Variance Reduction for Stable Feature Selection

VARIANCE REDUCTION FOR STABLE FEATURE SELECTION

Presenter: Yue HanAdvisor: Lei Yu

Department of Computer Science10/27/10

OUTLINE Introduction and Motivation Background and Related Work Preliminaries

Publications Theoretical Framework Empirical Framework : Margin Based Instance Weighting

Empirical Study Planned Tasks

INTRODUCTION AND MOTIVATIONFEATURE SELECTION APPLICATIONS

D1

D2

SportsT1 T2 ….…… TN

12 0 ….…… 6

DM

C

Travel

Jobs

… … …

Terms

Doc

umen

ts

3 10 ….…… 28

0 11 ….…… 16

…

Features(Genes or Proteins)

Sam

ple

s

Pixels Vs

Features

INTRODUCTION AND MOTIVATIONFEATURE SELECTION FROM HIGH-DIMENSIONAL DATA

p: # of features n: # of samplesHigh-dimensional data: p >> n

Feature Selection:Alleviating the effect of the curse of dimensionality.Enhancing generalization capability.Speeding up learning process.Improving model interpretability.

Curse of Dimensionality:•Effects on distance functions•In optimization and learning•In Bayesian statistics

High-Dimensional Data

Feature Selection AlgorithmMRMR, SVMRFE, Relief-F,

F-statistics, etc.

Low-Dimensional Data

Learning ModelsClassification, Clustering, etc.

Knowledge Discovery on High-dimensional Data

INTRODUCTION AND MOTIVATIONSTABILITY OF FEATURE SELECTION

Training Data Feature SubsetTraining Data Feature

SubsetTraining Data Feature Subset

Feature Selection Method

Consistent or not???

Stability of Feature Selection: the insensitivity of the result of a feature selection algorithm to variations to the training set.Training Data

Learning ModelTraining

Data Learning

ModelTraining Data

Learning Model

Learning Algorithm

Stability of Learning Algorithm isfirstly examined by Turney in 1995

Stability of feature selection was relatively neglected before and attracted interests from researchers in data mining recently.

Stability Issue of Feature Selection

INTRODUCTION AND MOTIVATIONMOTIVATION FOR STABLE FEATURE SELECTION

D1

D2

Features

Sam

ples Given Unlimited Sample Size of D:

Feature selection results from D1 and D2 are the sameSize of D is limited: (n<<p for high dimensional data)Feature selection results from D1 and D2 are differentChallenge: Increasing #of samples could be very costly or

impractical

Experts from Biology and Biomedicine are interested in:

not only the prediction accuracy but also the consistency of feature subsets;validating stable genes or proteins less sensitive to variations to training data; biomarkers to explain the observed phenomena.

BACKGROUND AND RELATED WORKFEATURE SELECTION METHODS

Subset Generation

Subset Evaluation

Stopping Criterion

Result Validation

Original set

Subset

Goodness of subset

no Yes

Evaluation Criteria Filter Model Wrapper Model Embedded Model

Search Strategies: Complete Search Sequential Search Random Search

Representative Algorithms Relief, SFS, MDLM, etc. FSBC, ELSA, LVW, etc. BBHFS, Dash-Liu’s, etc.

BACKGROUND AND RELATED WORKSTABLE FEATURE SELECTION

Comparison of Feature Selection Algorithms w.r.t. Stability(Davis et al. Bioinformatics, vol. 22, 2006; Kalousis et al. KAIS, vol. 12, 2007)Quantify the stability in terms of consistency on subset or weight;Algorithms varies on stability and equally well for classification;Choose the best with both stability and accuracy.

Bagging-based Ensemble Feature Selection (Saeys et al. ECML07)Different bootstrapped samples of the same training set;Apply a conventional feature selection algorithm;Aggregates the feature selection results.

Group-based Stable Feature Selection (Yu et al. KDD08; Loscalzo et al. KDD09)Explore the intrinsic feature correlations;Identify groups of correlated features;Select relevant feature groups.

