Dimensionality Reduction by Feature Selection in Machine Learning

Dunja MladeničJ.Stefan Institute, Slovenia

Reasons for dimensionality reduction

Dimensionality reduction in machine learning is usually performed to:

Improve the prediction performance Improve learning efficiency Provide faster predictors possibly requesting

less information on the original data Reduce complexity of the learned results,

enable better understanding of the underlying process

Approaches to dimensionality reduction

Map the original features onto the reduced dimensionality space by:

selecting a subset of the original features no feature transformation, just select a feature subset

constructing features to replace the original features

using methods from statistics, such as, PCA using background knowledge for constructing

new features to be used in addition/instead of the original features (can be followed by feature subset selection)

general background knowledge (sum or product of features,...) domain specific background knowledge (parser for text data to

get noun phrases, clustering of words, user-specified function,…)

Addressed here

Example for the problem Data set

Five Boolean features C = F1 V F2

F3 = ┐F2 , F5 = ┐F4

Optimal subset: {F1, F2} or {F1, F3}

optimization in space of all feature subsets ( possibilities)

(tutorial on genomics [Yu 2004])

F1 F2 F3 F4 F5 C

0 0 1 0 1 0

0 1 0 0 1 1

1 0 1 0 1 1

1 1 0 0 1 1

0 0 1 1 0 0

0 1 0 1 0 1

1 0 1 1 0 1

1 1 0 1 0 1

F2

Search for feature subset An example of search space (John & Kohavi 1997)

Forward selection

Backward elimination

Feature subset selection commonly used search strategies:

forward selection FSubset={}; greedily add features one at a time

forward stepwise selection FSubset={}; greedily add or remove features one at a time

backward elimination FSubset=AllFeatures; greedily remove features one at a time

backward stepwise elimination FSubset=AllFeatures; greedily add or remove features one at a

time random mutation

FSubset=RandomFeatures; greedily add or remove randomly selected feature one at a time stop after a given number of iterations

Approaches to feature subset selection

Filters - evaluation function independent of the learning algorithm

Wrappers - evaluation using model selection based on the machine learning algorithm

Embedded approaches - feature selection during learning

Simple Filters - assume feature independence (used for problems with large number of features, eg. text classification)

Filtering

Evaluation independent of ML algorithm

Filters: Distribution-based [Koller & Sahami 1996]

Idea: select a minimal subset of features that keeps class probability distribution close to the original distribution P(C|FeatureSet) is close to P(C|AllFeatures)

1. start with all the features2. use backward elimination to eliminate a

predefined number of features

evaluation: the next feature to be deleted is obtained using Cross-entropy measure

Evaluation of a feature subset1. represent examples using the feature subset2. on a random subset of examples calculate

average difference in distance from the nearest example of the same class and the nearest

example of the different class

F discrete F cont.

some extensions, empirical and theoretical analysis in [Robnik-Sikonja & Kononenko 2003]

Filters: Relief [Kira & Rendell 1992]

otherwise

FExFEx

;1

)()(;0 21

)min()max(

)()( 21

FF

FExFEx

FSubset

2121 ))(),(diff()Ex,(Ex1i

ii FExFEx

Filters: FOCUS [Almallim & Dietterich 1991]

Evaluation of a feature subset1. represent examples using the feature subset2. count conflicts in class value (two examples with the

same feature values and different class value)

Search: all the (promising) subsets of the same (increasing) size are evaluated until a sufficient (no conflicts) subset is found

assumes existence of a small sufficient subset --> not appropriate for tasks with many features

some extensions of the algorithm use heuristic search to avoid evaluating all the subsets of the same size

Illustration of FOCUS

F1 F2 F3 F4 F5 C

0 0 1 0 1 0

0 1 0 0 1 1

1 0 1 0 1 1

1 1 0 0 1 1

0 0 1 1 0 0

0 1 0 1 0 1

1 0 1 1 0 1

1 1 0 1 0 1

F4 F5 C

0 1 0

0 1 1

0 1 1

0 1 1

1 0 0

1 0 1

1 0 1

1 0 1

Conflict!

Conflict!

Filters: Random [Liu & Setiono

1996]

Evaluation of a feature subset1. represent examples using the feature subset2. calculate the inconsistency rate

(the average difference between the number of examples with equal feature values and the number of examples among them with the locally, most frequent class value)

3. select the smallest subset with inconsistency rate below the given threshold

Search: random sampling to search the space of feature subsets

evaluate the predetermined number of subsets noise handling by setting the threshold > 0 if threshold = 0, then the same evaluation as in

FOCUS

Filters: MDL-based [Pfahringer

1995]

Evaluation using Minimum Description Length

1. represent examples using the feature subset2. calculate MDL of a simple decision table

representing examples

Search: start with random feature subset and add or delete a feature, one at a time

performs at least as well as the wrapper approach applied on the simple decision tables and scales up better to large number of training examples

