Optimal Linear Discriminant Analysis for High-Dimensional Functional Data

Abstract

Most of existing methods of functional data classification deal with one or a few processes. In this work we tackle classification of high-dimensional functional data, in which each observation is potentially associated with a large number of functional processes, p, which is comparable to or even much larger than the sample size n. The challenge arises from the complex inter-correlation structures among multiple functional processes, instead of a diagonal correlation for a single process. Since truncation is often needed for approximation in functional data, another difficulty stems from the fact that the discriminant set of the infinite-dimensional optimal classifier may be different from that of the truncated optimal classifier, when multiple (especially a large number of) processes are involved. We bridge the gap by proposing a penalized classifier that achieves both near-perfect classification that is unique to functional data, and discriminant set inclusion consistency in the sense that the classification-responsible functional predictors include those of the underlying optimal classifier. Simulation study and real data application are carried out to demonstrate its favorable performance. Supplementary materials for this article are available online.

Keywords:

• Discriminant set inclusion
• Functional principal components
• Penalized classifier

Supplementary Materials

The supplementary material contains proofs of the main results, Theorems 1–2 as well as auxiliary lemmas and their proofs.

Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University), Ministry of Education;

Acknowledgments

The authors would like to thank the Editor, the Associate Editor and two Referees for their instructive comments that have lead improvement of the article.

Additional information

Funding

Fang Yao’s research is supported in part by the National Key R&D Program of China grants (no. 2022YFA1003801, 2020YFE0204200), the National Natural Science Foundation of China grants (no. 12292981, 11931001 and 11871080), the LMAM and the LMEQF. Kaijie Xue’s research is supported in part by the National Natural Science Foundation of China grants (no. 11901313 and 11871080), the KLMDASR, the LEBPS, and the LPMC. Jin Yang’s research is supported in part by an NIH Intramural Research Program.