Showing papers on "Mahalanobis distance published in 1973"

Automatic soil-boundary location from transect data

Abstract: Spatially distributed soil data possess a short-range erratic variation, an irregular longer range pattern, and maybe multivariate. In order to reveal a pattern or meaning in them, they are usually classified by drawing boundaries. A method is described for determining boundaries automatically on transects. A small portion of a sampled transect is taken and divided about its midpoint, and Mahalanobis' generalized distance, Dor D2,between the two halves calculated from the sample data. The procedure is repeated for portions of the same length at positions one-sampling interval apart along the transect. High peaks on the resultant series of D2 identify the boundaries. The length of portions is set equal to, or somewhat less than, the expected average distance between boundaries, and is determined by constructing correlograms of principal components. The lag distance over which fairly steady decay occurs is related closely to the distance between boundaries. The procedure is illustrated with data from a 6-km transect in Oxfordshire and shows good agreement with boundaries drawn by combined air-photo interpretation and field judgment. A means of extending the procedure to two dimensions is suggested.

An asymptotic expansion of the expectation of the estimated error rate in discriminant analysis1

Abstract: Summary When a sample discriminant function is computed, it is desired to estimate the error rate using this function. This is often done by computing G(-D/2), where G is the cumulative normal distribution and D2 is the estimated Mahalanobis' distance. In this paper an asymptotic expansion of the expectation of G(-D/2) is derived and is compared with existing Monte Carlo estimates. The asymptotic bias of G(-D/2) is derived also and the well-known practical result that G(-D/2) gives too favourable an estimate of the true error rate

Separation and probability of correct classification among two or more distributions

Abstract: A function on theK-fold product of a set in normed vector space will be called a separation measurement if, for any collection ofK points, the function is bounded below and above, respectively, by maximum and total distance between pairs of points in the collection. Separation measurements are relavent toK-sample hypothesis testing and also to discrimination amongK classes, and several examples are given. In particular, ordinaryL 1 distance between integrable functions can be generalized to a non-pairwise separation measurement for densitiesf 1,f 2,...,f K inL 1[μ]; and this separation is a linear transform of the optimal discriminant's probability of correct classification.

Distance, discrimination and error

Abstract: Publisher Summary This chapter discusses the ways of evaluating lower bounds for the probability of correct decision under the minimum-distance classification or topothetical rule. It also discusses the assumption that the populations are apart by certain given minimum distance. The desirable properties of decision rules based on sample analogues of the Mahalanobis distance result from the fact that it emerges as the natural measure of dissimilarity between homoscedastic normal populations. Thus, it is equivalent to the Kullback–Leibler information measure, to Jeffreys divergence, and to Bhattacharya's measure of divergence between two densities. Analogues of the Mahalanobis distance also appear in the discrimination problem for infinite-dimensional normal distributions, that is, Gaussian processes. A special type of stochastic process that admits the same treatment as the finite-dimensional normal case is the normal p-dimensional diffusion process or Wiener process with drift. The chapter also discusses the reduction of the corresponding topothetical or identification problem to the standard p-variate normal case.