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<center><h1>Week 45: Random Forests and Boosting</h1></center> <!-- document title -->
<p>
<!-- author(s): Morten Hjorth-Jensen -->
<center>
<b>Morten Hjorth-Jensen</b> [1, 2]
</center>
<p>
<!-- institution(s) -->
<center>[1] <b>Department of Physics, University of Oslo</b></center>
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Sep 16, 2020</h4></center> <!-- date -->
<br>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec0">Random forests </h2>
<p>
Random forests provide an improvement over bagged trees by way of a
small tweak that decorrelates the trees.
<p>
As in bagging, we build a
number of decision trees on bootstrapped training samples. But when
building these decision trees, each time a split in a tree is
considered, a random sample of \( m \) predictors is chosen as split
candidates from the full set of \( p \) predictors. The split is allowed to
use only one of those \( m \) predictors.
<p>
A fresh sample of \( m \) predictors is
taken at each split, and typically we choose
$$
m\approx \sqrt{p}.
$$
<p>
In building a random forest, at
each split in the tree, the algorithm is not even allowed to consider
a majority of the available predictors.
<p>
The reason for this is rather clever. Suppose that there is one very
strong predictor in the data set, along with a number of other
moderately strong predictors. Then in the collection of bagged
variable importance random forest trees, most or all of the trees will
use this strong predictor in the top split. Consequently, all of the
bagged trees will look quite similar to each other. Hence the
predictions from the bagged trees will be highly correlated.
Unfortunately, averaging many highly correlated quantities does not
lead to as large of a reduction in variance as averaging many
uncorrelated quantities. In particular, this means that bagging will
not lead to a substantial reduction in variance over a single tree in
this setting.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec1">Random Forest Algorithm </h2>
The algorithm described here can be applied to both classification and regression problems.
<p>
We will grow of forest of say \( B \) trees.
<ol>
<li> For \( b=1:B \)</li>
<ul>
<li> Draw a bootstrap sample of from the training data organized in our \( \boldsymbol{X} \) matrix.</li>
<li> We grow then a random forest tree \( T_b \) based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached</li>
<ol>
<li> we select \( m \le p \) variables at random from the \( p \) predictors/features</li>
<li> pick the best split point among the \( m \) features using either the CART algorithm or the ID3 for classification and create a new node</li>
<li> split the node into daughter nodes</li>
</ol>
</ul>
<li> Output then the ensemble of trees \( \{T_b\}_1^{B} \) and make predictions for either a regression type of problem or a classification type of problem.</li>
</ol>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec2">Random Forests Compared with other Methods on the Cancer Data </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.tree</span> <span style="color: #008000; font-weight: bold">import</span> DecisionTreeClassifier
<span style="color: #408080; font-style: italic"># Load the data</span>
cancer <span style="color: #666666">=</span> load_breast_cancer()
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
<span style="color: #008000; font-weight: bold">print</span>(X_train<span style="color: #666666">.</span>shape)
<span style="color: #008000; font-weight: bold">print</span>(X_test<span style="color: #666666">.</span>shape)
<span style="color: #408080; font-style: italic"># Logistic Regression</span>
logreg <span style="color: #666666">=</span> LogisticRegression(solver<span style="color: #666666">=</span><span style="color: #BA2121">&#39;lbfgs&#39;</span>)
logreg<span style="color: #666666">.</span>fit(X_train, y_train)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression: {:.2f}&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X_test,y_test)))
<span style="color: #408080; font-style: italic"># Support vector machine</span>
svm <span style="color: #666666">=</span> SVC(gamma<span style="color: #666666">=</span><span style="color: #BA2121">&#39;auto&#39;</span>, C<span style="color: #666666">=100</span>)
svm<span style="color: #666666">.</span>fit(X_train, y_train)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with SVM: {:.2f}&quot;</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test,y_test)))
<span style="color: #408080; font-style: italic"># Decision Trees</span>
deep_tree_clf <span style="color: #666666">=</span> DecisionTreeClassifier(max_depth<span style="color: #666666">=</span><span style="color: #008000">None</span>)
deep_tree_clf<span style="color: #666666">.</span>fit(X_train, y_train)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Decision Trees: {:.2f}&quot;</span><span style="color: #666666">.</span>format(deep_tree_clf<span style="color: #666666">.</span>score(X_test,y_test)))
