version 0.2, May 2016
This notebook is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. Special thanks goes to Kevin Markham
Why are we learning about ensembling?
The most typical form of an ensemble is made by combining $T$ different base classifiers.
Each base classifier $M(\mathcal{S}_j)$ is trained by applying algorithm $M$ to a random subset
$\mathcal{S}_j$ of the training set $\mathcal{S}$.
For simplicity we define $M_j \equiv M(\mathcal{S}_j)$ for $j=1,\dots,T$, and
$\mathcal{M}=\{M_j\}_{j=1}^{T}$ a set of base classifiers.
Then, these models are combined using majority voting to create the ensemble $H$ as follows
$$
f_{mv}(\mathcal{S},\mathcal{M}) = max_{c \in \{0,1\}} \sum_{j=1}^T
\mathbf{1}_c(M_j(\mathcal{S})).
$$
# read in and prepare the chrun data
# Download the dataset
import pandas as pd
import numpy as np
data = pd.read_csv('../datasets/churn.csv')
# Create X and y
# Select only the numeric features
X = data.iloc[:, [1,2,6,7,8,9,10]].astype(np.float)
# Convert bools to floats
X = X.join((data.iloc[:, [4,5]] == 'no').astype(np.float))
y = (data.iloc[:, -1] == 'True.').astype(np.int)
X.head()
Account Length | Area Code | VMail Message | Day Mins | Day Calls | Day Charge | Eve Mins | Int'l Plan | VMail Plan | |
---|---|---|---|---|---|---|---|---|---|
0 | 128.0 | 415.0 | 25.0 | 265.1 | 110.0 | 45.07 | 197.4 | 1.0 | 0.0 |
1 | 107.0 | 415.0 | 26.0 | 161.6 | 123.0 | 27.47 | 195.5 | 1.0 | 0.0 |
2 | 137.0 | 415.0 | 0.0 | 243.4 | 114.0 | 41.38 | 121.2 | 1.0 | 1.0 |
3 | 84.0 | 408.0 | 0.0 | 299.4 | 71.0 | 50.90 | 61.9 | 0.0 | 1.0 |
4 | 75.0 | 415.0 | 0.0 | 166.7 | 113.0 | 28.34 | 148.3 | 0.0 | 1.0 |
y.value_counts().to_frame('count').assign(percentage = lambda x: x/x.sum())
count | percentage | |
---|---|---|
0 | 2850 | 0.855086 |
1 | 483 | 0.144914 |
from sklearn.cross_validation import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=42)
Create 100 decision trees
n_estimators = 100
# set a seed for reproducibility
np.random.seed(123)
n_samples = X_train.shape[0]
# create bootstrap samples (will be used to select rows from the DataFrame)
samples = [np.random.choice(a=n_samples, size=n_samples, replace=True) for _ in range(n_estimators)]
from sklearn.tree import DecisionTreeClassifier
np.random.seed(123)
seeds = np.random.randint(1, 10000, size=n_estimators)
trees = {}
for i in range(n_estimators):
trees[i] = DecisionTreeClassifier(max_features="sqrt", max_depth=None, random_state=seeds[i])
trees[i].fit(X_train.iloc[samples[i]], y_train.iloc[samples[i]])
# Predict
y_pred_df = pd.DataFrame(index=X_test.index, columns=list(range(n_estimators)))
for i in range(n_estimators):
y_pred_df.ix[:, i] = trees[i].predict(X_test)
y_pred_df.head()
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
438 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
2674 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1345 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
1957 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
2148 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
5 rows × 100 columns
Predict using majority voting
y_pred_df.sum(axis=1)[:10]
438 2 2674 5 1345 35 1957 17 2148 3 3106 4 1786 22 321 6 3082 10 2240 5 dtype: int64
y_pred = (y_pred_df.sum(axis=1) >= (n_estimators / 2)).astype(np.int)
from sklearn import metrics
metrics.f1_score(y_pred, y_test)
0.52459016393442637
metrics.accuracy_score(y_pred, y_test)
0.89454545454545453
from sklearn.ensemble import BaggingClassifier
clf = BaggingClassifier(base_estimator=DecisionTreeClassifier(), n_estimators=100, bootstrap=True,
random_state=42, n_jobs=-1, oob_score=True)
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)
metrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)
(0.53600000000000003, 0.89454545454545453)
