Yue Han and Lei Yu Binghamton University

 Introduction, Motivation and Related Work  Theoretical Framework  Empirical Framework : Margin Based Instance Weighting  Empirical Study ◦ Synthetic Data ◦ Real-world Data  Conclusion and Future Work

D1D1 D2D2 Sports T 1 T 2 ….…… T N 12 0 ….…… 6 DMDM C Travel Jobs … …… Terms Documents 3 10 ….…… ….…… 16 … Features(Genes or Proteins) Samples Pixels Vs Features

Stability of Feature Selection : the insensitivity of the result of a feature selection algorithm to variations to the training set. Stability of feature selection was relatively neglected before and attracted interests from researchers in data mining recently. Stability of Learning Algorithm Learning Algorithm Training Data Learning Models Training D 1 Data Space Training D 2 Training D n Feature Subset R 1 Feature Subset R 2 Feature Subset R n Feature Selection Method Consistent or not??? Stability Issue of Feature Selection

Data Variations Stable Feature Selection Method Stable Feature SubsetLearning Results Learning Methods Closer to characteristic features (biomarkers) Better learning performance Largely different feature subsets Similarly good learning performance Domain experts (Biomedicine and Biology) also interested in: Biomarkers stable and insensitive to data variations Unstable feature selection method Dampen confidence for validation; Increase the cost for experiments

How to represent the underlying data distribution without increasing sample size? Challenge: Challenge: Increasing training sample size could be very costly or impractical Training D 1 Feature Weight Vector Data Space True Feature Weight Vector Feature Weight Vector Training D 2 Training D n Variance: fluctuation of n weight values around its central tendency Bias: loss of central tendency(average) from the true weight value Error: average loss of n weight values from the true weight value for

Error: Data Space: ; Training Data: D ; FS Result: r(D) ; True FS Result: r* Bias: Variance: Bias-Variance Decomposition of Feature Selection Error: o Reveals relationship between accuracy(opposite of error) and stability (opposite of variance); o Suggests a better trade-off between the bias and variance of feature selection. For each individual feature: weight value instead of 0/1 selection Average for all features

Feature Selection (Weighting)  Monte Carlo Estimator Reducing Variance of Monte Carlo Estimator: Importance Sampling ? Increasing sample size impractical and costly Importance Sampling Instance Weighting Intuition behind importance sampling: More instances draw from important regions Less instances draw from other regions Intuition behind instance weighting: Increase weights for instances from important regions Decrease weights for instances from other regions How to weight the instances? How important is each instance?

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

Original Space For each Margin Vector Feature Space Hypothesis Margin (along each dimension): hitmiss Nearest Hit Nearest 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.

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

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:

Time Complexity Analysis: o Dominated by Instance Weighting: o Efficient for High-dimensional Data with small sample size (n<<d)

Average Pair-wise Similarity: Kuncheva Index: Training D 1 Data Space Training D 2 Training D n Feature Subset R 1 Feature Subset R 2 Feature Subset R n Feature Selection Method Consistent or not??? Stability Issue of Feature Selection

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 from Leave-one-out Test Data: 5000 instances Method in Comparison: SVM-RFE: Recursively eliminate 10% features of previous iteration till 10 features remained. Measures: Variance, Bias, Error Subset 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 Weighting  better bias-variance tradeoff  increased subset stability  improved classification accuracy

Microarray Data: Methods in Comparison: SVM-RFE Ensemble SVM-RFE Instance Weighting SVM-RFE Measures: Variance Subset Stability Accuracies (KNN, SVM) Bootstrapped Training Data Feature Subset Aggregated Feature Subset 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. Prediction accuracy(via both KNN and SVM): non-discriminative among three approaches for all data sets

Accomplishments:  Establish a bias-variance decomposition framework for feature selection;  Propose an empirical framework for stable feature selection;  Develop an efficient margin-based instance weighting algorithm;  Comprehensive study through synthetic and real-world data. Future Work:  Extend current framework to other state-of-the-art feature selection algorithms;  Explore the relationship between stable feature selection and classification performance.

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.

Thank you and Questions?