302 lines
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HTML
302 lines
17 KiB
HTML
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<!-- navigation toc: --> <li><a href="._week45-bs001.html#overview-of-week-45" style="font-size: 80%;">Overview of week 45</a></li>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week45-bs007.html#iterative-fitting-classification-and-adaboost" style="font-size: 80%;">Iterative Fitting, Classification and AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs008.html#adaptive-boosting-adaboost" style="font-size: 80%;">Adaptive Boosting, AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs009.html#building-up-adaboost" style="font-size: 80%;">Building up AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs010.html#adaptive-boosting-adaboost-basic-algorithm" style="font-size: 80%;">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="#basic-steps-of-adaboost" style="font-size: 80%;">Basic Steps of AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs012.html#adaboost-examples" style="font-size: 80%;">AdaBoost Examples</a></li>
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<!-- 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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week45-bs015.html#steepest-descent-example" style="font-size: 80%;">Steepest Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs016.html#gradient-boosting-algorithm" style="font-size: 80%;">Gradient Boosting, algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs017.html#gradient-boosting-examples-of-regression" style="font-size: 80%;">Gradient Boosting, Examples of Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs018.html#gradient-boosting-classification-example" style="font-size: 80%;">Gradient Boosting, Classification Example</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs019.html#xgboost-extreme-gradient-boosting" style="font-size: 80%;">XGBoost: Extreme Gradient Boosting</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs020.html#regression-case" style="font-size: 80%;">Regression Case</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs021.html#xgboost-on-the-cancer-data" style="font-size: 80%;">Xgboost on the Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs022.html#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs023.html#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs024.html#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week45-bs026.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs027.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week45-bs029.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs030.html#code-example" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs031.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs032.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week45-bs034.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs035.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs036.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs037.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs038.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs039.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0011"></a>
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<!-- !split -->
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<h2 id="basic-steps-of-adaboost" class="anchor">Basic Steps of AdaBoost </h2>
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<p>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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</p>
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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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<p>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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