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Showing papers on "Mahalanobis distance published in 1968"


Journal ArticleDOI
TL;DR: This article showed that the increase in the probabilities of misclassification is directly related to shrinkage of the multiple correlation coefficient R2 in new samples and that these are related to the unbiased estimation of Mahalanobis' 62 using D2.
Abstract: When a sample discriminant function D, is computed, it is desired to estimate the chance of misclassification using D8. This is often done by classifying the sample with the help of D8 or by computing 4((ID), where b is the cumulative normal distribution, and D2 is Mahalanobis' distance. When D. is applied to a new sample, the observed probabilities of misclassification are usually found to be greater than those computed from the initial sample. The purposes of this paper are to show that this increase in the probabilities of misclassification is directly related to the 'shrinkage' of the multiple correlation coefficient R2 in new samples and that these are related to the unbiased estimation of Mahalanobis' 62 using D2.

154 citations


Journal ArticleDOI
TL;DR: It was shown that the use of transformations had very little effect on the relative distances and that highly significant heterogeneity — as measured by the multivariate extension of Bartlett's test — had little impact on the analysis of the non-transformed data.
Abstract: he racial means and the residual covariance matrix from the multivariate analysis of variance of an experiment based on a randomized block design involving 15 races of maize (Zea mays L.) from southeastern South America were used to calculate generalized distances between the races. Sixteen characters commonly used in taxonomic studies of the races of maize were employed. The effects of transformations designed to eliminate some of the heterogeneity among the withinrace, within-row covariance matrices were studied, and the effects of within-plot sampling were investigated. It was shown that the use of transformations had very little effect on the relative distances (and hence that highly significant heterogeneity — as measured by the multivariate extension of Bartlett's test — had little effect on the analysis of the non-transformed data). Similarly, essentially the same relative distances were obtained when only a single plant from each race was used per block (eight blocks were used). In all cases the distances obtained were relatively very similar (i.e., Spearman rank correlation coefficient between 0.91 and 0.99) to the distances obtained from the commonly employed Mahalanobis’ generalized distance technique.

26 citations


Journal ArticleDOI
TL;DR: In this paper, the authors compared three distance-squared statistics, i.e., D2, A2, and D*2, between two multivariate normal populations with unequal covariance matrices.
Abstract: Three statistics estimating distance squared between two multivariate normal populations with unequal covariance matrices are empirically compared using two sets of data. The data consist of samples of equal size from the populations. The three statistics are: a. the classical Mahalanobis distance-squared statistic, D2; b. the distance-squared statistic, A2 (Russian D2), introduced by Reyment [1962]; c. a distance-squared statistic, D*2, based on a minimax criterion of classification (Anderson and Bahadur [1962]). It is shown that 2 is not a useful statistic. D2 and D*2 are close in numerical value for the data considered. They can differ considerably for unequal sample sizes as shown in a theorem for a special case of l2 = c2s1. Some arguments in favor of D*2 as an appropriate distance statistic are presented.

20 citations