Feature selection methodsfrom correlation

to causality

Isabelle Guyon isabelle@clopinet.com

NIPS 2008 workshop on kernel learning

1) Feature Extraction, Foundations and ApplicationsI. Guyon, S. Gunn, et al.Springer, 2006.http://clopinet.com/fextract-book

2) Causal feature selectionI. Guyon, C. Aliferis, A. ElisseeffTo appear in “Computational Methods of Feature Selection”, Huan Liu and Hiroshi Motoda Eds., Chapman and Hall/CRC Press, 2007.http://clopinet.com/causality

Acknowledgements and references

http://clopinet.com/causality

Constantin Aliferis Alexander Statnikov

André Elisseeff Jean-Philippe Pellet

Gregory F. Cooper Peter Spirtes

Introduction

Feature Selection

• Thousands to millions of low level features: select the most relevant one to build better, faster, and easier to understand learning machines.

X

n

m

n’

Applications

Bioinformatics

Quality control

Machine vision

Customer knowledge

variables/features

examples

10

102

103

104

105

OCRHWR

MarketAnalysis

TextCategorization

Syst

em d

iagn

osis

10 102 103 104 105

106

Nomenclature

• Univariate method: considers one variable (feature) at a time.

• Multivariate method: considers subsets of variables (features) together.

• Filter method: ranks features or feature subsets independently of the predictor (classifier).

• Wrapper method: uses a classifier to assess features or feature subsets.

Univariate Filter

Methods

Univariate feature ranking

• Normally distributed classes, equal variance 2 unknown; estimated from data as 2

within.

• Null hypothesis H0: + = -

• T statistic: If H0 is true,

t= (+ - -)/(withinm++1/m-Studentm++m--d.f.

-1

- +

- +

P(Xi|Y=-1)

P(Xi|Y=1)

xi

• H0: X and Y are independent.

• Relevance index test statistic.

• Pvalue false positive rate FPR = nfp / nirr

• Multiple testing problem: use Bonferroni correction pval n pval

• False discovery rate: FDR = nfp / nsc FPR n/nsc

• Probe method: FPR nsp/np

pvalr0 r

Null distribution

Statistical tests ( chap. 2)

(Guyon, Dreyfus, 2006, )

Univariate Dependence

• Independence:

P(X, Y) = P(X) P(Y)

• Measure of dependence:

MI(X, Y) = P(X,Y) log dX dY

= KL( P(X,Y) || P(X)P(Y) )

P(X,Y)

P(X)P(Y)

A choice of feature selection ranking methods depending on the nature of:

• the variables and the target (binary, categorical, continuous)

• the problem (dependencies between variables, linear/non-linear relationships between variables and target)

• the available data (number of examples and number of variables, noise in data)

• the available tabulated statistics.

Other criteria ( chap. 3)

(Wlodzislaw Duch, 2006)

Multivariate Methods

Univariate selection may fail

Guyon-Elisseeff, JMLR 2004; Springer 2006

Filters,Wrappers, andEmbedded methods

All features FilterFeature subset Predictor

All features

Wrapper

Multiple Feature subsets

Predictor

All featuresEmbedded

method

Feature subset

Predictor

Relief

nearest hit

nearest miss

Dhit Dmiss

Relief=<Dmiss/Dhit>

Dhit

Dmiss

Kira and Rendell, 1992

Wrappers for feature selection

N features, 2N possible feature subsets!

Kohavi-John, 1997

• Exhaustive search.• Stochastic search (simulated annealing,

genetic algorithms)• Beam search: keep k best path at each

step. • Greedy search: forward selection or

backward elimination.• Floating search: Alternate forward and

backward strategies.

Search Strategies ( chap. 4)

(Juha Reunanen, 2006)

Forward Selection (wrapper)

n

n-1

n-2

1

…

Start

Also referred to as SFS: Sequential Forward Selection

Guided search: we do not consider alternative paths.Typical ex.: Gram-Schmidt orthog. and tree classifiers.

Forward Selection (embedded)

…

Start

n

n-1

n-2

1

Backward Elimination (wrapper)

1

n-2

n-1

n

…

Start

Also referred to as SBS: Sequential Backward Selection

Backward Elimination (embedded)

…

Start

1

n-2

n-1

n

Guided search: we do not consider alternative paths.Typical ex.: “recursive feature elimination” RFE-SVM.

Scaling Factors

Idea: Transform a discrete space into a continuous space.

• Discrete indicators of feature presence: i {0, 1}

• Continuous scaling factors: i IR

=[1, 2, 3, 4]

Now we can do gradient descent!

• Many learning algorithms are cast into a minimization of some regularized functional:

Empirical errorRegularization

capacity control

Formalism ( chap. 5)

(Lal, Chapelle, Weston, Elisseeff, 2006)

Justification of RFE and many other embedded methods.

Embedded method

• Embedded methods are a good inspiration to design new feature selection techniques for your own algorithms:– Find a functional that represents your prior knowledge about

what a good model is.– Add the weights into the functional and make sure it’s

either differentiable or you can perform a sensitivity analysis efficiently

– Optimize alternatively according to and – Use early stopping (validation set) or your own stopping

criterion to stop and select the subset of features

• Embedded methods are therefore not too far from wrapper techniques and can be extended to multiclass, regression, etc…

Causality

What can go wrong?

Guyon-Aliferis-Elisseeff, 2007

X2 X1

180 190 200 210 220 230 240 250 260

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What can go wrong?

20 40 60 80 100

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X2 X1

X1

X

2

X2 X1

180 190 200 210 220 230 240 250 260

20

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What can go wrong?

Guyon-Aliferis-Elisseeff, 2007

X2 Y

X1

Lung Cancer

Smoking Genetics

Coughing

AttentionDisorder

Allergy

Anxiety Peer Pressure

Yellow Fingers

Car Accident

Born an Even Day

Fatigue

Local causal graph

What works and why?

Bilevel optimization

1) For each feature subset, train predictor on training data.

2) Select the feature subset, which performs best on validation data.– Repeat and average if you

want to reduce variance (cross-validation).

3) Test on test data.

N variables/features

M s

ampl

es

m1

m2

m3

Split data into 3 sets:training, validation, and test set.

Complexity of Feature Selection

Method Number of subsets tried

Complexity C

Exhaustive search wrapper

2N N

Nested subsets Feature ranking

N(N+1)/2 or N

log N

Generalization_error Validation_error + (C/m2)

m2: number of validation examples, N: total number of features,n: feature subset size.

With high probability:

n

Error

Try to keep C of the order of m2.

Insensitivity to irrelevant features

Simple univariate predictive model, binary target and features, all relevant features correlate perfectly with the target, all irrelevant features randomly drawn. With 98% confidence, abs(feat_weight) < w and i wixi < v.

ng number of “good” (relevant) features

nb number of “bad” (irrelevant) features

m number of training examples.

Conclusion

• Feature selection focuses on uncovering subsets of variables X1, X2, … predictive of the target Y.

• Multivariate feature selection is in principle more powerful than univariate feature selection, but not always in practice.

• Taking a closer look at the type of dependencies in terms of causal relationships may help refining the notion of variable relevance.

• Feature selection and causal discovery may be more harmful than useful.

• Causality can help ML but ML can also help causality