Anuradha Roy University of Texas San Antonio: Discriminant Analysis for Multi-level Multivariate Observations

Abstract: Although devised in 1936 by Fisher, discriminant analysis is still rapidly evolving, as the complexity of contemporary data sets grows exponentially. Our classification rules explore these complexities by modeling various correlations in multi-level multivariate data. Furthermore, our classification rules are suitable to data sets where the number of response variables is comparable or larger than the number of observations. We assume that the multi-level multivariate observations have a doubly exchangeable covariance structure and different Kronecker product structures on the mean vectors. The main idea of this talk is to employ the information of the double exchangeability of a variance-covariance matrix for three-level data, which allows partitioning a covariance structure into three unstructured covariance matrices corresponding to each of the three levels. As a consequence, the number of estimated covariance parameters is substantially reduced, comparing to the Fisher's approach, which enables us to apply the proposed procedures even to a very small number of observations. The new discriminant functions are very efficient in discriminating individuals in a small sample scenario. Iterative algorithms are proposed to calculate the maximum likelihood estimates of the unknown population parameters as closed form solutions do not exist for these unknown parameters. The new discriminant functions are applied to a real data set as well as to simulated data sets. We compare our findings with the Fisher's discriminant function.

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