# Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods

As stated previously and seen in many of the examples discussed in the previous chapter about
a single decision tree, we often end up overfitting our training
data. This normally means that we have a high variance. Can we reduce
the variance of a statistical learning method?

This leads us to a set of different methods that can combine different
machine learning algorithms or just use one of them to construct
forests and jungles of trees, homogeneous ones or heterogenous
ones. These methods are recognized by different names which we will
try to explain here. These are

1. Voting classifiers

2. Bagging and Pasting

3. Random forests

4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)

We discuss these methods here.

### An Overview of Ensemble Methods

<!-- FIGURE: [DataFiles/ensembleoverview.png, width=600 frac=0.8] -->



## Bagging

The **plain** decision trees suffer from high
variance. This means that if we split the training data into two parts
at random, and fit a decision tree to both halves, the results that we
get could be quite different. In contrast, a procedure with low
variance will yield similar results if applied repeatedly to distinct
data sets; linear regression tends to have low variance, if the ratio
of $n$ to $p$ is moderately large. 

**Bootstrap aggregation**, or just **bagging**, is a
general-purpose procedure for reducing the variance of a statistical
learning method. 


Bagging typically results in improved accuracy
over prediction using a single tree. Unfortunately, however, it can be
difficult to interpret the resulting model. Recall that one of the
advantages of decision trees is the attractive and easily interpreted
diagram that results.

However, when we bag a large number of trees, it is no longer
possible to represent the resulting statistical learning procedure
using a single tree, and it is no longer clear which variables are
most important to the procedure. Thus, bagging improves prediction
accuracy at the expense of interpretability.  Although the collection
of bagged trees is much more difficult to interpret than a single
tree, one can obtain an overall summary of the importance of each
predictor using the MSE (for bagging regression trees) or the Gini
index (for bagging classification trees). In the case of bagging
regression trees, we can record the total amount that the MSE is
decreased due to splits over a given predictor, averaged over all $B$ possible
trees. A large value indicates an important predictor. Similarly, in
the context of bagging classification trees, we can add up the total
amount that the Gini index  is decreased by splits over a given
predictor, averaged over all $B$ trees.

heads_proba = 0.51
coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
plt.figure(figsize=(8,3.5))
plt.plot(cumulative_heads_ratio)
plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
plt.xlabel("Number of coin tosses")
plt.ylabel("Heads ratio")
plt.legend(loc="lower right")
plt.axis([0, 10000, 0.42, 0.58])
save_fig("votingsimple")
plt.show()

from sklearn.model_selection import train_test_split
from sklearn.datasets import make_moons

X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

from sklearn.ensemble import RandomForestClassifier
from sklearn.ensemble import VotingClassifier
from sklearn.linear_model import LogisticRegression
from sklearn.svm import SVC

log_clf = LogisticRegression(solver="liblinear", random_state=42)
rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
svm_clf = SVC(gamma="auto", random_state=42)

voting_clf = VotingClassifier(
    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
    voting='hard')

voting_clf.fit(X_train, y_train)

from sklearn.metrics import accuracy_score

for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
    clf.fit(X_train, y_train)
    y_pred = clf.predict(X_test)
    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))

log_clf = LogisticRegression(solver="liblinear", random_state=42)
rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
svm_clf = SVC(gamma="auto", probability=True, random_state=42)

voting_clf = VotingClassifier(
    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
    voting='soft')
voting_clf.fit(X_train, y_train)

from sklearn.metrics import accuracy_score

for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
    clf.fit(X_train, y_train)
    y_pred = clf.predict(X_test)
    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))

from sklearn.model_selection import train_test_split
from sklearn.datasets import make_moons

X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
from sklearn.ensemble import RandomForestClassifier
from sklearn.ensemble import VotingClassifier
from sklearn.linear_model import LogisticRegression
from sklearn.svm import SVC

log_clf = LogisticRegression(random_state=42)
rnd_clf = RandomForestClassifier(random_state=42)
svm_clf = SVC(random_state=42)

voting_clf = VotingClassifier(
    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
    voting='hard')
voting_clf.fit(X_train, y_train)

from sklearn.metrics import accuracy_score

for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
    clf.fit(X_train, y_train)
    y_pred = clf.predict(X_test)
    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))

log_clf = LogisticRegression(random_state=42)
rnd_clf = RandomForestClassifier(random_state=42)
svm_clf = SVC(probability=True, random_state=42)

voting_clf = VotingClassifier(
    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
    voting='soft')
voting_clf.fit(X_train, y_train)

from sklearn.metrics import accuracy_score

for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
    clf.fit(X_train, y_train)
    y_pred = clf.predict(X_test)
    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))

