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Minimum Redundancy and Maximum Relevance Feature Selection Hang Xiao

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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

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Background Relevance between features – Correlation – F-statistic – Mutual information Independent : p(x,y) = p(x)p(y) I(x,y) = 0 p(x,y) : joint distribution function of X and Y p(x), p(y) : marginal probability distribution functions

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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

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mRMR: Discrete Variables Maximize Relevance: Minimal Redundancy: S is the set of features I(i,j) is mutual information between feature i and j

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mRMR: Continuous Variables Maximum relevance: F-statistic F(i,h) Minimum redundancy : Correlation cor(i,j)

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Combine Relevance and Redundancy Additive combination Multiplicative combination

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Most Related Methods Most used feature selection methods: top- ranking 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

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Class Prediction Methods Naive Bayes (NB) classifier {g 1, g 2, …, g m } gene expression level p(g i |h k ) is conditional table (density) Support Vector Machine SVM – Draw an optimal hyperplane in the feature vector space

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Class Prediction Methods Logistic Regression (LR) – a linear combination of the feature variables – transformed into probabilities by a logistic function Linear Discriminant Analysis (LDA) – Find a linear combination of feature – ANOVA, regression analysis

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Microarray Gene Expression Data Sets for Cancer Classification

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LOOCV : Leave- One-Out Cross Validation Baseline feature : based solely on maximum relevance

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(a) Relevance V I, and (b) Redundancy for MRMR features on discretized NCI dataset. (c) The respective LOOCV errors obtained using the Naive Bayes classifier The role of redundancy reduction

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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

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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.

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Comparison with other work

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Theoretical basis of mRMR Maximum Dependency Criterion – Statistic association – Definition : mutual information I(S m,h) Mutual Information – For two variables x and y – For multivariate variable S m and the target h

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High-Dimensional Mutual Information For multivariate variable Sm and the target h Estimate high-dimensional I(S m,h) is so difficult – An ill-posed problem to find inverse of large co- variance matrix – Insufficient number of samples – Combinatorial time complex O(C(|Ω|,|S|))

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Factorize the Mutual Information Mutual information for multivariate variable S m and the target h Define: It can be proved:

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Factorize I(S m,h) Relevance of S={x 1,x 2, …} and h, or RL(S,h) Redundancy among variables {x 1,x 2,...}, or RD(S) For incremental search, max I(S,h) is “equivalent” to max [RL(S,h) – RD(S)], so called min-Redundancy- Max-Relevance(mRMR)

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Advantages of mRMR Both relevance and redundancy estimation are low- dimensional problems (i.e. involving only 2 variables). This is much easier than directly estimating multivariate density or mutual information in the high- dimensional space! Fast speed More reliable estimation mRMR is an optimal first-order approximation of I(.) maximization Relevance-only ranking only maximizes J(.)!

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Search Algorithm of mRMR Greedy search algorithm – In the pool Ω, find the variable x 1 that has the largest I(x 1,h). Exclude x 1 from Ω – Search x 2 so that it maximizes I(x 2,h) - ∑I(.,x 2 )/|Ω| – Iterate this process until an expected number of variables have been obtained, or other constraints are satisfied Complexity O(|S|*|Ω|)

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Comparing Max-Dep and mRMR: Complexity of Feature Selection

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Comparing Max-Dep and mRMR: Accuracy of Feature Selected in Classification Leave-One-Out cross validation of feature classification accuracies of mRMR and MaxDep

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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

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Use Wrappers to Refine Features Forward wrappers (incremental selection) Backward wrappers (decremental selection) NCI Data

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Conclusions The Max-Dependency feature selection can be efficiently implemented as the mRMR algorithm Significantly outperforms the widely used max- relevance 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. The programs are ready!

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