Wrapper

Evaluation uses the same ML algorithm that is used after the feature selection

Wrappers: Instance-based learning

Evaluation using instance-based learning represent examples using the feature subset estimate model quality using cross-validation

Search [Aha & Bankert 1994] start with random feature subset use beam search with backward elimination

Search [Skalak 1994] start with random feature subset use random mutation

Wrappers: Decision tree induction

Evaluation using decision tree induction represent examples using the feature subset estimate model quality using cross-validation

Search [Bala et al 1995], [Cherkauer & Shavlik 1996] using genetic algorithm

Search [Caruana & Freitag 1994] adding and removing features (backward stepwise

elimination) additionally, at each step removes all the features that were

not used in the decision tree induced for the evaluation of the current feature subset

Metric-based model selection

Idea:poor models behave differently on training and other dataEvaluation using machine learning algorithm represent examples using the feature subset generate model using some ML algorithm estimate model quality comparing the performance of two models

on training and on unlabeled data, chose the largest subset that satisfies triangular inequality with all the smaller subsets

Combine metric and cross-validation [Bengio & Chapados 2003] based on their disagreement on testing examples (higher

disagreement means lower trust to cross-validation) Intuition: cross-validation provides good results but has high

variance and should benefit from a combination with model selection having with lower variance

ijPfdPfdffd xYjTxYiTjiU 0),...,(),(),( ||

Embedded

Feature selection as integral part of model generation

Embedded

at each iteration of the incremental optimization of the model

use a fast gradient-based heuristic to find the most promising feature [Perkins et al 2003]

Idea: features that are relevant to the concept should affect the generalization error bound of non-linear SVM more than irrelevant features

use backward elimination based on the criteria derived from generalization error bounds of the SVM theory (the weight vector norm or, using upper bounds of the leave-one-out error) [Rakotomamonjy 2003]

Embedded: in filters [Cardie 1993]

Use embedded feature selection as pre-processing

evaluation and search using the process embedded in decision tree induction

the final feature subset contains only the features that appear in the induced decision tree

used for learning using Nearest Neighbor algorithm

Simple Filtering

Evaluation independent of ML algorithm

Feature subset selection on text data – commonly used methods

Simple filtering using some scoring measure to evaluate individual feature

supervised measures: information gain, cross entropy for text (information gain on

only one feature value), mutual information for text supervised measures for binary class

odds ratio (target class vs. the rest), bi-normal separation unsupervised measures:

term frequency, document frequency Simple filtering using embedded approach to score

the features scoring measure equal to weights in the normal to the

hyperplane of linear SVM trained on all the features [Brank et al 2002]

learning using linear SVM, Perceptron, Naïve Bayes

Scoring individual feature

InformationGain: CrossEntropyTxt: MutualInfoTxt: OddsRatio: Frequency: Bi-NormalSeparation:F - Normal distribution cumulative probability function

)|())|(1(

))|(1()|(log

negWPposWP

negWPposWP

)(WFreq

WWF negposC CP

FCPFCPFP

, , )(

)|(log)|()(

negposC CP

WCPWCPWP

, )(

)|(log)|()(

negposC WP

CWPCP

, )(

)|(log)(

))|(())|(( 11 negWPFposWPF

Influence of feature selection on the classification performance

Some ML algorithms are more sensitive to the feature subset than other Naïve Bayes on document categorization

sensitive to the feature subset Linear SVM has embedded weighting of

features that partially compensates for feature selection

Illustration of feature selection

Naïve Bayes on Yahoo! hierarchy data Comparison of different feature scoring

measures in simple filtering Linear SVM on standard Reuters-

2000 news data Comparison of scoring measures

including embedded SVM-normal and perceptron used as pre-processing

Illustration on 5 datasets from Yahoo! hierarchy using Naïve Bayes [Mladenic & Grobelnik 2003]

Yahoocategory

# of sub-ctgs.(nodes)

# of words/phrases(1-grams+…5grams)

# ofdocuments

Avg. nodedistance

Avg. # ofwords perdocument

Entertain-ment

8,081 30,998(15144+11211+2970+1059+505)

79,011 7.56 60

Arts andHumanities

3,085 11,473(7380+3538+463+75+17)

27,765 6.59 65

Computersand Internet

2,652 7,631(5049+2276+261+38+7)

23,105 6.77 55

Education 349 3,198(1919+1061+184+28+6)

5,406 4.54 100

Reference 129 928(701+196+28+3+0)

