© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Ensemble Methods l An ensemble...

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1

Ensemble Methods

An ensemble method constructs a set of base classifiers from the training data

– Ensemble or Classifier Combination

Predict class label of previously unseen records by aggregating predictions made by multiple classifiers


General Idea

OriginalTraining data

....D1D2 Dt-1 Dt

D

Step 1:Create Multiple

Data Sets

C1 C2 Ct -1 Ct

Step 2:Build Multiple

Classifiers

C*Step 3:

CombineClassifiers


Why does it work?

Suppose there are 25 base classifiers

– Each classifier has error rate, = 0.35

– Assume classifiers are independent

– Probability that the ensemble classifier makes a wrong prediction:

25

13

25 06.0)1(25

i

ii

i


Methods

By manipulating the training dataset: a classifier is built for a sampled subset of the training dataset.

– Two ensemble methods: bagging (bootstrap averaging) and boosting.


Characteristics

Ensemble methods work better with unstable classifiers.

– Base classifiers that are sensitive to minor perturbations in the training set. For example, decision trees and ANNs.

– The variability among training examples is one of the primary sources of errors in a classifier.


Bias-Variance Decomposition

Consider the trajectories of a projectile launched at a particular angle. The observed distance can be divided into 3 components.

– Force (f) and angle (θ)

– Suppose the target is t, but the projectile hits at x at distance d away from t.

tff NoiseVarianceBaistyd ),(,


Two Decision Trees (1)


Two Decision Trees (2)

Bias: The stronger the assumptions made by a classifier about the nature of its decision boundary, the larger the classifier’s bias will be. – A smaller tree has a stronger assumption.– An algorithm cannot learn the target.

Variance: Variability in the training data affects the expected error, because different compositions of the training set may lead to different decision boundaries.

Intrinsic noise in the target class.– Target class for some domain can be non-

deterministic.– Same attributes values with different class labels.


Bagging

Sampling with replacement

Build classifier on each bootstrap sample

Each sample has probability 1 - (1 – 1/n)n of being selected. When n is large, a bootstrap sample Di contains about 63.2% of the training data.

Original Data 1 2 3 4 5 6 7 8 9 10Bagging (Round 1) 7 8 10 8 2 5 10 10 5 9Bagging (Round 2) 1 4 9 1 2 3 2 7 3 2Bagging (Round 3) 1 8 5 10 5 5 9 6 3 7


Bagging Algorithm


A Bagging Example (1)

Consider a one-level binary decision tree x <= k where k is a split point to minimize the entropy.

Without bagging, the best decision stump is

– x <= 0.35 or x >= 0.75


Summary on Bagging

Bagging improves generalization error by reducing the variance of the base classifier.

Bagging depends on the stability of the base classifier.

If a base classifier is unstable, bagging helps to reduce the errors associated with random fluctuations in the training data.

If a base classifier is stable, then the error of the ensemble is primarily caused by bias in the base classifier. Bagging may make error larger, because the sample size is 37% smaller.


Boosting

An iterative procedure to adaptively change distribution of training data by focusing more on previously misclassified records

– Initially, all N records are assigned equal weights

– Unlike bagging, weights may change at the end of boosting round

– The weights can be used by a base classifier to learn a model that is biased toward higher-weight examples.


Boosting

Records that are wrongly classified will have their weights increased

Records that are classified correctly will have their weights decreased

Original Data 1 2 3 4 5 6 7 8 9 10Boosting (Round 1) 7 3 2 8 7 9 4 10 6 3Boosting (Round 2) 5 4 9 4 2 5 1 7 4 2Boosting (Round 3) 4 4 8 10 4 5 4 6 3 4

• Example 4 is hard to classify

• Its weight is increased, therefore it is more likely to be chosen again in subsequent rounds


Example: AdaBoost

Base classifiers: C1, C2, …, CT

Error rate:

Importance of a classifier:

N

jjjiji yxCw

1

)(

i

ii

1ln

2

1


Example: AdaBoost

Weight update:

If any intermediate rounds produce error rate higher than 50%, the weights are reverted back to 1/n and the resampling procedure is repeated

Classification:

( )( 1)

(j 1)i

i

exp if ( )

exp if ( )

where is the normalization factor to ensure w 1.

j

j

jj i ij i

ij j i i

j

C x yww

Z C x y

Z

T

jjj

yyxCxC

1

)(maxarg)(*


A Boosting Example (1)

Consider a one-level binary decision tree x <= k where k is a split point to minimize the entropy.

Without bagging, the best decision stump is

– x <= 0.35 or x >= 0.75

© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Ensemble Methods l An ensemble...

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