## Bagging Examples

from sklearn.ensemble import BaggingClassifier
from sklearn.tree import DecisionTreeClassifier

bag_clf = BaggingClassifier(
    DecisionTreeClassifier(random_state=42), n_estimators=500,
    max_samples=100, bootstrap=True, n_jobs=-1, random_state=42)
bag_clf.fit(X_train, y_train)
y_pred = bag_clf.predict(X_test)

from sklearn.metrics import accuracy_score
print(accuracy_score(y_test, y_pred))

tree_clf = DecisionTreeClassifier(random_state=42)
tree_clf.fit(X_train, y_train)
y_pred_tree = tree_clf.predict(X_test)
print(accuracy_score(y_test, y_pred_tree))

%matplotlib inline

from matplotlib.colors import ListedColormap

def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):
    x1s = np.linspace(axes[0], axes[1], 100)
    x2s = np.linspace(axes[2], axes[3], 100)
    x1, x2 = np.meshgrid(x1s, x2s)
    X_new = np.c_[x1.ravel(), x2.ravel()]
    y_pred = clf.predict(X_new).reshape(x1.shape)
    custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
    plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
    if contour:
        custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
        plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha)
    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha)
    plt.axis(axes)
    plt.xlabel(r"$x_1$", fontsize=18)
    plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
plt.figure(figsize=(11,4))
plt.subplot(121)
plot_decision_boundary(tree_clf, X, y)
plt.title("Decision Tree", fontsize=14)
plt.subplot(122)
plot_decision_boundary(bag_clf, X, y)
plt.title("Decision Trees with Bagging", fontsize=14)
save_fig("baggingtree")
plt.show()

### Making your own Bootstrap: Changing the Level of the Decision Tree

Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with
a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points $n$).


import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.pipeline import make_pipeline
from sklearn.utils import resample
from sklearn.tree import DecisionTreeRegressor

n = 100
n_boostraps = 100
maxdepth = 8

# Make data set.
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
error = np.zeros(maxdepth)
bias = np.zeros(maxdepth)
variance = np.zeros(maxdepth)
polydegree = np.zeros(maxdepth)
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)

from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

# we produce a simple tree first as benchmark
simpletree = DecisionTreeRegressor(max_depth=3) 
simpletree.fit(X_train_scaled, y_train)
simpleprediction = simpletree.predict(X_test_scaled)
for degree in range(1,maxdepth):
    model = DecisionTreeRegressor(max_depth=degree) 
    y_pred = np.empty((y_test.shape[0], n_boostraps))
    for i in range(n_boostraps):
        x_, y_ = resample(X_train_scaled, y_train)
        model.fit(x_, y_)
        y_pred[:, i] = model.predict(X_test_scaled)#.ravel()

    polydegree[degree] = degree
    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
    print('Polynomial degree:', degree)
    print('Error:', error[degree])
    print('Bias^2:', bias[degree])
    print('Var:', variance[degree])
    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
 
mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2)
print(mse_simpletree)
plt.xlim(1,maxdepth)
plt.plot(polydegree, error, label='MSE')
plt.plot(polydegree, bias, label='bias')
plt.plot(polydegree, variance, label='Variance')
plt.legend()
save_fig("baggingboot")
plt.show()

## Random forests

Random forests provide an improvement over bagged trees by way of a
small tweak that decorrelates the trees. 

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. 

A fresh sample of $m$ predictors is
taken at each split, and typically we choose

$$
m\approx \sqrt{p}.
$$

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. 

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.


The algorithm described here can be applied to both classification and regression problems.

We will grow of forest of say $B$ trees.
1. For $b=1:B$

  * Draw a bootstrap sample from the training data organized in our $\boldsymbol{X}$ matrix.

  * 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

1. we select $m \le p$ variables at random from the $p$ predictors/features

2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node

3. split the node into daughter nodes



4. 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.

