mhjensen
274 lines
12 KiB
HTML
274 lines
12 KiB
HTML
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{'highest level': 2,
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'sections': [('Random forests', 2, None, '___sec0'),
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('Random Forest Algorithm', 2, None, '___sec1'),
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('Random Forests Compared with other Methods on the Cancer Data',
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('Compare Bagging on Trees with Random Forests',
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("Boosting, a Bird's Eye View", 2, None, '___sec4'),
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('Iterative Fitting, Regression and Squared-error Cost Function',
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('Squared-Error Example and Iterative Fitting',
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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="#___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> <!-- end of navigation bar -->
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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="part0008"></a>
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<!-- !split -->
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<h2 id="___sec7" class="anchor">Squared-Error Example and Iterative Fitting </h2>
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<p>
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To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
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<p>
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For simplicity we assume also that our functions \( b(x;\gamma)=1+\gamma x \).
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<p>
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This means that for every iteration \( m \), we need to optimize
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$$
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(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
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$$
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<p>
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We start our iteration by simply setting \( f_0(x)=0 \).
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Taking the derivatives with respect to \( \beta \) and \( \gamma \) we obtain
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$$
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\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
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$$
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and
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$$
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\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
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$$
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We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector)
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$$
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\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,
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$$
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which gives us \( \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}) \). Similarly we have
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$$
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\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
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$$
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<p>
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which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting
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for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.
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<p>
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The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as
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\( f_1(x) = \beta_1(1+\gamma_1x) \). Doing this \( M \) times results in our final estimate for the function \( f \).
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<p>
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<p>
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<li><a href="">...</a></li>
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