mhjensen
250 lines
11 KiB
HTML
250 lines
11 KiB
HTML
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<title>Week 45: Random Forests and Boosting</title>
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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="#___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="._week45-bs013.html#___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="part0007"></a>
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<!-- !split -->
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<h2 id="___sec6" class="anchor">Iterative Fitting, Regression and Squared-error Cost Function </h2>
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<p>
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The way we proceed is as follows (here we specialize to the squared-error cost function)
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<ol>
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<li> Establish a cost function, here \( {\cal C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2 \) with \( f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m) \).</li>
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<li> Initialize with a guess \( f_0(x) \). It could be one or even zero or some random numbers.</li>
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<li> For \( m=1:M \)
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<ol type="a"></li>
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<li> minimize \( \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2 \) wrt \( \gamma \) and \( \beta \)</li>
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<li> This gives the optimal values \( \beta_m \) and \( \gamma_m \)</li>
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<li> Determine then the new values \( f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m) \)</li>
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</ol>
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</ol>
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We could use any of the algorithms we have discussed till now. If we
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use trees, \( \gamma \) parameterizes the split variables and split points
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at the internal nodes, and the predictions at the terminal nodes.
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<p>
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<li><a href="._week45-bs006.html">«</a></li>
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<li><a href="._week45-bs000.html">1</a></li>
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<li><a href="._week45-bs001.html">2</a></li>
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<li><a href="._week45-bs002.html">3</a></li>
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<li><a href="._week45-bs003.html">4</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week45-bs025.html">26</a></li>
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