1,995 4.29 45

OddsRatio

CrossEntropy

MutualInf

Random

InfGain

• Feature subset size importantly influences the performance

• Some measures more sensitive than other

Scoring Rank F2 Precision Recall Ctgs

Ent. OddsRatio TermFq CrossEntTxt MutInfTxt InfGain Rnd

16 49 4103 4197 4054 4050

0.59 0.49 0.39 0.27 0.22 0.002

0.44 0.41 0.35 0.57 0.86 0.99

0.80 0.71 0.69 0.38 0.21 0.001

115 756 1612 1461 1423 30

Arts. OddsRatio TermFq CrossEntTxt MutInfTxt InfGain Rnd

9.9 15.2 1560 1620 1592 1549

0.59 0.58 0.44 0.35 0.21 0.001

0.40 0.48 0.33 0.56 0.94 0.99

0.83 0.77 0.75 0.46 0.20 0.001

120 299 626 558 343 24

Comp OddsRatio TermFq CrossEntT MutInfT InfGain Rnd

11.9 89.2 1355 1448 1393 1333

0.60 0.58 0.49 0.38 0.17 0.01

0.40 0.50 0.35 0.62 0.94 0.99

0.84 0.74 0.75 0.45 0.15 0.004

138 511 757 739 450 27

Edu. TermFq OddsRatio MutInfTxt CrossEntTxt InfGain Rnd

29.5 9.4 43 10.8 113 178

0.55 0.48 0.46 0.46 0.11 0.01

0.57 0.36 0.48 0.28 0.98 0.99

0.65 0.81 0.59 0.82 0.11 0.003

113 35 158 162 144 18

Ref. OddsRatio CrossEntTxt MutInfTxt TermFq InfGain Rnd

3.8 3.3 6.9 3.8 27 65.8

0.64 0.60 0.55 0.53 0.22 0.06

0.51 0.62 0.73 0.78 0.99 0.99

0.81 0.71 0.60 0.57 0.21 0.05

15 46 51 38 38 8

Rank of the correct category in the list of all categories

F2-measure combining precision and recall emphases on recall

Ctgs – number of categories looking promising (testing example needs to be classified by their models)

best results: Odds ratio using only a small number

of features (50-100, 0.2%-5%) improves performance of

Naïve Bayes surprisingly good results:

unsupervised Term frequency

poor results: Information gain

probably because it is not compatible with Naïve Bayes (selects mostly features representative for neg. class and features informative when not occurring in the document)

Illustration on Reuters-2000 Data [Brank et al 2002]

Reuters-2000 Data used in the experiments 16 categories covering the range of break-event point

(estimated on a sample) and class distribution Training: sample of 118,294 articles from the training

period Testing: 302,323 articles from the test period

~810,000 News articles 103 Categories~810,000 News articles 103 Categories

20. Aug,199620. Aug,1996 19. Aug,199714. April,199714. April,1997

Training PeriodTraining Period Test PeriodTest Period

504,468 articles504,468 articles 302,323 articles302,323 articles

Naïve Bayes, Macroaverages

0.34

0.39

0.44

0.49

0.54

10 100 1000 10000 100000

Number of features kept

Av

era

ge

F1

ov

er

the

te

st

se

t

odds ratio information gain normal-1 perceptron-1

Experiments with Naïve Bayes Classifier

• Benefits from feature selection

• SVM-normal gives the best performance

InfoGain

OddsRatio

SVM Normal

PercNormal

Perceptron, Macroaverages

0.05

0.15

0.25

0.35

0.45

0.55

10 100 1000 10000 100000

Number of features kept

Av

era

ge

F1

ov

er

the

te

st

se

t

odds ratio information gain normal-1 perceptron-1

Experiments with Perceptron Classifier

• Does not benefit from feature selection

• Perceptron and SVM Normal feature selection give comparable performance

InfoGain

OddsRatio

SVM Normal

PercNormal

Linear SVM, Macroaverages

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

10 100 1000 10000 100000

Number of features kept

Av

era

ge

F1

ov

er

the

te

st

se

t

odds ratio information gain normal-1 perceptron-1

Experiments with the Linear SVM Classifier

• Does not benefit from feature selection

• SVM-normal the best performance

InfoGain

OddsRatioSVM Normal

PercNormal

DiscussionUsing discarded features can help

The features that harm performance if used as input were found to improve performance if used as additional output

obtain additional information by introducing mapping from the selected features to the discarded features (the multitask learning setting [Caruana de Sa 2003])

experiments on synthetic regression and classification problems and real-world medical data have shown improvements in performance

Intuition: transfer of information occurs inside the model, when in addition to the class value it models also additional output consisting of the discarded features

DiscussionFeature subset selection as pre-processing ignore interaction with the target learning algorithm

Simple Filters – work for large number of features assume feature independence, limited results the size of feature subset to be determined

Filters – search space of size , can not handle many features relay on general data characteristics (consistency, distance, class

distribution) use the target learning algorithm for evaluation

Wrappers – high accuracy, computationally expensive use model selection with cross-validation of the target algorithm, similar

to metric-based model selection (eg., comparing output on training and on unlabeled data)

Feature subset selection during learning use the target learning algorithm during feature selection

Embedded – can be used by filters to find the feature subset

F2