FYS-STK4155/doc/pub/week45/ipynb/week45.ipynb

Week 45: Random Forests and Boosting

Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

Date: Nov 6, 2020

Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license

Overview of week 45

• Thursday: Wrapping up from last week. Bagging and Random forests. Boosting methods. Link to video of lecture

• Friday: Boosting and gradient boosting. Link to video of lecture

Geron's chapter 7. See also lecture from STK-IN4300, lecture 9. Chapter 10 (sections 10.1-10.10 are the most relevant ones) of Hastie et al contains also a good discussion.

Video on boosting methods by Hastie (the textbook author).

Thursday

Bagging, voting and random forests. The material on bagging and voting is a repeat from last week and can be found in the slides from week 44. We repeat here the voting approach since this will serve as a motivation for boosting methods later.

Why Voting?

The idea behind boosting, and voting as well can be phrased as follows: Can a group of people somehow arrive at highly reasoned decisions, despite the weak judgement of the individual members?

The aim is to create a good classifier by combining several weak classifiers. A weak classifier is a classifier which is able to produce results that are only slightly better than guessing at random.

The basic approach is to apply repeatedly (in boosting this is done in an iterative way) a weak classifier to modifications of the data. In voting we simply apply the law of large numbers while in boosting we give more weight to misclassified data in each iteration.

Decision trees play an important role as our weak classifier. They serve as the basic method.

Tossing coins

The simplest case is a so-called voting ensemble. To illustrate this, think of yourself tossing coins with a biased outcome of 51 per cent for heads and 49% for tails. With only few tosses, you may not clearly see this distribution for heads and tails. However, after some thousands of tosses, there will be a clear majority of heads. With 2000 tosses you should see approximately 1020 heads and 980 tails.

We can then state that the outcome is a clear majority of heads. If you do this ten thousand times, it is easy to see that there is a 97% likelihood of a majority of heads.

Another example would be to collect all polls before an election. Different polls may show different likelihoods for a candidate winning with say a majority of the popular vote. The majority vote would then consist in many polls indicating that this candidate will actually win.

The example here shows how we can implement the coin tossing case, clealry demostrating that after some tosses we see the law of large numbers kicking in.

Standard imports first

In [1]:

%matplotlib inline

# Common imports
from IPython.display import Image 
from pydot import graph_from_dot_data
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.tree import DecisionTreeClassifier
from sklearn.model_selection import train_test_split
from sklearn.tree import export_graphviz
from sklearn.preprocessing import StandardScaler, OneHotEncoder
from sklearn.compose import ColumnTransformer
from IPython.display import Image 
from pydot import graph_from_dot_data
import os

# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"

if not os.path.exists(PROJECT_ROOT_DIR):
    os.mkdir(PROJECT_ROOT_DIR)

if not os.path.exists(FIGURE_ID):
    os.makedirs(FIGURE_ID)

if not os.path.exists(DATA_ID):
    os.makedirs(DATA_ID)

def image_path(fig_id):
    return os.path.join(FIGURE_ID, fig_id)

def data_path(dat_id):
    return os.path.join(DATA_ID, dat_id)

def save_fig(fig_id):
    plt.savefig(image_path(fig_id) + ".png", format='png')

Simple Voting Example, head or tail

In [2]:

# Common imports
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
plt.rcParams['axes.labelsize'] = 14
plt.rcParams['xtick.labelsize'] = 12
plt.rcParams['ytick.labelsize'] = 12

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()

Using the Voting Classifier

We can use the voting classifier on other data sets, here the exciting binary case of two distinct objects using the make moons functionality of Scikit-Learn.

In [3]:

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))

LogisticRegression 0.864
RandomForestClassifier 0.872
SVC 0.888
VotingClassifier 0.896
LogisticRegression 0.864
RandomForestClassifier 0.872
SVC 0.888
VotingClassifier 0.912

Voting and Bagging

In [4]:

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)

Out [4]:

VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
                             ('rf', RandomForestClassifier(random_state=42)),
                             ('svc', SVC(random_state=42))])

In [5]:

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))

LogisticRegression 0.864
RandomForestClassifier 0.896
SVC 0.896
VotingClassifier 0.912

In [6]:

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)

Out [6]:

VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
                             ('rf', RandomForestClassifier(random_state=42)),
                             ('svc', SVC(probability=True, random_state=42))],
                 voting='soft')

In [7]:

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))

LogisticRegression 0.864
RandomForestClassifier 0.896
SVC 0.896
VotingClassifier 0.92

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.

Random Forest Algorithm

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.