BACKGROUND AND RELATED WORKMARGIN BASED FEATURE SELECTION

Sample Margin: how much canan instance travel before it hitsthe decision boundaryHypothesis Margin: how much can the hypothesis travel before it hits an instance (Distance between the hypothesis and the opposite hypothesis of an instance)

Representative Algorithms: Relief, Relief-F, G-flip, Simba, etc. margin is used for feature weighting or feature selection (totally different use in our study)

PUBLICATIONS Yue Han and Lei Yu. An Empirical Study on Stability of Feature

Selection Algorithms. Technical Report from Data Mining Research Laboratory, Binghamton University, 2009.

Yue Han and Lei Yu. Margin Based Sample Weighting for Stable Feature Selection. In Proceedings of the 11th International Conference on Web-Age Information Management (WAIM2010), pages 680-691, Jiuzhaigou, China, July 15-17, 2010.

Yue Han and Lei Yu. A Variance Reduction Framework for Stable Feature Selection. In Proceedings of the 10th IEEE International Conference on Data Mining (ICDM2010), Sydney, Australia, December 14-17, 2010, To Appear.

Lei Yu, Yue Han and Michael E. Berens. Stable Gene Selection from Microarray Data via Sample Weighting. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 2010, Major Revision Under Review.

THEORETICAL FRAMEWORKBIAS-VARIANCE DECOMPOSITION OF FEATURE SELECTION ERROR

Expected Loss(Error):

Training Data: D; Data Space: ; FS Result: r(D); True FS Result: r*

Bias:Variance:

Bias-Variance Decomposition of Feature Selection Error:

o Reveals relationship between accuracy(opposite of loss) and stability (opposite of variance);

o Suggests a better trade-off between the bias and variance of feature selection.

THEORETICAL FRAMEWORKVARIANCE REDUCTION VIA IMPORTANCE SAMPLING Feature Selection (Weighting) Monte Carlo EstimatorRelevance Score: Monte Carlo Estimator:

Variance of Monte Carlo Estimator:Impact Factor: feature selection algorithm and sample size? Increasing sample size impractical and costlyImportance Sampling A good importance sampling function h(x)

Instance Weightin

g

Intuition behind h(x) :More instances draw from important regionsLess instances draw from other regions

Intuition behind instance weight :Increase weights for instances from important regionsDecrease weights for instances from other regions

EMPIRICAL FRAMEWORKOVERALL FRAMEWORK

Challenges:

How to produce weights for instances from the point view of feature selection stability;

How to present weighted instances to conventional feature selection algorithms.

Margin Based Instance Weighting for Stable Feature Selection

EMPIRICAL FRAMEWORKMARGIN VECTOR FEATURE SPACE

Original SpaceFor each

Margin Vector Feature Space

Hypothesis Margin:

hit miss

Nearest HitNearest Miss

captures the local profile of feature relevance for all features at

Instances exhibit different profiles of feature relevance; Instances influence feature selection results differently.

EMPIRICAL FRAMEWORKAN ILLUSTRATIVE EXAMPLE

Hypothesis-Margin based Feature Space Transformation:(a) Original Feature Space (b) Margin Vector Feature Space.

(a) (b)

EMPIRICAL FRAMEWORKMARGIN BASED INSTANCE WEIGHTING ALGORITHM

Instance

exhibits different profiles of feature relevance

influence feature selection results differently

Instance Weighting

Higher Outlying Degree Lower Weight

Lower Outlying Degree Higher Weight

Review: Variance reduction via Importance Sampling

More instances draw from important regions

Less instances draw from other regions

Weighting:

Outlying Degree:

EMPIRICAL FRAMEWORKALGORITHM ILLUSTRATION

Time Complexity Analysis:

o Dominated by Instance Weighting:

o Efficient for High-dimensional Data with small sample size (n<<d)

EMPIRICAL STUDYSUBSET STABILITY MEASURES

Average Pair-wise Similarity:

Kuncheva Index:

Feature SubsetJaccard Index;nPOGR;SIMv.

