The use of a distance measure in regularised discriminant analysis
type: Item
instance of: scholarly article
title: The use of a distance measure in regularised discriminant analysis
zbMATH Open document ID: 1010.62053
author: J. P. Koolaard
author: S. Ganesalingam
author: C. R. O. Lawoko
published in: Computational Statistics
publication date: 2003-02-06T00:00:00Z
review text: Let \(X\) be \(p\)-dimensional measurements on each object, where each object belongs to one of \(K\) classes, and let the distribution of measurements from each class be the multivariate normal distribution with population mean \(\mu_{i}\) and covariance function \(\Sigma_{i}\) for class \(i\). For the sample regularized discriminant function (SRDF) an estimation of the covariance matrix is proposed of the form \[ \hat\Sigma_{i}(\lambda,\gamma)=(1-\gamma)\hat\Sigma_{i}(\lambda)+ \gamma p^{-1}tr\left(\hat\Sigma_{i}(\lambda)\right)I, \] where \[ \hat\Sigma_{i}(\lambda)=[(1-\lambda)(n_{i}-1)S_{i}+\lambda S_{p}][(1-\lambda)(n_{i}-1)+\lambda(N-K)]^{-1},\quad 0\leq\lambda\leq 1; \] where \(n_{i}\) is the sample size, \(N=\sum_{i=1}^{K}n_{i}\), \(S_{p}=(N-K)^{-1}\sum_{i=1}^{K}S_{i}(n_{i}-1)\), \(S_{i}\) is the sample covariance matrix for class \(i\), and \(I\) is the identity matrix. The authors investigate the use of the Bhattacharyya distance measure \(B\) as an alternative to the cross-validation procedure of the SRDF in determining the optimal estimated values of \(\hat\lambda\) and \(\hat\gamma\). A classification rule which uses regularization parameters obtained from \(B\) is presented for two and three populations cases. Using simulation, experiment comparisons of commonly used discriminant functions and SRDF are carried out. Various rules are compared in terms of their performances in correctly classifying observations from several real data sets.
Mathematics Subject Classification ID: 62H30
zbMATH DE Number: 1865447
zbMATH Keywords: regularized discriminant analysis
zbMATH Keywords: estimates
zbMATH Keywords: Bhattacharyya distance measure
zbMATH Keywords: regularization parameters
reviewed by: Aleksandr D. Borisenko
MaRDI profile type: Publication
full work available at URL: s001800200101
OpenAlex ID: W2911951322
cites work: Q3796553
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DOI: 10.1007/S001800200101
description: scientific article; zbMATH DE number 1865447