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.svm import SVC
from sklearn.linear_model import LogisticRegression
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import BaggingClassifier

# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
# Support vector machine
svm = SVC(gamma='auto', C=100)
svm.fit(X_train, y_train)
print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
# Decision Trees
deep_tree_clf = DecisionTreeClassifier(max_depth=None)
deep_tree_clf.fit(X_train, y_train)
print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Logistic Regression
logreg.fit(X_train_scaled, y_train)
print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Support Vector Machine
svm.fit(X_train_scaled, y_train)
print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Decision Trees
deep_tree_clf.fit(X_train_scaled, y_train)
print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))


from sklearn.ensemble import RandomForestClassifier
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import cross_validate
# Data set not specificied
#Instantiate the model with 500 trees and entropy as splitting criteria
Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
Random_Forest_model.fit(X_train_scaled, y_train)
#Cross validation
accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
print(accuracy)
print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))


import scikitplot as skplt
y_pred = Random_Forest_model.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = Random_Forest_model.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
plt.show()

Recall that the cumulative gains curve shows the percentage of the
overall number of cases in a given category *gained* by targeting a
percentage of the total number of cases.

Similarly, the receiver operating characteristic curve, or ROC curve,
displays the diagnostic ability of a binary classifier system as its
discrimination threshold is varied. It plots the true positive rate against the false positive rate.


### Compare  Bagging on Trees with Random Forests

bag_clf = BaggingClassifier(
    DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
    n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)

bag_clf.fit(X_train, y_train)
y_pred = bag_clf.predict(X_test)
from sklearn.ensemble import RandomForestClassifier
rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
rnd_clf.fit(X_train, y_train)
y_pred_rf = rnd_clf.predict(X_test)
np.sum(y_pred == y_pred_rf) / len(y_pred)

## Boosting, a Bird's Eye View

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.

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.



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

$$
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
$$

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 $\gamma_m$.

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

$$
\sigma(t) = \frac{1}{1+\exp{(-t)}},
$$

where $t=\gamma_0+\gamma_1 x$ and the parameters $\gamma_0$ and
$\gamma_1$ were determined by the Logistic Regression fitting
algorithm.

As another example, consider the cost function we defined for linear regression

$$
C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$

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

$$
\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
$$

In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\beta_m$ and $\gamma_m$.


### Iterative Fitting, Regression and Squared-error Cost Function

The way we proceed is as follows (here we specialize to the squared-error cost function)

1. 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 $f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)$.

2. Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers.

3. For $m=1:M$

a. minimize $\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2$ wrt $\gamma$ and $\beta$

b. This gives the optimal values $\beta_m$ and $\gamma_m$

c. Determine then the new values $f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)$


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.



To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.

For simplicity we assume also that our functions $b(x;\gamma)=1+\gamma x$. 

This means that for every iteration $m$, we need to optimize

$$
(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
$$

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

$$
\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
$$

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. 

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 $f$. 



### Iterative Fitting, Classification and AdaBoost

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 
$\{-1,1\}$.

The error rate of the training sample is then

$$
\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
$$

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 $G_m(x)$.

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

$$
G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
$$

In our iterative procedure we define thus

$$
f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
$$

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

$$
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
$$

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 $w_i^m= \exp{(-y_if_{m-1}(x_i))}$.



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 $y$.

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

$$
w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
$$

### Adaptive boosting: AdaBoost, Basic Algorithm

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 $\boldsymbol{y}$. 

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. 


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.
1. 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 $\sum_{i=0}^{n-1}w_i = 1$.

2. We rewrite the misclassification error as

$$
\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
$$

1. 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.

a. Fit then a given classifier to the training set using the weights $w_i$.

b. Compute then $\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly.

c. Define a quantity $\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}$

d. Set the new weights to $w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}$.


5. Compute the new classifier $G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)$.

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.




Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here.

from sklearn.ensemble import AdaBoostClassifier

ada_clf = AdaBoostClassifier(
    DecisionTreeClassifier(max_depth=1), n_estimators=200,
    algorithm="SAMME.R", learning_rate=0.5, random_state=42)
ada_clf.fit(X_train, y_train)

from sklearn.ensemble import AdaBoostClassifier

ada_clf = AdaBoostClassifier(
    DecisionTreeClassifier(max_depth=1), n_estimators=200,
    algorithm="SAMME.R", learning_rate=0.5, random_state=42)
ada_clf.fit(X_train_scaled, y_train)
y_pred = ada_clf.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = ada_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
plt.show()

## Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent

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.

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.


