Abstract
For the case where all multivariate normal parameters are known, we derive a new linear dimension reduction (LDR) method to determine a low-dimensional subspace that preserves or nearly preserves the original feature-space separation of the individual populations and the Bayes probability of misclassification. We also give necessary and sufficient conditions which provide the smallest reduced dimension that essentially retains the Bayes probability of misclassification from the original full-dimensional space in the reduced space. Moreover, our new LDR procedure requires no computationally expensive optimization procedure. Finally, for the case where parameters are unknown, we devise a LDR method based on our new theorem and compare our LDR method with three competing LDR methods using Monte Carlo simulations and a parametric bootstrap based on real data.
Highlights
The fact that the Bayes probability of misclassification (BPMC) of a statistical classification rule does not increase as the dimension or feature space increases, provided the class-conditional probability densities are known, is well-known
While all population parameters are known, we have presented a simple and flexible algorithm for a low-dimensional representation of data from multiple multivariate normal populations with different parametric configurations
We have provided a constructive proof for obtaining a low-dimensional representation space when certain population parameter conditions are satisfied
Summary
The fact that the Bayes probability of misclassification (BPMC) of a statistical classification rule does not increase as the dimension or feature space increases, provided the class-conditional probability densities are known, is well-known. Extensions of LDA that incorporate information on the differences in covariance matrices are known as heteroscedastic linear dimension reduction (HLDR) methods. Using the Bayes classification procedure in which we assume equal costs of misclassification and that all class parameters are known, we determine the reduced dimension q < p that is the smallest reduced dimension for which there exists a LDR matrix B ∈ q× p that preserves all of the classification information originally contained in the p-dimensional feature space.
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