<span style="color: #408080; font-style: italic">#now scale the data</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
scaler <span style="color: #666666">=</span> StandardScaler()
scaler<span style="color: #666666">.</span>fit(X_train)
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
<span style="color: #408080; font-style: italic"># Logistic Regression</span>
logreg<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy Logistic Regression with scaled data: {:.2f}&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
<span style="color: #408080; font-style: italic"># Support Vector Machine</span>
svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy SVM with scaled data: {:.2f}&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
<span style="color: #408080; font-style: italic"># Decision Trees</span>
deep_tree_clf<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Decision Trees and scaled data: {:.2f}&quot;</span><span style="color: #666666">.</span>format(deep_tree_clf<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> RandomForestClassifier
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> LabelEncoder
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_validate
<span style="color: #408080; font-style: italic"># Data set not specificied</span>
<span style="color: #408080; font-style: italic">#Instantiate the model with 500 trees and entropy as splitting criteria</span>
Random_Forest_model <span style="color: #666666">=</span> RandomForestClassifier(n_estimators<span style="color: #666666">=500</span>,criterion<span style="color: #666666">=</span><span style="color: #BA2121">&quot;entropy&quot;</span>)
Random_Forest_model<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
<span style="color: #408080; font-style: italic">#Cross validation</span>
accuracy <span style="color: #666666">=</span> cross_validate(Random_Forest_model,X_test_scaled,y_test,cv<span style="color: #666666">=10</span>)[<span style="color: #BA2121">&#39;test_score&#39;</span>]
<span style="color: #008000; font-weight: bold">print</span>(accuracy)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Random Forests and scaled data: {:.2f}&quot;</span><span style="color: #666666">.</span>format(Random_Forest_model<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
y_pred <span style="color: #666666">=</span> Random_Forest_model<span style="color: #666666">.</span>predict(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000">True</span>)
plt<span style="color: #666666">.</span>show()
y_probas <span style="color: #666666">=</span> Random_Forest_model<span style="color: #666666">.</span>predict_proba(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
plt<span style="color: #666666">.</span>show()
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec3">Compare Bagging on Trees with Random Forests </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>bag_clf <span style="color: #666666">=</span> BaggingClassifier(
DecisionTreeClassifier(splitter<span style="color: #666666">=</span><span style="color: #BA2121">&quot;random&quot;</span>, max_leaf_nodes<span style="color: #666666">=16</span>, random_state<span style="color: #666666">=42</span>),
n_estimators<span style="color: #666666">=500</span>, max_samples<span style="color: #666666">=1.0</span>, bootstrap<span style="color: #666666">=</span><span style="color: #008000">True</span>, n_jobs<span style="color: #666666">=-1</span>, random_state<span style="color: #666666">=42</span>)
</pre></div>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>bag_clf<span style="color: #666666">.</span>fit(X_train, y_train)
y_pred <span style="color: #666666">=</span> bag_clf<span style="color: #666666">.</span>predict(X_test)
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> RandomForestClassifier
rnd_clf <span style="color: #666666">=</span> RandomForestClassifier(n_estimators<span style="color: #666666">=500</span>, max_leaf_nodes<span style="color: #666666">=16</span>, n_jobs<span style="color: #666666">=-1</span>, random_state<span style="color: #666666">=42</span>)
rnd_clf<span style="color: #666666">.</span>fit(X_train, y_train)
y_pred_rf <span style="color: #666666">=</span> rnd_clf<span style="color: #666666">.</span>predict(X_test)
np<span style="color: #666666">.</span>sum(y_pred <span style="color: #666666">==</span> y_pred_rf) <span style="color: #666666">/</span> <span style="color: #008000">len</span>(y_pred)
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec4">Boosting, a Bird's Eye View </h2>
<p>
The basic idea is to combine weak classifiers in order to create a good
classifier. With a weak classifier we often intend a classifier which
produces results which are only slightly better than we would get by
random guesses.
