Stable Feature Selection: Theory and Algorithms

Stable Feature Selection: Theory and Algorithms Presenter: Yue Han Advisor: Lei Yu Ph.D. Dissertation • 4/26/2012

Outline • Introduction and Motivation • Background and Related Work • Major Contributions • Publications • Theoretical Framework for Stable Feature Selection • Empirical Framework : Margin Based Instance Weighting • Empirical Study • General Experimental Setup • Experiments on Synthetic Data • Experiments on Real-World Data • Conclusion and Future Work

Feature Selection Applications Pixel Selection Gene Selection Sports Travel Politics Tech Artist Life Science Internet. Business Health Elections Word Selection

Feature Selection from High-dimensional Data p: # of features n: # of samples High-dimensional data: p >> n • Curse of Dimensionality: • Effects on distance functions • In optimization and learning • In Bayesian statistics • Feature Selection: • Alleviating the effect of the curse of dimensionality. • Enhancing generalization capability. • Speeding up learning process. • Improving model interpretability. Knowledge Discovery on High-dimensional Data

Stability of Feature Selection Feature Selection Method Training Data Training Data Feature Subset Training Data Feature Subset Consistent or not??? Feature Subset Stability Issue of Feature Selection Stability of Feature Selection: the insensitivity of the result of a feature selection algorithm to variations to the training set. Training Data Learning Model Stability of feature selection was relatively neglected before and attracted interests from researchers in data mining recently. Training Data Learning Model Training Data Learning Model Learning Algorithm Stability of Learning Algorithm is firstly examined by Turney in 1995

Motivation for Stable Feature Selection Given Unlimited Sample Size: Feature selection results from D1 and D2 are the same Given Limited Sample Size: (n<<p for high dimensional data) Feature selection results from D1 and D2 are different • Biologists cares about: • Prediction accuracy & Consistency of feature subsets; • Confidence for biological validation ; • Biomarkers to explain the observed phenomena.

Feature Selection Methods Subset Generation Subset Evaluation Original set Subset Goodness of subset Stopping Criterion no Yes Result Validation • Representative Algorithms • Relief, SFS, MDLM, etc. • FSBC, ELSA, LVW, etc. • BBHFS, Dash-Liu’s, etc. • Evaluation Criteria • Filter Model • Wrapper Model • Embedded Model • Search Strategies: • Complete Search • Sequential Search • Random Search

Stable 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.

Margin based Feature Selection Sample Margin: how much can an instance travel before it hits the decision boundary Hypothesis 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 based on HM: 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. 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), pages 205-215, Sydney, Australia, December 14-17, 2010. • 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), pages 262-272, vol. 9 no. 1, 2012. • Yue Han and Lei Yu. A Variance Reduction Framework for Stable Feature Selection. Statistical Analysis and Data Mining(SADM), Accepted, 2012.

Bias-variance Decomposition of Feature Selection Error Training Data: D Data Space: FS Result: r(D) True FS Result: r* Bias-Variance Decomposition of Feature Selection Error: • Relationship between accuracy(opposite of loss)&stability(opposite of variance); • Suggests a better trade-off between the bias and variance of feature selection.

Bias, Variance and Error of Monte Carlo Estimator Feature Selection (Weighting)  Monte Carlo Estimator Relevance Score: Monte Carlo Estimator: Impact Factor: feature selection algorithm and sample size

Variance Reduction via Sample Weighting Probability density function Importance Sampling A good importance sampling function h(x)

Overall 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

Margin Vector Feature Space For each For each Hypothesis Margin: captures the local profile of feature relevance for all features at Nearest Hit Nearest Miss • Instances exhibit different profiles of feature relevance; • Instances influence feature selection results differently. miss hit

An Illustrative Example (a) Original Feature Space (b) Margin Vector Feature Space.

Extension for Hypothesis Margin of 1NN • To reduce the effect of noise or outliers

Margin Based Instance Weighting Algorithm exhibits different profiles of feature relevance • Review: • Variance reduction via Importance Sampling • More instances draw • from important regions • Less instances draw from other regions Instance influence feature selection results differently Higher Outlying Degree Lower Weight Instance Weighting Lower Outlying Degree Higher Weight Weighting: Outlying Degree:

Iterative Margin Based Instance Weighting Assumption: Instances are equally important in original feature space Margin Vector Feature Space Original Feature Space Weighted Feature Space Margin Vector Feature Space Weighted Feature Space Margin Vector Feature Space • The iterative procedure always Converges fast; • There exists little difference in terms of learned weights; • Overall a stable procedure. Instance Weight Final Instance Weight Updated Instance Weight

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

Objective of Empirical Study Feature Selection Method Training Data Training Data Feature Subset Training Data Feature Subset • to demonstrate the bias-variance decomposition in theoretical framework; • to verify the effectiveness of the proposed instance weighting framework on variance reduction; • to study the impacts of variance reduction on the stability and predictive performance of the selected subsets. Consistent or not??? Feature Subset Stability of Feature Selection

Algorithms in Comparison

Stability Measures, Predictive Accuracy Measures nPOGR: Stability Measures • Feature Subset • Jaccard Index; • nPOGR; • SIMv; • Kuncheva Index. • Feature Ranking: • Spearman Coefficient • Feature Weighting: • Pearson Correlation Coefficient Kuncheva Index: • AUC Accuracy: • the area under the receiver operating characteristic (ROC) Curve Predictive Accuracy • CV Accuracy: • Prediction Accuracy base on Cross-validation

Experiments on Synthetic Data 500 Training Data: 100 instances with 50 from and 50 from Leave-one-out Test Data: 5000 instances 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 Method in Comparison: SVM-RFE: Recursively eliminate 10% features of previous iteration till 10 features remained. Measures: Variance, Bias, Error Subset Stability (Kuncheva Index) CV Accuracy (SVM)

Experiments on Synthetic Data • Observations: • Error is equal to the sum of bias and variancefor 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.

Experiments on Synthetic Data • Conclusion: • Variance Reduction via Margin Based Instance Weighting • better bias-variance tradeoff • increased subset stability • improved classification accuracy

Experiments on Synthetic Data • Observations: • the sample size dependency of the performance of SVM-RFE • the effectiveness of instance weighting on alleviating such dependency

Experiments on Real-world Data Microarray Data: 10-time 10-fold Cross-Validation Methods in Comparison: SVM-RFE Ensemble SVM-RFE Instance Weighting SVM-RFE Training Data 10 fold ... Test Data 100 bootstrapped(random repetition) Bootstrapped Training Data 2/3 TrainingData 1/3 Test Data 100 ... Measures: Subset Stability: Kenchuva Index CV Accuracies (KNN, SVM) Measures: Subset Stability: nPOGR AUC Accuracies (KNN, SVM)

Experiments on Real-world Data • Observations: • Non-discriminative during early iterations; • SVM-RFE sharply increase as # of features approaches 10; • IW SVM-RFE shows significantly slower rate of increase. Note: 40 iterations starting from about 1000 features till 10 features remain

Experiments on Real-world Data • 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. Consistent Results showed under random repetition setting and not included here

Experiments on Real-world Data • Observations: • Instance Weighting enables the selection of more genes with higher frequency • Instance Weighting produce much bigger consensus gene signatures

Experiments on Real-world Data • 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. Consistent Results showed under random repetition setting(also ReliefF) and results can be found in the dissertation but not included here for conciseness.

Conclusion and Future Work • Conclusion: • Theoretical Framework for Stable Feature Selection; • Empirical Weighting Framework for Stable Feature Selection; • Effective and Efficient Margin Based Instance Weighting Approaches; • Extensive Study on Proposed Theoretical And Empirical Frameworks; • Extensive Study on Proposed Weighting Approaches; • Extensive Study on Sample Size Effect on Feature Selection Stability. • Future Work: • Explore Other Weighting Approaches; • Study the Relationship Between Feature Selection and Classification w.r.t. Bias-Variance Properties.

Thank you and Questions?

Stable Feature Selection: Theory and Algorithms