Sensible Functional Linear Discriminant Analysis

TL;DR

This paper extends Fisher's linear discriminant analysis to functional and longitudinal data, proposing an efficient method for optimal projections and demonstrating asymptotic perfect classification under certain conditions.

Contribution

It introduces a novel approach for functional LDA applicable to both dense and sparse data, addressing covariance noninvertibility and projection challenges.

Findings

Asymptotic perfect classification is achievable in certain scenarios.

The proposed method effectively handles sparse and dense functional data.

Numerical examples validate the approach's performance.

Abstract

The focus of this paper is to extend Fisher's linear discriminant analysis (LDA) to both densely re-corded functional data and sparsely observed longitudinal data for general $c$-category classification problems. We propose an efficient approach to identify the optimal LDA projections in addition to managing the noninvertibility issue of the covariance operator emerging from this extension. A conditional expectation technique is employed to tackle the challenge of projecting sparse data to the LDA directions. We study the asymptotic properties of the proposed estimators and show that asymptotically perfect classification can be achieved in certain circumstances. The performance of this new approach is further demonstrated with numerical examples.