mhjensen
276 lines
12 KiB
HTML
276 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="#___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="._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="part0006"></a>
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<!-- !split -->
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<h2 id="___sec5" class="anchor">What is boosting? Additive Modelling/Iterative Fitting </h2>
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<p>
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Boosting is a way of fitting an additive expansion in a set of
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elementary basis functions like for example some simple polynomials.
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Assume for example that we have a function
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$$
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f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
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$$
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<p>
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where \( \beta_m \) are the expansion parameters to be determined in a
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minimization process and \( b(x;\gamma_m) \) are some simple functions of
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the multivariable parameter \( x \) which is characterized by the
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parameters \( \gamma_m \).
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<p>
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As an example, consider the Sigmoid function we used in logistic
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regression. In that case, we can translate the function
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\( b(x;\gamma_m) \) into the Sigmoid function
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$$
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\sigma(t) = \frac{1}{1+\exp{(-t)}},
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$$
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<p>
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where \( t=\gamma_0+\gamma_1 x \) and the parameters \( \gamma_0 \) and
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\( \gamma_1 \) were determined by the Logistic Regression fitting
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algorithm.
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<p>
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As another example, consider the cost function we defined for linear regression
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$$
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C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
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$$
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<p>
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In this case the function \( f(x) \) was replaced by the design matrix
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\( \boldsymbol{X} \) and the unknown linear regression parameters \( \boldsymbol{\beta} \),
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that is \( \boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta} \). In linear regression we can
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simply invert a matrix and obtain the parameters \( \beta \) by
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$$
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\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
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$$
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<p>
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In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \( \beta_m \) and \( \gamma_m \).
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<p>
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<p>
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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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