Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria
type: Item
instance of: scholarly article
title: Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria
zbMATH Open document ID: 0591.62053
DOI: 10.1016/0047-259X(85)90092-2
author: Yasunori Fujikoshi
published in: Journal of Multivariate Analysis
publication date: 1985-01-01T00:00:00Z
review text: The author considers two criteria for selecting the ''best'' subset of variables for the linear discriminant function in the case of two p- variate normal populations \(\Pi_ 1\), \(\Pi_ 2\) with different means and a common covariance matrix, the means and the matrix being unknown and are to be estimated by random samples of unequal sizes \(N_ 1\), \(N_ 2.\) One criterion is based on minimizing \textit{G. J. McLachlan's} asymptotic unbiased estimate [Biometrics 36, 501-510 (1980; Zbl 0442.62046)] for the error rate of misclassification \[ M(j)=\Phi [-2^{-1}D_ j+2^{- 1}(k_ j-1)(N_ 1^{-1}+N_ 2^{-1})/D_ j+\quad \{32(N_ 1+N_ 2-2)\}^{-1}D_ j\{4(4k_ j-1)-D^ 2_ j\}] \] where \(D_ j\) is the j-subset sample Mahalanobis distance between \(\Pi_ 1\) and \(\Pi_ 2\), and \(k_ j\) is the dimension of this subset. The other selection criterion is based on a ''no additional information'' model minimizing Akaike's information criterion \[ A(j)=(N_ 1+N_ 2)\log \{1+(p-k_ j)F(j)/(N_ 1+N_ 2-p-1)\}+2(k_ j-p), \] \[ where\quad F(j)=\{(N_ 1+N_ 2-p-1)/(p-k_ j)\}(D^ 2-D^ 2_ j)/\{(N_ 1+N_ 2-2)(N_ 1^{-1\quad}+N_ 2^{-1})+D_ j^ 2\}, \] D being the p-variate Mahalanobis distance. It is shown that the expected error rate is closely related to the no additional information model. The asymptotic distributions and error rate risks of both criteria are obtained and are shown to be identical for these criteria, so in this sense the two criteria considered are asymptotically equivalent.
Mathematics Subject Classification ID: 62H30
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62F07
zbMATH DE Number: 3949518
zbMATH Keywords: two-group discriminant analysis
zbMATH Keywords: selection of variables
zbMATH Keywords: linear discriminant function
zbMATH Keywords: p-variate normal populations
zbMATH Keywords: different means
zbMATH Keywords: common covariance matrix
zbMATH Keywords: asymptotic unbiased estimate
zbMATH Keywords: error rate of misclassification
zbMATH Keywords: Mahalanobis distance
zbMATH Keywords: selection
zbMATH Keywords: Akaike's information criterion
zbMATH Keywords: no additional information model
MaRDI profile type: Publication
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full work available at URL: 0047-259x(85)90092-2
OpenAlex ID: W2061038390
description: scientific article; zbMATH DE number 3949518