Random Forests Compared with other Methods on the Cancer Data

In [8]:

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()

(426, 30)
(143, 30)
Test set accuracy with Logistic Regression: 0.95
Test set accuracy with SVM: 0.63
Test set accuracy with Decision Trees: 0.89
Test set accuracy Logistic Regression with scaled data: 0.96
Test set accuracy SVM with scaled data: 0.96
Test set accuracy with Decision Trees and scaled data: 0.89

/Users/MortenImac/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/_logistic.py:762: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  extra_warning_msg=_LOGISTIC_SOLVER_CONVERGENCE_MSG)

[1.         0.73333333 0.93333333 1.         1.         0.92857143
 1.         0.92857143 0.92857143 1.        ]
Test set accuracy with Random Forests and scaled data: 0.98

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

In [9]:

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)

In [10]:

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)

Out [10]:

0.9790209790209791

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.

What is boosting? Additive Modelling/Iterative Fitting

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.

Squared-Error Example and Iterative Fitting

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).

Adaptive Boosting, AdaBoost

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))}.

Building up AdaBoost

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.

Basic Steps of AdaBoost

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.

AdaBoost Examples

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

In [11]:

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()

Additive boosting for Regression

Here we present Drucker's AdaBoost tailored for regression.

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.

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:

1. 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].

2. We choose then a regression machine (for example plain linear regression or a simple decision tree). A given regression machine makes then a hypothesis.

3. Using every member of the training set with the chosen regression machine we obtain then a prediction \tilde{y}_i.

4. 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

L_i\in [0,1].

Gradient boosting: Basics with Steepest 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.

The Squared-Error again! Steepest Descent

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.

Steepest Descent Example

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.

Gradient Boosting, algorithm

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)+\nu h_m(u_m,x);

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

Gradient Boosting, Examples of Regression

In [12]:

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()

/Users/MortenImac/anaconda3/lib/python3.6/site-packages/sklearn/utils/validation.py:73: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  return f(**kwargs)
/Users/MortenImac/anaconda3/lib/python3.6/site-packages/sklearn/utils/validation.py:73: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  return f(**kwargs)
/Users/MortenImac/anaconda3/lib/python3.6/site-packages/sklearn/utils/validation.py:73: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  return f(**kwargs)
/Users/MortenImac/anaconda3/lib/python3.6/site-packages/sklearn/utils/validation.py:73: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  return f(**kwargs)
/Users/MortenImac/anaconda3/lib/python3.6/site-packages/sklearn/utils/validation.py:73: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
  return f(**kwargs)

Max depth: 1
Error: 0.5692982587408999
Bias^2: 0.2744795520281314
Var: 0.2948187067127685
0.5692982587408999 >= 0.2744795520281314 + 0.2948187067127685 = 0.5692982587408999
Max depth: 2
Error: 0.5623134325774894
Bias^2: 0.2741269616463988
Var: 0.2881864709310905
0.5623134325774894 >= 0.2741269616463988 + 0.2881864709310905 = 0.5623134325774893
Max depth: 3
Error: 0.5623132958108783
Bias^2: 0.27412708964346316
Var: 0.2881862061674152
0.5623132958108783 >= 0.27412708964346316 + 0.2881862061674152 = 0.5623132958108783
Max depth: 4
Error: 0.5623132958108783
Bias^2: 0.27412708964346316
Var: 0.2881862061674152
0.5623132958108783 >= 0.27412708964346316 + 0.2881862061674152 = 0.5623132958108783
Max depth: 5
Error: 0.5623132958108783
Bias^2: 0.27412708964346316
Var: 0.2881862061674152
0.5623132958108783 >= 0.27412708964346316 + 0.2881862061674152 = 0.5623132958108783

Gradient Boosting, Classification Example

In [13]:

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()

(426, 30)
(143, 30)
[1.         0.86666667 0.93333333 1.         1.         0.92857143
 1.         0.92857143 0.92857143 0.92857143]
Test set accuracy with Random Forests and scaled data: 0.97

XGBoost: Extreme Gradient Boosting

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.

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

In [14]:

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()

[11:32:43] WARNING: /Users/travis/build/dmlc/xgboost/src/learner.cc:480: 
Parameters: { colsaobjective } might not be used.

  This may not be accurate due to some parameters are only used in language bindings but
  passed down to XGBoost core.  Or some parameters are not used but slip through this
  verification. Please open an issue if you find above cases.


Max depth: 0
Error: 0.3122934461614368
Bias^2: 0.31229345725057706
Var: 3.552713678800501e-15
0.3122934461614368 >= 0.31229345725057706 + 3.552713678800501e-15 = 0.3122934572505806
[11:32:43] WARNING: /Users/travis/build/dmlc/xgboost/src/learner.cc:480: 
Parameters: { colsaobjective } might not be used.

  This may not be accurate due to some parameters are only used in language bindings but
  passed down to XGBoost core.  Or some parameters are not used but slip through this
  verification. Please open an issue if you find above cases.


Max depth: 1
Error: 0.3630793080432832
Bias^2: 0.3249573668730529
Var: 0.038121938705444336
0.3630793080432832 >= 0.3249573668730529 + 0.038121938705444336 = 0.3630793055784972
[11:32:43] WARNING: /Users/travis/build/dmlc/xgboost/src/learner.cc:480: 
Parameters: { colsaobjective } might not be used.