<p>
This is done by applying in an iterative way a weak (or a standard
classifier like decision trees) to modify the data. In each iteration
we emphasize those observations which are misclassified by weighting
them with a factor.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec5">What is boosting? Additive Modelling/Iterative Fitting </h2>
<p>
Boosting is a way of fitting an additive expansion in a set of
elementary basis functions like for example some simple polynomials.
Assume for example that we have a function
$$
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
$$
<p>
where \( \beta_m \) are the expansion parameters to be determined in a
minimization process and \( b(x;\gamma_m) \) are some simple functions of
the multivariable parameter \( x \) which is characterized by the
parameters \( \gamma_m \).
<p>
As an example, consider the Sigmoid function we used in logistic
regression. In that case, we can translate the function
\( b(x;\gamma_m) \) into the Sigmoid function
$$
\sigma(t) = \frac{1}{1+\exp{(-t)}},
$$
<p>
where \( t=\gamma_0+\gamma_1 x \) and the parameters \( \gamma_0 \) and
\( \gamma_1 \) were determined by the Logistic Regression fitting
algorithm.
<p>
As another example, consider the cost function we defined for linear regression
$$
C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$
<p>
In this case the function \( f(x) \) was replaced by the design matrix
\( \boldsymbol{X} \) and the unknown linear regression parameters \( \boldsymbol{\beta} \),
that is \( \boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta} \). In linear regression we can
simply invert a matrix and obtain the parameters \( \beta \) by
$$
\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
$$
<p>
In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \( \beta_m \) and \( \gamma_m \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec6">Iterative Fitting, Regression and Squared-error Cost Function </h2>
<p>
The way we proceed is as follows (here we specialize to the squared-error cost function)
<ol>
<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>
<li> Initialize with a guess \( f_0(x) \). It could be one or even zero or some random numbers.</li>
<li> For \( m=1:M \)
<ol type="a"></li>
<li> minimize \( \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2 \) wrt \( \gamma \) and \( \beta \)</li>
<li> This gives the optimal values \( \beta_m \) and \( \gamma_m \)</li>
<li> Determine then the new values \( f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m) \)</li>
</ol>
</ol>
We could use any of the algorithms we have discussed till now. If we
use trees, \( \gamma \) parameterizes the split variables and split points
at the internal nodes, and the predictions at the terminal nodes.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec7">Squared-Error Example and Iterative Fitting </h2>
<p>
To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
<p>
For simplicity we assume also that our functions \( b(x;\gamma)=1+\gamma x \).
<p>
This means that for every iteration \( m \), we need to optimize
$$
(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
$$
<p>
We start our iteration by simply setting \( f_0(x)=0 \).
Taking the derivatives with respect to \( \beta \) and \( \gamma \) we obtain
$$
\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
$$
and
$$
\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
$$
We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector)
$$
\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,
$$
which gives us \( \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}) \). Similarly we have
$$
\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
$$
<p>
which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting
for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.
<p>
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
\( f_1(x) = \beta_1(1+\gamma_1x) \). Doing this \( M \) times results in our final estimate for the function \( f \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec8">Iterative Fitting, Classification and AdaBoost </h2>
<p>
Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
observations. We define a classification function \( G(x) \) which produces a prediction taking one or the other of the two values
\( \{-1,1\} \).
<p>
The error rate of the training sample is then
$$
\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
$$
<p>
The iterative procedure starts with defining a weak classifier whose
error rate is barely better than random guessing. The iterative
procedure in boosting is to sequentially apply a weak
classification algorithm to repeatedly modified versions of the data
producing a sequence of weak classifiers \( G_m(x) \).
<p>
Here we will express our function \( f(x) \) in terms of \( G(x) \). That is
$$
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
$$
will be a function of
$$
G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec9">Adaptive Boosting, AdaBoost </h2>
<p>
In our iterative procedure we define thus
$$
f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
$$
<p>
The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
exponential cost/loss function defined as
$$
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
$$
<p>
We optimize \( \beta \) and \( G \) for each value of \( m=1:M \) as we did in the regression case.
This is normally done in two steps. Let us however first rewrite the cost function as
$$
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
$$
where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec10">Building up AdaBoost </h2>
<p>
First, for any \( \beta > 0 \), we optimize \( G \) by setting
$$
G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
$$
which is the classifier that minimizes the weighted error rate in predicting \( y \).
<p>
We can do this by rewriting
$$
\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
$$
which can be rewritten as
$$
(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
$$
which leads to
$$
\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
$$
where we have redefined the error as
$$
\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
$$
which leads to an update of
$$
f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
$$
This leads to the new weights
$$
w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec11">Adaptive boosting: AdaBoost, Basic Algorithm </h2>
<p>
The algorithm here is rather straightforward. Assume that our weak
classifier is a decision tree and we consider a binary set of outputs
with \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
observations. Our design matrix is given in terms of the
feature/predictor vectors
\( \boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}] \). Finally, we define also a
classifier determined by our data via a function \( G(x) \). This function tells us how well we are able to classify our outputs/targets \( \boldsymbol{y} \).
<p>
We have already defined the misclassification error \( \mathrm{err} \) as
$$
\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
$$
where the function \( I() \) is one if we misclassify and zero if we classify correctly.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec12">Basic Steps of AdaBoost </h2>
<p>
With the above definitions we are now ready to set up the algorithm for AdaBoost.
The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
<ol>
<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>
<li> We rewrite the misclassification error as</li>
</ol>
$$
\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
$$
<ol>
<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.
<ol type="a"></li>
<li> Fit then a given classifier to the training set using the weights \( w_i \).</li>
<li> Compute then \( \mathrm{err} \) and figure out which events are classified properly and which are classified wrongly.</li>
<li> Define a quantity \( \alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m} \)</li>
<li> Set the new weights to \( w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)} \).</li>
</ol>
<li> Compute the new classifier \( G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i) \).</li>
</ol>
For the iterations with \( m \le 2 \) the weights are modified
individually at each steps. The observations which were misclassified
at iteration \( m-1 \) have a weight which is larger than those which were
classified properly. As this proceeds, the observations which were
difficult to classifiy correctly are given a larger influence. Each
new classification step \( m \) is then forced to concentrate on those
observations that are missed in the previous iterations.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec13">AdaBoost Examples </h2>
<p>
Using <b>Scikit-Learn</b> it is easy to apply the adaptive boosting algorithm, as done here.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> AdaBoostClassifier
ada_clf <span style="color: #666666">=</span> AdaBoostClassifier(
DecisionTreeClassifier(max_depth<span style="color: #666666">=1</span>), n_estimators<span style="color: #666666">=200</span>,
algorithm<span style="color: #666666">=</span><span style="color: #BA2121">&quot;SAMME.R&quot;</span>, learning_rate<span style="color: #666666">=0.5</span>, random_state<span style="color: #666666">=42</span>)
ada_clf<span style="color: #666666">.</span>fit(X_train, y_train)
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> AdaBoostClassifier
ada_clf <span style="color: #666666">=</span> AdaBoostClassifier(
DecisionTreeClassifier(max_depth<span style="color: #666666">=1</span>), n_estimators<span style="color: #666666">=200</span>,
algorithm<span style="color: #666666">=</span><span style="color: #BA2121">&quot;SAMME.R&quot;</span>, learning_rate<span style="color: #666666">=0.5</span>, random_state<span style="color: #666666">=42</span>)
ada_clf<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
y_pred <span style="color: #666666">=</span> ada_clf<span style="color: #666666">.</span>predict(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000">True</span>)
plt<span style="color: #666666">.</span>show()
y_probas <span style="color: #666666">=</span> ada_clf<span style="color: #666666">.</span>predict_proba(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
plt<span style="color: #666666">.</span>show()
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec14">AdaBoost for Regression </h2>
<p>
Here we present <a href="https://pdfs.semanticscholar.org/8d49/e2dedb817f2c3330e74b63c5fc86d2399ce3.pdf" target="_blank">Drucker's AdaBoost</a> tailored for regression.
