An extension of Fisher's discriminant analysis for stochastic processes

Abstract

In this paper we present a general notion of Fisher's linear discriminant analysis that extends the classical multivariate concept to situations that allow for function-valued random elements. The development uses a bijective mapping that connects a second order process to the reproducing kernel Hilbert space generated by its within class covariance kernel. This approach provides a seamless transition between Fisher's original development and infinite dimensional settings that lends itself well to computation via smoothing and regularization. Simulation results and real data examples are provided to illustrate the methodology.

On local times, density estimation and supervised classification from functional data

“…Recently, Shin [39] proposed an extension of Fisher discriminant analysis for stochastic processes, refer to InfFLD, which uses a bijective mapping that connects a second-order stationary process with the reproducing kernel Hilbert space generated by its within class covariance kernel. In particular, he provides the results of a simulation study comparing the InfFLD method with the classical multivariate Fisher's (FLD), penalized discriminant analysis (PDA) using both the ridge penalty (PDA/Ridge) and a penalty matrix for cubic spline smoothing (PDA/Spline) principal components analysis (NPCD/PCA) and multivariate partial least-squares regression (NPCD/MPLSR) (nonparametric curve discrimination methods proposed by Ferraty and Vieu [16]).…”

“…In Table 1 we reproduce the results given in [39] and add the result obtained with our method, referred to as NPDE. As we can see, the NPDE method behaves slightly worse than InfFLD but better than the other methods.…”

An RKHS framework for functional data analysis

“…Shin (2008) has used an RKHS formulation to generalize the classical Fisher's method for linear discriminant analysis (LDA) to the FDA setting. Specifically, she defines the discriminant functions to be random variables ' 2 L 2 X \f0g that maximize VarðE½'jGÞ=E½Varð'jGÞ.…”

Research of Face Recognition with Fisher Linear Discriminant

“…Facial recognition is one of the pattern recognition approaches for personal identification purposes in addition to other biometric approaches such as fingerprint recognition, signature, and retina of the eye and so on [2]. The recognition of the face image is related to an object that is never the same, because of the parts that can change [2] [3] [4]. These changes can be caused by facial expressions, shooting angles, or changes in accessories on the face.…”