The majority voting approach gives the same weight to each classfier regardless of the performance of each one. Why not take into account the oob performance of each classifier
First, in the traditional approach, a similar comparison of the votes of the base classifiers is made, but giving a weight $\alpha_j$ to each classifier $M_j$ during the voting phase $$ f_{wv}(\mathcal{S},\mathcal{M}, \alpha) =\max_{c \in \{0,1\}} \sum_{j=1}^T \alpha_j \mathbf{1}_c(M_j(\mathcal{S})), $$ where $\alpha=\{\alpha_j\}_{j=1}^T$. The calculation of $\alpha_j$ is related to the performance of each classifier $M_j$. It is usually defined as the normalized misclassification error $\epsilon$ of the base classifier $M_j$ in the out of bag set $\mathcal{S}_j^{oob}=\mathcal{S}-\mathcal{S}_j$ \begin{equation} \alpha_j=\frac{1-\epsilon(M_j(\mathcal{S}_j^{oob}))}{\sum_{j_1=1}^T 1-\epsilon(M_{j_1}(\mathcal{S}_{j_1}^{oob}))}. \end{equation}
Select each oob sample
samples_oob = []
# show the "out-of-bag" observations for each sample
for sample in samples:
samples_oob.append(sorted(set(range(n_samples)) - set(sample)))
Estimate the oob error of each classifier
errors = np.zeros(n_estimators)
for i in range(n_estimators):
y_pred_ = trees[i].predict(X_train.iloc[samples_oob[i]])
errors[i] = 1 - metrics.accuracy_score(y_train.iloc[samples_oob[i]], y_pred_)
%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
plt.scatter(range(n_estimators), errors)
plt.xlim([0, n_estimators])
plt.title('OOB error of each tree')
<matplotlib.text.Text at 0x7f17f02ed550>
Estimate $\alpha$
alpha = (1 - errors) / (1 - errors).sum()
weighted_sum_1 = ((y_pred_df) * alpha).sum(axis=1)
weighted_sum_1.head(20)
438 0.019993 2674 0.050009 1345 0.350236 1957 0.170230 2148 0.030047 3106 0.040100 1786 0.219819 321 0.059707 3082 0.100178 2240 0.050128 1910 0.180194 2124 0.190111 2351 0.049877 1736 0.950014 879 0.039378 785 0.219632 2684 0.010104 787 0.710568 170 0.220390 1720 0.020166 dtype: float64
y_pred = (weighted_sum_1 >= 0.5).astype(np.int)
metrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)
(0.52674897119341557, 0.8954545454545455)
clf = BaggingClassifier(base_estimator=DecisionTreeClassifier(), n_estimators=100, bootstrap=True,
random_state=42, n_jobs=-1, oob_score=True)
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)
metrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)
(0.53600000000000003, 0.89454545454545453)
errors = np.zeros(clf.n_estimators)
y_pred_all_ = np.zeros((X_test.shape[0], clf.n_estimators))
for i in range(clf.n_estimators):
oob_sample = ~clf.estimators_samples_[i]
y_pred_ = clf.estimators_[i].predict(X_train.values[oob_sample])
errors[i] = metrics.accuracy_score(y_pred_, y_train.values[oob_sample])
y_pred_all_[:, i] = clf.estimators_[i].predict(X_test)
alpha = (1 - errors) / (1 - errors).sum()
y_pred = (np.sum(y_pred_all_ * alpha, axis=1) >= 0.5).astype(np.int)
metrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)
(0.55335968379446643, 0.89727272727272722)
The staking method consists in combining the different base classifiers by learning a second level algorithm on top of them. In this framework, once the base classifiers are constructed using the training set $\mathcal{S}$, a new set is constructed where the output of the base classifiers are now considered as the features while keeping the class labels.
Even though there is no restriction on which algorithm can be used as a second level learner, it is common to use a linear model, such as $$ f_s(\mathcal{S},\mathcal{M},\beta) = g \left( \sum_{j=1}^T \beta_j M_j(\mathcal{S}) \right), $$ where $\beta=\{\beta_j\}_{j=1}^T$, and $g(\cdot)$ is the sign function $g(z)=sign(z)$ in the case of a linear regression or the sigmoid function, defined as $g(z)=1/(1+e^{-z})$, in the case of a logistic regression.