<p>
In bagging, each training example is equally likely to be
picked. In boosting, the probability of a particular
example being in the training set of a particular machine
depends on the performance of the prior machines on
that example. The following is a modification of
Adaboost by Drucker.
<p>
Start by selecting a set of training data \( n \) and assign to each entry a weight \( w_i=1 \) for \( i=1,2,\dots,n \). As we have done earlier, we could pick say \( 80\% \) of the data set for training. The algorithm runs as follows:
<ol>
<li> We define the probability that the training sample \( i \) is in the set by \( p_i = w_i/\sum_iw_i \). We pick \( n \) samples (with replacement) to form our training set. We pick a number uniformly in the range \( [0,\sum_iw_i] \).</li>
<li> We choose then a regression machine (for example plain linear regression or a simple decision tree). A given regression machine makes then a hypothesis.</li>
<li> Using every member of the training set with the chosen regression machine we obtain then a prediction \( \tilde{y}_i \).</li>
<li> We calculate then the loss function \( L_i \) for each training sample. We can use various types of loss function as long as we have a value</li>
</ol>
\( L_i\in [0,1] \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec15">Gradient boosting: Basics with Steepest Descent </h2>
<p>
Gradient boosting is again a similar technique to Adaptive boosting,
it combines so-called weak classifiers or regressors into a strong
method via a series of iterations.
<p>
In order to understand the method, let us illustrate its basics by
bringing back the essential steps in linear regression, where our cost
function was the least squares function.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec16">The Squared-Error again! Steepest Descent </h2>
<p>
We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize
This means that for every iteration, we need to optimize
$$
(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$
<p>
We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as
$$
f_M(x) = \sum_{m=0}^M h_m(x).
$$
<p>
In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as
$$
g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
$$
<p>
With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that
the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \).
<p>
Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have
$$
(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec17">Steepest Descent Example </h2>
<p>
Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that
$$
f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.
$$
We can then proceed and compute
$$
g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,
$$
and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \( m=M \). We can modify the steepest descent method, or steepest boosting, by introducing what is called <b>gradient boosting</b>.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec18">Gradient Boosting, algorithm </h2>
<p>
Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function
$$
C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$
<p>
The way we proceed in an iterative fashion is to
<ol>
<li> Initialize our estimate \( f_0(x) \).</li>
<li> For \( m=1:M \), we
<ol type="a"></li>
<li> compute the negative gradient vector \( \boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x) \) at \( f(x) = f_{m-1}(x) \);</li>
<li> fit the so-called base-learner to the negative gradient \( h_m(u_m,x) \);</li>
<li> update the estimate \( f_m(x) = f_{m-1}(x)+\nu h_m(u_m,x) \);</li>
</ol>
<li> The final estimate is then \( f_M(x) = \sum_{m=1}^M\nu h_m(u_m,x) \).</li>
</ol>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec19">Gradient Boosting Example, Regression </h2>
<p>
We discuss here the difference between the steepest descent approach and gradient boosting by repeating our simple regression example above.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec20">Gradient Boosting, Examples of Regression </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> GradientBoostingRegressor
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">6</span>
<span style="color: #408080; font-style: italic"># Make data set.</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
scaler <span style="color: #666666">=</span> StandardScaler()
scaler<span style="color: #666666">.</span>fit(X_train)
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,maxdegree):
model <span style="color: #666666">=</span> GradientBoostingRegressor(max_depth<span style="color: #666666">=</span>degree, n_estimators<span style="color: #666666">=100</span>, learning_rate<span style="color: #666666">=1.0</span>)
