Fisher’s Discriminant for Functional Data

Abstract

Functional linear discriminant analysis has been discussed for decades. Most existing methods may rely on some impractical and unverifiable model assumptions, for example, the observed functions lie on C2 space and can be represented by a very mall number of bases (eigenfunctions). However, those assumptions may not hold in general, especially for images. We propose a Bayesian Gaussian process representation in order to extend Fisher’s discriminant analysis to functional data, especially for images. The probability structure for our extended Fisher's discriminant is explicitly formulated, and we utilize the smoothness assumptions of functional data as prior probabilities. The vague assumption on domain of observed functions is explicitly formulated as prior probabilities, and we propose a guideline to determine an appropriate prior motivated by previous studies on natural image statistics. The proposed framework can deal with complicated functions as well by careful design of prior probabilities. We also propose a maximum a posteriori probability (MAP) approach to estimate the unknowns. Existing methods which directly employ the smoothness assumption of functional data can be shown as special cases within this framework given corresponding priors while their estimates of the unknowns are one-step approximations to the proposed MAP estimates. Experimental results on the Yale face database and the ETH-80 object categorization dataset show that the proposed method significantly outperforms the other Fisher's discriminant methods for general image data. ?