NORMIX: COMPUTATIONAL METHODS FOR ESTIMATING THE PARAMETERS OF MULTIVARIATE NORMAL MIXTURES OF DISTRIBUTIONS.
NAVAL PERSONNEL RESEARCH ACTIVITY SAN DIEGO CALIF
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Procedures are described for estimating the means, covariances, and mixing proportions of a mixture of multivariate-normal distributions. First, it is shown that the maximum-likelihood estimates must satisfy a certain set of simultaneous equations. The coefficients necessary for a complete Newton-Raphson iterative solution of the equations are presented. Since these coefficients are rather complicated, a simpler and more intuitively-appealing iterative method is presented and partially justified on the basis that the two iterative methods are approximately the same when the component distributions are well-separated. The formulas for the simplified iteration involve the familiar statistics of sums and sums of squares and cross-products with the modification that each sample value is weighted by its relative likelihood of membership in the type whose parameters are being estimated. Since the estimation procedure is basically maximum-likelihood, tests of hypotheses on the number of component distributions can be developed using likelihood ratios. The computational feasibility of the iteration methods is demonstrated in an example in which a computer program to perform the iterations was run on an artificially-constructed mixture of three bivariate normal distributions. A discussion of potential applications indicates that the method has promise in the fields of personality typology, social class analysis, biological taxonomy, information retrieval and artificial intelligence.
- Statistics and Probability