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

$$
(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$

We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as

$$
f_M(x) = \sum_{m=0}^M h_m(x).
$$

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

$$
g_m(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
$$

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 $g_m(x_i) = -2(y_i-f(x_i))$.

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

$$
(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
$$

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 **gradient boosting**. 


Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points,
so we do not learn a function that can generalize. However, we can modify the algorithm by
fitting a weak learner to approximate the negative gradient signal. 

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

$$
C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
$$

The way we proceed in an iterative fashion is to
1. Initialize our estimate $f_0(x)$.

2. For $m=1:M$, we

a. compute the negative gradient vector $\boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x)$ at $f(x) = f_{m-1}(x)$;

b. fit the so-called base-learner to the negative gradient $h_m(u_m,x)$;

c. update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$;


4. The final estimate is then $f_M(x) = \sum_{m=1}^M h_m(u_m,x)$.

## Gradient Boosting, Examples of Regression

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.preprocessing import StandardScaler
import scikitplot as skplt
from sklearn.metrics import mean_squared_error

n = 100
maxdegree = 6

# Make data set.
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)

error = np.zeros(maxdegree)
bias = np.zeros(maxdegree)
variance = np.zeros(maxdegree)
polydegree = np.zeros(maxdegree)
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

for degree in range(1,maxdegree):
    model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)  
    model.fit(X_train_scaled,y_train)
    y_pred = model.predict(X_test_scaled)
    polydegree[degree] = degree
    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
    variance[degree] = np.mean( np.var(y_pred) )
    print('Max depth:', degree)
    print('Error:', error[degree])
    print('Bias^2:', bias[degree])
    print('Var:', variance[degree])
    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))

plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
plt.plot(polydegree, bias, label='bias')
plt.plot(polydegree, variance, label='Variance')
plt.legend()
save_fig("gdregression")
plt.show()

## Gradient Boosting, Classification Example

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
import scikitplot as skplt
from sklearn.ensemble import GradientBoostingClassifier
from sklearn.model_selection import cross_validate

# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0)  
gd_clf.fit(X_train_scaled, y_train)
#Cross validation
accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
print(accuracy)
print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))

import scikitplot as skplt
y_pred = gd_clf.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("gdclassiffierconfusion")
plt.show()
y_probas = gd_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
save_fig("gdclassiffierroc")
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
save_fig("gdclassiffiercgain")
plt.show()

## XGBoost: Extreme Gradient Boosting


[XGBoost](https://github.com/dmlc/xgboost) 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 [article by Chen and Guestrin](https://arxiv.org/abs/1603.02754).

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.

It is now the algorithm which wins essentially all ML competitions!!!

## Regression Case

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
import xgboost as xgb
from sklearn.preprocessing import StandardScaler
import scikitplot as skplt
from sklearn.metrics import mean_squared_error

n = 100
maxdegree = 6

# Make data set.
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)

error = np.zeros(maxdegree)
bias = np.zeros(maxdegree)
variance = np.zeros(maxdegree)
polydegree = np.zeros(maxdegree)
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

for degree in range(maxdegree):
    model =  xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)

    model.fit(X_train_scaled,y_train)
    y_pred = model.predict(X_test_scaled)
    polydegree[degree] = degree
    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
    variance[degree] = np.mean( np.var(y_pred) )
    print('Max depth:', degree)
    print('Error:', error[degree])
    print('Bias^2:', bias[degree])
    print('Var:', variance[degree])
    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))

plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
plt.plot(polydegree, bias, label='bias')
plt.plot(polydegree, variance, label='Variance')
plt.legend()
plt.show()

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.


import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import cross_validate
import scikitplot as skplt
import xgboost as xgb
# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)

xg_clf = xgb.XGBClassifier()
xg_clf.fit(X_train_scaled,y_train)

y_test = xg_clf.predict(X_test_scaled)

print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))

import scikitplot as skplt
y_pred = xg_clf.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("xdclassiffierconfusion")
plt.show()
y_probas = xg_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
save_fig("xdclassiffierroc")
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
save_fig("gdclassiffiercgain")
plt.show()


xgb.plot_tree(xg_clf,num_trees=0)
plt.rcParams['figure.figsize'] = [50, 10]
save_fig("xgtree")
plt.show()

xgb.plot_importance(xg_clf)
plt.rcParams['figure.figsize'] = [5, 5]
save_fig("xgparams")
plt.show()