Lets first get a new training set consisting of the output of every classifier
X_train_2 = pd.DataFrame(index=X_train.index, columns=list(range(n_estimators)))
for i in range(n_estimators):
X_train_2[i] = trees[i].predict(X_train)
X_train_2.head()
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2360 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1412 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1404 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
626 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
347 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
5 rows × 100 columns
from sklearn.linear_model import LogisticRegressionCV
lr = LogisticRegressionCV()
lr.fit(X_train_2, y_train)
LogisticRegressionCV(Cs=10, class_weight=None, cv=None, dual=False, fit_intercept=True, intercept_scaling=1.0, max_iter=100, multi_class='ovr', n_jobs=1, penalty='l2', random_state=None, refit=True, scoring=None, solver='lbfgs', tol=0.0001, verbose=0)
lr.coef_
array([[ 0.10093102, 0.1042197 , 0.09431205, 0.09652843, 0.09709429, 0.09902616, 0.11100235, 0.09662288, 0.09340919, 0.09112994, 0.10012606, 0.09821902, 0.09383543, 0.09553507, 0.09147579, 0.09649564, 0.08965686, 0.09196857, 0.09684012, 0.09020758, 0.09839592, 0.09513808, 0.1044603 , 0.10028703, 0.09671603, 0.09725639, 0.10912207, 0.10590827, 0.10275491, 0.10275279, 0.10607316, 0.09803225, 0.10319411, 0.0926599 , 0.09702325, 0.09524124, 0.088848 , 0.09960894, 0.09053403, 0.09010282, 0.0990557 , 0.0987997 , 0.10538386, 0.09584352, 0.09633964, 0.09001206, 0.09181887, 0.08995095, 0.10130986, 0.10827168, 0.10064992, 0.09771002, 0.08922346, 0.10078438, 0.10173442, 0.1052274 , 0.09743252, 0.09597317, 0.08932798, 0.10033609, 0.10346122, 0.10145004, 0.09017084, 0.10348697, 0.09335995, 0.09795824, 0.10166729, 0.09306547, 0.09538575, 0.10997592, 0.09352845, 0.09860336, 0.1059772 , 0.09583408, 0.09823145, 0.09995048, 0.10224689, 0.10065135, 0.10208938, 0.11257989, 0.09956423, 0.11515946, 0.09798322, 0.10092449, 0.10150098, 0.10275192, 0.09180693, 0.0990442 , 0.10016612, 0.10145948, 0.09848122, 0.10322931, 0.09913907, 0.08925477, 0.09950337, 0.10277594, 0.09249331, 0.0954106 , 0.1053263 , 0.09849884]])
y_pred = lr.predict(y_pred_df)
metrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)
(0.53658536585365846, 0.89636363636363636)
y_pred_all_ = np.zeros((X_test.shape[0], clf.n_estimators))
X_train_3 = np.zeros((X_train.shape[0], clf.n_estimators))
for i in range(clf.n_estimators):
X_train_3[:, i] = clf.estimators_[i].predict(X_train)
y_pred_all_[:, i] = clf.estimators_[i].predict(X_test)
lr = LogisticRegressionCV()
lr.fit(X_train_3, y_train)
y_pred = lr.predict(y_pred_all_)
metrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)
(0.56250000000000011, 0.89818181818181819)
vs using only one dt
dt = DecisionTreeClassifier()
dt.fit(X_train, y_train)
y_pred = dt.predict(X_test)
metrics.f1_score(y_pred, y_test), metrics.accuracy_score(y_pred, y_test)
(0.44510385756676557, 0.82999999999999996)
While boosting is not algorithmically constrained, most boosting algorithms consist of iteratively learning weak classifiers with respect to a distribution and adding them to a final strong classifier. When they are added, they are typically weighted in some way that is usually related to the weak learners' accuracy. After a weak learner is added, the data is reweighted: examples that are misclassified gain weight and examples that are classified correctly lose weight (some boosting algorithms actually decrease the weight of repeatedly misclassified examples, e.g., boost by majority and BrownBoost). Thus, future weak learners focus more on the examples that previous weak learners misclassified. (Wikipedia)
from IPython.display import Image
Image(url= "http://vision.cs.chubu.ac.jp/wp/wp-content/uploads/2013/07/OurMethodv81.png", width=900)
AdaBoost (adaptive boosting) is an ensemble learning algorithm that can be used for classification or regression. Although AdaBoost is more resistant to overfitting than many machine learning algorithms, it is often sensitive to noisy data and outliers.
AdaBoost is called adaptive because it uses multiple iterations to generate a single composite strong learner. AdaBoost creates the strong learner (a classifier that is well-correlated to the true classifier) by iteratively adding weak learners (a classifier that is only slightly correlated to the true classifier). During each round of training, a new weak learner is added to the ensemble and a weighting vector is adjusted to focus on examples that were misclassified in previous rounds. The result is a classifier that has higher accuracy than the weak learners’ classifiers.