model<span style="color: #666666">.</span>fit(X_train_scaled,y_train)
y_pred <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(X_test_scaled)
polydegree[degree] <span style="color: #666666">=</span> degree
error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>) )
bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred))<span style="color: #666666">**2</span> )
variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred) )
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Max depth:&#39;</span>, degree)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error[degree])
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias[degree])
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance[degree])
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;{} &gt;= {} + {} = {}&#39;</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
plt<span style="color: #666666">.</span>xlim(<span style="color: #666666">1</span>,maxdegree<span style="color: #666666">-1</span>)
plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Error&#39;</span>)
plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;bias&#39;</span>)
plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Variance&#39;</span>)
plt<span style="color: #666666">.</span>legend()
save_fig(<span style="color: #BA2121">&quot;gdregression&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec21">Gradient Boosting, Classification Example </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> GradientBoostingClassifier
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_validate
<span style="color: #408080; font-style: italic"># Load the data</span>
cancer <span style="color: #666666">=</span> load_breast_cancer()
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
<span style="color: #008000; font-weight: bold">print</span>(X_train<span style="color: #666666">.</span>shape)
<span style="color: #008000; font-weight: bold">print</span>(X_test<span style="color: #666666">.</span>shape)
<span style="color: #408080; font-style: italic">#now scale the data</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
scaler <span style="color: #666666">=</span> StandardScaler()
scaler<span style="color: #666666">.</span>fit(X_train)
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
gd_clf <span style="color: #666666">=</span> GradientBoostingClassifier(max_depth<span style="color: #666666">=3</span>, n_estimators<span style="color: #666666">=100</span>, learning_rate<span style="color: #666666">=1.0</span>)
gd_clf<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
<span style="color: #408080; font-style: italic">#Cross validation</span>
accuracy <span style="color: #666666">=</span> cross_validate(gd_clf,X_test_scaled,y_test,cv<span style="color: #666666">=10</span>)[<span style="color: #BA2121">&#39;test_score&#39;</span>]
<span style="color: #008000; font-weight: bold">print</span>(accuracy)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Random Forests and scaled data: {:.2f}&quot;</span><span style="color: #666666">.</span>format(gd_clf<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
y_pred <span style="color: #666666">=</span> gd_clf<span style="color: #666666">.</span>predict(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000">True</span>)
save_fig(<span style="color: #BA2121">&quot;gdclassiffierconfusion&quot;</span>)
plt<span style="color: #666666">.</span>show()
y_probas <span style="color: #666666">=</span> gd_clf<span style="color: #666666">.</span>predict_proba(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
save_fig(<span style="color: #BA2121">&quot;gdclassiffierroc&quot;</span>)
plt<span style="color: #666666">.</span>show()
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
save_fig(<span style="color: #BA2121">&quot;gdclassiffiercgain&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec22">XGBoost: Extreme Gradient Boosting </h2>
<p>
<a href="https://github.com/dmlc/xgboost" target="_blank">XGBoost</a> or Extreme Gradient
Boosting, is an optimized distributed gradient boosting library
designed to be highly efficient, flexible and portable. It implements
machine learning algorithms under the Gradient Boosting
framework. XGBoost provides a parallel tree boosting that solve many
data science problems in a fast and accurate way. See the <a href="https://arxiv.org/abs/1603.02754" target="_blank">article by Chen and Guestrin</a>.
<p>
The authors design and build a highly scalable end-to-end tree
boosting system. It has a theoretically justified weighted quantile
sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.
<p>
It is now the algorithm which wins essentially all ML competitions!!!
