LINEAR DISCRIMINANT RULES for HIGH-DIMENSIONAL CORRELATED DATA: ASYMPTOTIC and FINITE SAMPLE RESULTS

A new class of linear discrimination rules, designed for problems with many correlated variables, is proposed. This proposal tries to incorporate the most important patterns revealed by the empirical correlations and accurately approximate the optimal Bayes rule as the number of variables increases. In order to achieve this goal, the new rules rely on covariance matrix estimates derived from Gaussian factor models with small intrinsic dimensionality. Asymptotic results, based on a analysis that allows the number of variables to grow faster than the number of observations, show that the worst possible expected error rate of the proposed rules converges to the error of the optimal Bayes rule when the postulated model is true, and to a slightly larger constant when this model is a close approximation to the data generating process. Simulation results suggest that, in the data conditions they were designed for, the new rules can clearly outperform both Fisher's and naive linear discriminant rules.Discriminant Analysis, High Dimensionality, Expected Misclassification Rate, Min-Max Regret