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Penalized classification using Fisher's linear discriminant
Author(s) -
Witten Daniela M.,
Tibshirani Robert
Publication year - 2011
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/j.1467-9868.2011.00783.x
Subject(s) - linear discriminant analysis , optimal discriminant analysis , kernel fisher discriminant analysis , interpretability , discriminant , mathematics , pattern recognition (psychology) , artificial intelligence , fisher kernel , multiple discriminant analysis , covariance , covariance matrix , computer science , statistics , facial recognition system
Summary. We consider the supervised classification setting, in which the data consist of p features measured on n observations, each of which belongs to one of K classes. Linear discriminant analysis (LDA) is a classical method for this problem. However, in the high dimensional setting where p ≫ n , LDA is not appropriate for two reasons. First, the standard estimate for the within‐class covariance matrix is singular, and so the usual discriminant rule cannot be applied. Second, when p is large, it is difficult to interpret the classification rule that is obtained from LDA, since it involves all p features. We propose penalized LDA , which is a general approach for penalizing the discriminant vectors in Fisher's discriminant problem in a way that leads to greater interpretability. The discriminant problem is not convex, so we use a minorization–maximization approach to optimize it efficiently when convex penalties are applied to the discriminant vectors. In particular, we consider the use of L 1 and fused lasso penalties. Our proposal is equivalent to recasting Fisher's discriminant problem as a biconvex problem. We evaluate the performances of the resulting methods on a simulation study, and on three gene expression data sets. We also survey past methods for extending LDA to the high dimensional setting and explore their relationships with our proposal.