisionTrees-bs045.html#___sec44" style="font-size: 80%;">Random Forests Compared with other Methods on the Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs046.html#___sec45" style="font-size: 80%;">Compare Bagging on Trees with Random Forests</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs047.html#___sec46" style="font-size: 80%;">Boosting, a Bird'e Eye</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs048.html#___sec47" style="font-size: 80%;">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="#___sec48" style="font-size: 80%;">Basic Steps of AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs050.html#___sec49" style="font-size: 80%;">AdaBoost Examples</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs051.html#___sec50" style="font-size: 80%;">Gradient boosting: Basics</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs052.html#___sec51" style="font-size: 80%;">Gradient Boosting, algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs053.html#___sec52" style="font-size: 80%;">Gradient Boosting, Examples</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs054.html#___sec53" style="font-size: 80%;">Gradient Boots with Early Stopping</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs055.html#___sec54" style="font-size: 80%;">XGBoost: Extreme Gradient Boosting</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs056.html#___sec55" style="font-size: 80%;">Regression Case</a></li>
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<!-- navigation toc: --> <li><a href="._DecisionTrees-bs057.html#___sec56" style="font-size: 80%;">Xgboost on the Cancer Data</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="___sec48" class="anchor">Basic Steps of AdaBoost </h2>
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<p>
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With the above definitions we are now ready to set up the algorithm for AdaBoost.
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The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
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<ol>
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<li> We start by initializing all weights to \( w_i = 1/n \), with \( i=0,1,2,\dots n-1 \). It is to see then that \( \sum_{i=0}^{n-1}w_i = 1 \).</li>
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<li> We rewrite the misclassification error as</li>
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</ol>
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$$
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\mathrm{err}=\frac{\sum_{i=0}^{n-1}w_iI(y_i\ne G(\boldsymbol{X}_{i*})}{\sum_{i=0}^{n-1}w_i},
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$$
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<ol>
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<li> Then we start looping over all attempts at classifying, namely we start an iterative process for \( m=1:M \), where \( M \) is the final number of classifications. Our given classifier could for example be a plain decision tree.
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<ol type="a"></li>
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<li> Fit then a given classifier to the training using the weights \( w_i \).</li>
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<li> Compute then \( \mathrm{err} \) and figure out which events are classified properly and which are classified wrongly.</li>
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<li> Define a quantity $\alpha_{m} = \log{(1-\mathrm{err})/\mathrm{err}}</li>
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<li> Set the new weights to $w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(\boldsymbol{X}_{i*})}.</li>
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</ol>
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<li> Compute the new classifier $G(\boldsymbol{X})= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(\boldsymbol{X}_{i*}).</li>
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</ol>
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For the iterations with \( m \le 2 \) the weights are modified
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individually at each steps. The obersvations which were misclassified
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at iteration \( m-1 \) have a weight which is larger than those which were
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classified properly. As this proceeds, the observations which were
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difficult to classifiy correctly are given a larger influence. Each
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new classification step \( m \) is then forced to concentrate on those
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observations that are missed in the previous iterations.
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<p>
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<p>
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<!-- copyright only on the titlepage -->
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</center>
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</body>
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</html>
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