Showing papers on "Mahalanobis distance published in 1971"

The effect of nonnormality on some multivariate tests and robustness to nonnormality in the linear model

Abstract: SUMMARY The effect of nonnormality on multivariate regression tests, on thle one-way multivariate analysis of variance and on tests of equality of covariance matrices is studied following the approach of Box & Watson (1962). In the nonnormal case, an approximation to the distribution of a generalized Mahalanobis distance type of statistic for the multivariate regression problem is derived. It is shown that sensitivity to nonnormality in the multivariate observations is determined by the extent of nonnormality of the regressors. The randomization distribution of the generalized Mahalanobis distance is deduced. The multivariate analysis of variance is found to be robust to nonnormality whereas the tests for equality of covariance matrices are found to be sensitive to nonnormality. An explanation for this varying degree of sensitivity to nonnormality is given.

Some Expected Values for Probabilities of Correct Classification in Discriminant Analysis

Abstract: Monte Carlo estimates have been obtained for two quantities of interest in a discriminant analysis involving the usual linear discriminant function. The first is the unconditional probability of correct classification; the second is the expected value of its estimate based on the calculated Mahalanobis distance. These two quantities are shown in tables and graphs versus the population Mahalanobis distance. Equal sample sizes of 25, 50, and 100 have been used in forming the discriminant functions; 2, 6, 10, 15, 20, and 30 variates have been used. A comparison is made between the Monte Carlo estimates of the unconditional probability of correct classification and an approximation suggested by Lachenbruch [41].

Improved distance measure for pattern recognition

Abstract: The Euclidean distance has often been used to measure the similarity between patterns represented by multidimensional vectors. The Euclidean distance is expensive to implement in hardware, and alternatives have been sought. The letter proposes a new distance measure which is a weighted sum of the city block and square distances. This new distance is a more accurate predictor of the Euclidean distance than are either of its components.

Analysis of Growth Curves under the Hierarchical Models and Spline Regression, I,

Abstract: : An analysis of longitudinal data (i.e., 'growth curves'), where there is not an equal number of observations recorded on each experimental unit, is developed. The method utilizes a step-down procedure which is similar to that presented in J. Roy ((1958). Ann. Math. Statist. 29, 1177-1187) and S. N. Roy and Srivastava ((1964). Mahalanobis Dedicatory Volume. Indian Statistical Institute, Calcutta), and (for a 'testable' hypothesis) yields exact tests for all sample sizes. Also included is a discussion of polynomial spline regression, which appears to be potentially useful in the analysis of 'growth curves'. Finally it is shown that for a certain class of linear hypotheses the method leads to some interesting and unsolved problems in the standard multiresponse model (i.e., in the standard multivariate analysis of variance). (Author)