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mhjensen 0e5076dfbf update week 45
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<!-- navigation toc: --> <li><a href="._week45-bs001.html#___sec0" style="font-size: 80%;">Overview of week 45</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs002.html#___sec1" style="font-size: 80%;">Decision trees, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs003.html#___sec2" style="font-size: 80%;">Basics of a tree</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs004.html#___sec3" style="font-size: 80%;">A Sketch of a Tree, Regression problem</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs005.html#___sec4" style="font-size: 80%;">A Sketch of a Tree, Classification problem</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs006.html#___sec5" style="font-size: 80%;">A typical Decision Tree with its pertinent Jargon, Classification Problem</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs007.html#___sec6" style="font-size: 80%;">General Features</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs008.html#___sec7" style="font-size: 80%;">How do we set it up?</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs009.html#___sec8" style="font-size: 80%;">Decision trees and Regression</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs010.html#___sec9" style="font-size: 80%;">Building a tree, regression</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs011.html#___sec10" style="font-size: 80%;">A top-down approach, recursive binary splitting</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs012.html#___sec11" style="font-size: 80%;">Making a tree</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs013.html#___sec12" style="font-size: 80%;">Pruning the tree</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs014.html#___sec13" style="font-size: 80%;">Cost complexity pruning</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs015.html#___sec14" style="font-size: 80%;">Schematic Regression Procedure</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs016.html#___sec15" style="font-size: 80%;">A Classification Tree</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs017.html#___sec16" style="font-size: 80%;">Growing a classification tree</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs018.html#___sec17" style="font-size: 80%;">Classification tree, how to split nodes</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs019.html#___sec18" style="font-size: 80%;">Visualizing the Tree, Classification</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs020.html#___sec19" style="font-size: 80%;">Visualizing the Tree, The Moons</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs021.html#___sec20" style="font-size: 80%;">Other ways of visualizing the trees</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs022.html#___sec21" style="font-size: 80%;">Printing out as text</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs023.html#___sec22" style="font-size: 80%;">Algorithms for Setting up Decision Trees</a></li>
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">The CART algorithm for Classification</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs025.html#___sec24" style="font-size: 80%;">The CART algorithm for Regression</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs026.html#___sec25" style="font-size: 80%;">Computing the Gini index</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs027.html#___sec26" style="font-size: 80%;">Simple Python Code to read in Data and perform Classification</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs028.html#___sec27" style="font-size: 80%;">Computing the Gini Factor</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs029.html#___sec28" style="font-size: 80%;">Entropy and the ID3 algorithm</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs030.html#___sec29" style="font-size: 80%;">Cancer Data again now with Decision Trees and other Methods</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs031.html#___sec30" style="font-size: 80%;">Another example, the moons again</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs032.html#___sec31" style="font-size: 80%;">Playing around with regions</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs033.html#___sec32" style="font-size: 80%;">Regression trees</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs034.html#___sec33" style="font-size: 80%;">Final regressor code</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs035.html#___sec34" style="font-size: 80%;">Pros and cons of trees, pros</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs036.html#___sec35" style="font-size: 80%;">Disadvantages</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs037.html#___sec36" style="font-size: 80%;">Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs038.html#___sec37" style="font-size: 80%;">An Overview of Ensemble Methods</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs039.html#___sec38" style="font-size: 80%;">Bagging</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs040.html#___sec39" style="font-size: 80%;">More bagging</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs041.html#___sec40" style="font-size: 80%;">Making your own Bootstrap: Changing the Level of the Decision Tree</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs042.html#___sec41" style="font-size: 80%;">Why Voting?</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs043.html#___sec42" style="font-size: 80%;">Tossing coins</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs044.html#___sec43" style="font-size: 80%;">Standard imports first</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs045.html#___sec44" style="font-size: 80%;">Simple Voting Example, head or tail</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs046.html#___sec45" style="font-size: 80%;">Using the Voting Classifier</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs047.html#___sec46" style="font-size: 80%;">Voting and Bagging</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs048.html#___sec47" style="font-size: 80%;">Random forests</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs049.html#___sec48" style="font-size: 80%;">Random Forest Algorithm</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs050.html#___sec49" style="font-size: 80%;">Random Forests Compared with other Methods on the Cancer Data</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs051.html#___sec50" style="font-size: 80%;">Compare Bagging on Trees with Random Forests</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs052.html#___sec51" style="font-size: 80%;">Boosting, a Bird's Eye View</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs053.html#___sec52" style="font-size: 80%;">What is boosting? Additive Modelling/Iterative Fitting</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs054.html#___sec53" style="font-size: 80%;">Iterative Fitting, Regression and Squared-error Cost Function</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs055.html#___sec54" style="font-size: 80%;">Squared-Error Example and Iterative Fitting</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs056.html#___sec55" style="font-size: 80%;">Iterative Fitting, Classification and AdaBoost</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs057.html#___sec56" style="font-size: 80%;">Adaptive Boosting, AdaBoost</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs058.html#___sec57" style="font-size: 80%;">Building up AdaBoost</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs059.html#___sec58" style="font-size: 80%;">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs060.html#___sec59" style="font-size: 80%;">Basic Steps of AdaBoost</a></li>
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<!-- navigation toc: --> <li><a href="._week45-bs062.html#___sec61" style="font-size: 80%;">Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs063.html#___sec62" style="font-size: 80%;">The Squared-Error again! Steepest Descent</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs064.html#___sec63" style="font-size: 80%;">Steepest Descent Example</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs065.html#___sec64" style="font-size: 80%;">Gradient Boosting, algorithm</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs066.html#___sec65" style="font-size: 80%;">Gradient Boosting, Examples of Regression</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs067.html#___sec66" style="font-size: 80%;">Gradient Boosting, Classification Example</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs068.html#___sec67" style="font-size: 80%;">XGBoost: Extreme Gradient Boosting</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs069.html#___sec68" style="font-size: 80%;">Regression Case</a></li>
<!-- navigation toc: --> <li><a href="._week45-bs070.html#___sec69" style="font-size: 80%;">Xgboost on the Cancer Data</a></li>
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<h2 id="___sec23" class="anchor">The CART algorithm for Classification </h2>
<p>
For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \).
This could be for example a threshold set by a number below a certain circumference of a malign tumor.
<p>
How do we find these two quantities?
We search for the pair \( (k,t_k) \) that produces the purest subset using for example the <b>gini</b> factor \( G \).
The cost function it tries to minimize is then
$$
C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}},
$$
where \( G_{\mathrm{left/right}} \) measures the impurity of the left/right subset and \( m_{\mathrm{left/right}} \)
is the number of instances in the left/right subset
<p>
Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets
and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the
\( max\_depth \) hyperparameter), or if it cannot find a split that will reduce impurity. A few other
hyperparameters control additional stopping conditions such as the \( min\_samples\_split \),
\( min\_samples\_leaf \), \( min\_weight\_fraction\_leaf \), and \( max\_leaf\_nodes \).
<p>
<p>
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