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec23">Regression Case </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">xgboost</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">xgb</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">6</span>
<span style="color: #408080; font-style: italic"># Make data set.</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
scaler <span style="color: #666666">=</span> StandardScaler()
scaler<span style="color: #666666">.</span>fit(X_train)
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
model <span style="color: #666666">=</span> xgb<span style="color: #666666">.</span>XGBRegressor(objective <span style="color: #666666">=</span><span style="color: #BA2121">&#39;reg:squarederror&#39;</span>, colsaobjective <span style="color: #666666">=</span><span style="color: #BA2121">&#39;reg:squarederror&#39;</span>, colsample_bytree <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>, learning_rate <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>,max_depth <span style="color: #666666">=</span> degree, alpha <span style="color: #666666">=</span> <span style="color: #666666">10</span>, n_estimators <span style="color: #666666">=</span> <span style="color: #666666">200</span>)
model<span style="color: #666666">.</span>fit(X_train_scaled,y_train)
y_pred <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(X_test_scaled)
polydegree[degree] <span style="color: #666666">=</span> degree
error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>) )
bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred))<span style="color: #666666">**2</span> )
variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred) )
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Max depth:&#39;</span>, degree)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error[degree])
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias[degree])
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance[degree])
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;{} &gt;= {} + {} = {}&#39;</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
plt<span style="color: #666666">.</span>xlim(<span style="color: #666666">1</span>,maxdegree<span style="color: #666666">-1</span>)
plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Error&#39;</span>)
plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;bias&#39;</span>)
plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Variance&#39;</span>)
plt<span style="color: #666666">.</span>legend()
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec24">Xgboost on the Cancer Data </h2>
<p>
As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> LabelEncoder
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_validate
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">xgboost</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">xgb</span>
<span style="color: #408080; font-style: italic"># Load the data</span>
cancer <span style="color: #666666">=</span> load_breast_cancer()
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
<span style="color: #008000; font-weight: bold">print</span>(X_train<span style="color: #666666">.</span>shape)
<span style="color: #008000; font-weight: bold">print</span>(X_test<span style="color: #666666">.</span>shape)
<span style="color: #408080; font-style: italic">#now scale the data</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
scaler <span style="color: #666666">=</span> StandardScaler()
scaler<span style="color: #666666">.</span>fit(X_train)
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
xg_clf <span style="color: #666666">=</span> xgb<span style="color: #666666">.</span>XGBClassifier()
xg_clf<span style="color: #666666">.</span>fit(X_train_scaled,y_train)
y_test <span style="color: #666666">=</span> xg_clf<span style="color: #666666">.</span>predict(X_test_scaled)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Random Forests and scaled data: {:.2f}&quot;</span><span style="color: #666666">.</span>format(xg_clf<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
y_pred <span style="color: #666666">=</span> xg_clf<span style="color: #666666">.</span>predict(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000">True</span>)
save_fig(<span style="color: #BA2121">&quot;xdclassiffierconfusion&quot;</span>)
plt<span style="color: #666666">.</span>show()
y_probas <span style="color: #666666">=</span> xg_clf<span style="color: #666666">.</span>predict_proba(X_test_scaled)
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
save_fig(<span style="color: #BA2121">&quot;xdclassiffierroc&quot;</span>)
plt<span style="color: #666666">.</span>show()
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
save_fig(<span style="color: #BA2121">&quot;gdclassiffiercgain&quot;</span>)
plt<span style="color: #666666">.</span>show()
xgb<span style="color: #666666">.</span>plot_tree(xg_clf,num_trees<span style="color: #666666">=0</span>)
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">&#39;figure.figsize&#39;</span>] <span style="color: #666666">=</span> [<span style="color: #666666">50</span>, <span style="color: #666666">10</span>]
save_fig(<span style="color: #BA2121">&quot;xgtree&quot;</span>)
plt<span style="color: #666666">.</span>show()
xgb<span style="color: #666666">.</span>plot_importance(xg_clf)
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">&#39;figure.figsize&#39;</span>] <span style="color: #666666">=</span> [<span style="color: #666666">5</span>, <span style="color: #666666">5</span>]
save_fig(<span style="color: #BA2121">&quot;xgparams&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
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