DETERMINING STOCHASTIC DEPENDENCE FOR NORMALLY DISTRIBUTED VECTORS USING THE CHI-SQUARED METRIC

A fundamental problem in information theory and pattern recognition in volves computing and estimating the probability density function assoc iated with a set of random variables. In estimating this density funct ion, one can either assume that the form of the density function is kn own, and that we are merely estimating parameters that characterize th e distribution or that no information about the density function is av ailable. This problem has been extensively studied if the random varia bles are independent. If the random variables are dependent and are of the discrete sort, the problem of capturing this dependence between v ariables has been studied in Chow and Liu (IEEE Trans. Inf. Theory 14, 462-467 (May 1968)). The analogous problem for normally distributed c ontinuous random variables has been tackled in Chow et al. (Comput. Bi omed. Res. 12, 589-613 (1979)). In both these instances, the determina tion of the best dependence tree hinges on the well-known Expected Mut ual Information Measure (EMIM) Metric. Recently Valiveti and Oommen st udied the suitability of the chi-squared based metric in-lieu of the E MIM metric, for the discrete variable case (Pattern Recognition 25. 13 89-1400 (1992)). In this paper, we generalize the latter result and st udy the use of the chi-squared metric for determining dependence trees for normally distributed random vectors. We show that for such vector s, the chi-squared metric yields the optimal tree and that it is ident ical to the one obtained using the EMIM metric. The computation of the maximum likelihood estimate of the dependence tree is also discussed.