Feature Ranking:Spearman Coefficient

Feature Weighting:Pearson Correlation Coefficient

Training Data Feature SubsetTraining Data Feature

SubsetTraining Data Feature Subset

Feature Selection Method

Consistent or not???

Stability of Feature Selection

EMPIRICAL STUDYEXPERIMENTS ON SYNTHETIC DATA

Synthetic Data Generation: Feature Value:two multivariate normal distributions

Covariance matrix

is a 10*10 square matrix with elements 1 along the diagonal and 0.8 off diagonal.100 groups and 10 feature each

Class label: a weighted sum of all feature values with optimal feature weight vector

500 Training Data:100 instances with 50 from and 50 fromLeave-one-out Test Data:5000 instancesMethod in Comparison:SVM-RFE: Recursively eliminate 10% features of previous iteration till 10 features remained.

Measures:Variance, Bias, ErrorSubset Stability (Kuncheva Index)Accuracy (SVM)


Observations: Error is equal to the sum of bias and variance for both versions of SVM-RFE; Error is dominated by bias during early iterations and is dominated by variance during later iterations; IW SVM-RFE exhibits significantly lower bias, variance and error than SVM-RFE when the number of remaining features approaches 50.


Conclusion: Variance Reduction via Margin Based Instance Weightingbetter bias-variance tradeoffincreased subset stabilityimproved classification accuracy

EMPIRICAL STUDYEXPERIMENTS ON REAL-WORLD DATA

Microarray Data:

Experiment Setup:

Methods in Comparison:SVM-RFEEnsemble SVM-RFEInstance Weighting SVM-RFE

Measures:VarianceSubset StabilityAccuracies (KNN, SVM)

10 fold ... TrainingData

Test Data

10-fold Cross-Validation

Bootstrapped Training Data

Feature Subset

Aggregated Feature

Subset20

...

Bootstrapped Training Data

...

Feature Subset

20-Ensemble SVM-RFE


Note: 40 iterations starting from about 1000 features till 10 features remain

Observations:Non-discriminative during early iterations;

SVM-RFE sharply increase as # of features approaches 10;

IW SVM-RFE shows significantly slower rate of increase.


Observations:Both ensemble and instance weighting approaches improve stability consistently;

Ensemble is not as significant as instance weighting;

As # of features increases, stability score decreases because of the larger correction factor.


Conclusions:Improves stability of feature selection without sacrificing prediction accuracy;

Performs much better than ensemble approach and more efficient;

Leads to significantly increased stability with slight extra cost of time.

PLANNED TASKSOVERALL FRAMEWORK

Theoretical Framework of Feature Selection Stability

Empirical Instance Weighting Framework

Margin-based Instance Weighting

Representative FS

Algorithms

SVM

-RFE

Relie

f-F

F-st

atist

ics

HHSV

M

Various Real-world Data

Set

Gene

Dat

a

Text

Dat

a

Iterative ApproachState-of-the-art

Weighting Schemes

Relationship Between Feature Selection

Stability and Classification Accuracy

PLANNED TASKSLISTED TASKS

A Extensive Study on Instance Weighting FrameworkA1 Extension to Various Feature Selection AlgorithmsA2 Study on Datasets from Different Domains

B Development of Algorithms under Instance Weighting FrameworkB1 Development of Instance Weighting SchemesB2 Iterative Approach for Margin Based Instance Weighting

C Investigation on the Relationship between Stable Feature Selection and Classification AccuracyC1 How Bias-Variance Properties of Feature Selection Affect Classification AccuracyC2 Study on Various Factors for Stability of Feature Selection

Oct-Dec 2010

Jan-Mar 2011 April-June2011 July-Aug 2011

A1A2B1B2C1C2

Thank you and

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Variance Reduction for Stable Feature Selection

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