mhjensen
268 lines
12 KiB
HTML
268 lines
12 KiB
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<a class="navbar-brand" href="week45-bs.html">Week 45: Random Forests and Boosting</a>
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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="._week45-bs001.html#___sec0" style="font-size: 80%;">Random forests</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs002.html#___sec1" style="font-size: 80%;">Random Forest Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs003.html#___sec2" 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="._week45-bs004.html#___sec3" style="font-size: 80%;">Compare Bagging on Trees with Random Forests</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs005.html#___sec4" style="font-size: 80%;">Boosting, a Bird's Eye View</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs006.html#___sec5" style="font-size: 80%;">What is boosting? Additive Modelling/Iterative Fitting</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs007.html#___sec6" style="font-size: 80%;">Iterative Fitting, Regression and Squared-error Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs008.html#___sec7" style="font-size: 80%;">Squared-Error Example and Iterative Fitting</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs009.html#___sec8" style="font-size: 80%;">Iterative Fitting, Classification and AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs010.html#___sec9" style="font-size: 80%;">Adaptive Boosting, AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs011.html#___sec10" style="font-size: 80%;">Building up AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs012.html#___sec11" style="font-size: 80%;">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">Basic Steps of AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs014.html#___sec13" style="font-size: 80%;">AdaBoost Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs015.html#___sec14" style="font-size: 80%;">AdaBoost for Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs016.html#___sec15" style="font-size: 80%;">Gradient boosting: Basics with Steepest Descent</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs017.html#___sec16" style="font-size: 80%;">The Squared-Error again! Steepest Descent</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs018.html#___sec17" style="font-size: 80%;">Steepest Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs019.html#___sec18" style="font-size: 80%;">Gradient Boosting, algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#___sec19" style="font-size: 80%;">Gradient Boosting Example, Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs021.html#___sec20" style="font-size: 80%;">Gradient Boosting, Examples of Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs022.html#___sec21" style="font-size: 80%;">Gradient Boosting, Classification Example</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs023.html#___sec22" style="font-size: 80%;">XGBoost: Extreme Gradient Boosting</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs024.html#___sec23" style="font-size: 80%;">Regression Case</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs025.html#___sec24" style="font-size: 80%;">Xgboost on the Cancer Data</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0013"></a>
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<!-- !split -->
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<h2 id="___sec12" 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 easy to see that we must have \( \sum_{i=0}^{n-1}w_i = 1 \).</li>
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<li> We rewrite the misclassification error as</li>
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</ol>
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$$
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\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
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$$
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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 set 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 \( \alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m} \)</li>
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<li> Set the new weights to \( w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)} \).</li>
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</ol>
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<li> Compute the new classifier \( G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i) \).</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 observations 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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