  This may not be accurate due to some parameters are only used in language bindings but
  passed down to XGBoost core.  Or some parameters are not used but slip through this
  verification. Please open an issue if you find above cases.


Max depth: 2
Error: 0.3604382790875038
Bias^2: 0.32184437765562457
Var: 0.03859391063451767
0.3604382790875038 >= 0.32184437765562457 + 0.03859391063451767 = 0.36043828829014224
[11:32:43] WARNING: /Users/travis/build/dmlc/xgboost/src/learner.cc:480: 
Parameters: { colsaobjective } might not be used.

  This may not be accurate due to some parameters are only used in language bindings but
  passed down to XGBoost core.  Or some parameters are not used but slip through this
  verification. Please open an issue if you find above cases.


Max depth: 3
Error: 0.360435136192909
Bias^2: 0.32182779880247675
Var: 0.03860734403133392
0.360435136192909 >= 0.32182779880247675 + 0.03860734403133392 = 0.3604351428338107
[11:32:43] WARNING: /Users/travis/build/dmlc/xgboost/src/learner.cc:480: 
Parameters: { colsaobjective } might not be used.

  This may not be accurate due to some parameters are only used in language bindings but
  passed down to XGBoost core.  Or some parameters are not used but slip through this
  verification. Please open an issue if you find above cases.


Max depth: 4
Error: 0.360435136192909
Bias^2: 0.32182779880247675
Var: 0.03860734403133392
0.360435136192909 >= 0.32182779880247675 + 0.03860734403133392 = 0.3604351428338107
[11:32:43] WARNING: /Users/travis/build/dmlc/xgboost/src/learner.cc:480: 
Parameters: { colsaobjective } might not be used.

  This may not be accurate due to some parameters are only used in language bindings but
  passed down to XGBoost core.  Or some parameters are not used but slip through this
  verification. Please open an issue if you find above cases.


Max depth: 5
Error: 0.360435136192909
Bias^2: 0.32182779880247675
Var: 0.03860734403133392
0.360435136192909 >= 0.32182779880247675 + 0.03860734403133392 = 0.3604351428338107

Xgboost on the Cancer Data

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.

In [16]:

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 xgboost 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()

(426, 30)
(143, 30)
Test set accuracy with xgboost and scaled data: 1.00

---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
~/anaconda3/lib/python3.6/site-packages/graphviz/backend.py in run(cmd, input, capture_output, check, encoding, quiet, **kwargs)
    163     try:
--> 164         proc = subprocess.Popen(cmd, startupinfo=get_startupinfo(), **kwargs)
    165     except OSError as e:

~/anaconda3/lib/python3.6/subprocess.py in __init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, encoding, errors)
    728                                 errread, errwrite,
--> 729                                 restore_signals, start_new_session)
    730         except:

~/anaconda3/lib/python3.6/subprocess.py in _execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, start_new_session)
   1363                             err_msg += ': ' + repr(err_filename)
-> 1364                     raise child_exception_type(errno_num, err_msg, err_filename)
   1365                 raise child_exception_type(err_msg)

FileNotFoundError: [Errno 2] No such file or directory: 'dot': 'dot'

During handling of the above exception, another exception occurred:

ExecutableNotFound                        Traceback (most recent call last)
<ipython-input-16-ca36fc2e2adf> in <module>
     41 
     42 
---> 43 xgb.plot_tree(xg_clf,num_trees=0)
     44 plt.rcParams['figure.figsize'] = [50, 10]
     45 save_fig("xgtree")

~/anaconda3/lib/python3.6/site-packages/xgboost/plotting.py in plot_tree(booster, fmap, num_trees, rankdir, ax, **kwargs)
    246 
    247     s = BytesIO()
--> 248     s.write(g.pipe(format='png'))
    249     s.seek(0)
    250     img = image.imread(s)

~/anaconda3/lib/python3.6/site-packages/graphviz/files.py in pipe(self, format, renderer, formatter, quiet)
    136         out = backend.pipe(self._engine, format, data,
    137                            renderer=renderer, formatter=formatter,
--> 138                            quiet=quiet)
    139 
    140         return out

~/anaconda3/lib/python3.6/site-packages/graphviz/backend.py in pipe(engine, format, data, renderer, formatter, quiet)
    242     """
    243     cmd, _ = command(engine, format, None, renderer, formatter)
--> 244     out, _ = run(cmd, input=data, capture_output=True, check=True, quiet=quiet)
    245     return out
    246 

~/anaconda3/lib/python3.6/site-packages/graphviz/backend.py in run(cmd, input, capture_output, check, encoding, quiet, **kwargs)
    165     except OSError as e:
    166         if e.errno == errno.ENOENT:
--> 167             raise ExecutableNotFound(cmd)
    168         else:
    169             raise

ExecutableNotFound: failed to execute ['dot', '-Tpng'], make sure the Graphviz executables are on your systems' PATH

In [ ]: