Report

Minimum Redundancy and Maximum Relevance Feature Selection Hang Xiao Background • Feature – a feature is an individual measurable heuristic property of a phenomenon being observed – In character recognition: horizontal and vertical profiles, number of internal holes, stroke detection – In speech recognition: noise ratios, length of sounds, relative power, filter matches – In microarray : genes expression Background • Relevance between features – Correlation – F-statistic – Mutual information p(x,y) : joint distribution function of X and Y p(x), p(y) : marginal probability distribution functions Independent : p(x,y) = p(x)p(y) I(x,y) = 0 Feature Selection Problem • Maximal relevance – selecting the features with the highest relevance to the target class c, based on mutual info., F-test, etc. without considering relationships among features • Minimal Redundancy – Selected features are correlated – Selected features cover narrow regions in space mRMR: Discrete Variables • Maximize Relevance: S is the set of features I(i,j) is mutual information between feature i and j • Minimal Redundancy: mRMR: Continuous Variables • Maximum relevance: F-statistic F(i,h) • Minimum redundancy : Correlation cor(i,j) Combine Relevance and Redundancy • Additive combination • Multiplicative combination Most Related Methods • Most used feature selection methods: topranking features without considering relationships among features. • Yu & Liu, 2003/2004. information gain, essentially similar approach • Wrapper: not filter approach, classifier-involved and thus features do not generalize well • PCA and ICA: Feature are orthogonal or independent, but not in the original feature space Class Prediction Methods • Naive Bayes (NB) classifier {g1, g2, …, gm} gene expression level p(gi|hk) is conditional table (density) • Support Vector Machine SVM – Draw an optimal hyperplane in the feature vector space Class Prediction Methods • Linear Discriminant Analysis (LDA) – Find a linear combination of feature – ANOVA , regression analysis • Logistic Regression (LR) – a linear combination of the feature variables – transformed into probabilities by a logistic function Microarray Gene Expression Data Sets for Cancer Classification LOOCV : Leave- One-Out Cross Validation Baseline feature : based solely on maximum relevance The role of redundancy reduction (a) Relevance VI, and (b) Redundancy for MRMR features on discretized NCI dataset. (c) The respective LOOCV errors obtained using the Naive Bayes classifier Do mRMR Features Generalize Well on Unseen Data? Child Leukemia data (7 classes, 215 training samples, 112 testing samples) testing errors. M is the number of features used in classification What is the Relationship of mRMR Features and Various Data Discretization Schemes? LOOCV testing results classifier(#error) for binarized NCI and Lymphoma data using SVM classifier. Comparison with other work Theoretical basis of mRMR • Maximum Dependency Criterion – Statistic association – Definition : mutual information I(Sm,h) • Mutual Information – For two variables x and y – For multivariate variable Sm and the target h High-Dimensional Mutual Information • For multivariate variable Sm and the target h • Estimate high-dimensional I(Sm,h) is so difficult – An ill-posed problem to find inverse of large covariance matrix – Insufficient number of samples – Combinatorial time complex O(C(|Ω|,|S|)) Factorize the Mutual Information • Mutual information for multivariate variable Sm and the target h Define: It can be proved: Factorize I(Sm,h) • Relevance of S={x1,x2, …} and h, or RL(S,h) • Redundancy among variables {x1,x2,...}, or RD(S) • For incremental search, max I(S,h) is “equivalent” to max [RL(S,h) – RD(S)], so called min-RedundancyMax-Relevance(mRMR) Advantages of mRMR • Both relevance and redundancy estimation are lowdimensional problems (i.e. involving only 2 variables). This is much easier than directly estimating multivariate density or mutual information in the highdimensional space! • Fast speed • More reliable estimation • mRMR is an optimal first-order approximation of I(.) maximization • Relevance-only ranking only maximizes J(.)! Search Algorithm of mRMR • Greedy search algorithm – In the pool Ω, find the variable x1 that has the largest I(x1,h). Exclude x1 from Ω – Search x2 so that it maximizes I(x2,h) - ∑I(.,x2)/|Ω| – Iterate this process until an expected number of variables have been obtained, or other constraints are satisfied • Complexity O(|S|*|Ω|) Comparing Max-Dep and mRMR: Complexity of Feature Selection Comparing Max-Dep and mRMR: Accuracy of Feature Selected in Classification • Leave-One-Out cross validation of feature classification accuracies of mRMR and MaxDep Use Wrappers to Refine Features • mRMR is a filter approach – Fast – Features might be redundant – Independent of the classifier • Wrappers seek to minimize the number of errors directly – – – – Slow Features are less robust Dependent on classifier Better prediction accuracy • Use mRMR first to generate a short feature pool and use wrappers to get a least redundant feature set with better accuracy Use Wrappers to Refine Features Forward wrappers (incremental selection) NCI Data Backward wrappers (decremental selection) Conclusions • The Max-Dependency feature selection can be efficiently implemented as the mRMR algorithm • Significantly outperforms the widely used maxrelevance selection method: mRMR features cover a broader feature space with less features • mRMR is very efficient and useful for gene selection and many other applications. 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