Algorithm:
# read in and prepare the chrun data
# Download the dataset
import pandas as pd
import numpy as np
data = pd.read_csv('../datasets/churn.csv')
# Create X and y
# Select only the numeric features
X = data.iloc[:, [1,2,6,7,8,9,10]].astype(np.float)
# Convert bools to floats
X = X.join((data.iloc[:, [4,5]] == 'no').astype(np.float))
y = (data.iloc[:, -1] == 'True.').astype(np.int)
from sklearn.cross_validation import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=42)
n_samples = X_train.shape[0]
n_estimators = 10
weights = pd.DataFrame(index=X_train.index, columns=list(range(n_estimators)))
t = 0
weights[t] = 1 / n_samples
Train the classifier
from sklearn.tree import DecisionTreeClassifier
trees = []
trees.append(DecisionTreeClassifier(max_depth=1))
trees[t].fit(X_train, y_train, sample_weight=weights[t].values)
DecisionTreeClassifier(class_weight=None, criterion='gini', max_depth=1, max_features=None, max_leaf_nodes=None, min_samples_leaf=1, min_samples_split=2, min_weight_fraction_leaf=0.0, presort=False, random_state=None, splitter='best')
Estimate error
y_pred_ = trees[t].predict(X_train)
error = []
error.append(1 - metrics.accuracy_score(y_pred_, y_train))
error[t]
0.13613972234661886
alpha = []
alpha.append(np.log((1 - error[t]) / error[t]))
alpha[t]
1.8477293114995077
Update weights
weights[t + 1] = weights[t]
filter_ = y_pred_ != y_train
weights.loc[filter_, t + 1] = weights.loc[filter_, t] * np.exp(alpha[t])
Normalize weights
weights[t + 1] = weights[t + 1] / weights[t + 1].sum()
Iteration 2 - n_estimators
for t in range(1, n_estimators):
trees.append(DecisionTreeClassifier(max_depth=1))
trees[t].fit(X_train, y_train, sample_weight=weights[t].values)
y_pred_ = trees[t].predict(X_train)
error.append(1 - metrics.accuracy_score(y_pred_, y_train))
alpha.append(np.log((1 - error[t]) / error[t]))
weights[t + 1] = weights[t]
filter_ = y_pred_ != y_train
weights.loc[filter_, t + 1] = weights.loc[filter_, t] * np.exp(alpha[t])
weights[t + 1] = weights[t + 1] / weights[t + 1].sum()
error
[0.13613972234661886, 0.15629198387819077, 0.84370801612180923, 0.84370801612180923, 0.84370801612180923, 0.84370801612180923, 0.84370801612180923, 0.84370801612180923, 0.84370801612180923, 0.84370801612180923]
Only classifiers when error < 0.5
new_n_estimators = np.sum([x<0.5 for x in error])
y_pred_all = np.zeros((X_test.shape[0], new_n_estimators))
for t in range(new_n_estimators):
y_pred_all[:, t] = trees[t].predict(X_test)
y_pred = (np.sum(y_pred_all * alpha[:new_n_estimators], axis=1) >= 1).astype(np.int)
metrics.f1_score(y_pred, y_test.values), metrics.accuracy_score(y_pred, y_test.values)
(0.51051051051051044, 0.85181818181818181)
from sklearn.ensemble import AdaBoostClassifier
clf = AdaBoostClassifier()
clf
AdaBoostClassifier(algorithm='SAMME.R', base_estimator=None, learning_rate=1.0, n_estimators=50, random_state=None)
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)
metrics.f1_score(y_pred, y_test.values), metrics.accuracy_score(y_pred, y_test.values)
(0.29107981220657275, 0.86272727272727268)
from sklearn.ensemble import GradientBoostingClassifier
clf = GradientBoostingClassifier()
clf
GradientBoostingClassifier(init=None, learning_rate=0.1, loss='deviance', max_depth=3, max_features=None, max_leaf_nodes=None, min_samples_leaf=1, min_samples_split=2, min_weight_fraction_leaf=0.0, n_estimators=100, presort='auto', random_state=None, subsample=1.0, verbose=0, warm_start=False)
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)
metrics.f1_score(y_pred, y_test.values), metrics.accuracy_score(y_pred, y_test.values)
(0.52892561983471076, 0.89636363636363636)