Functional Nonlinear Learning

Abstract

Using representations of functional data can be more convenient and beneficial in subsequent statistical models than direct observations. These representations, in a lower-dimensional space, extract and compress information from individual curves. The existing representation learning approaches in functional data analysis usually use linear mapping in parallel to those from multivariate analysis, for example, functional principal component analysis (FPCA). However, functions, as infinite-dimensional objects, sometimes have nonlinear structures that cannot be uncovered by linear mapping. Linear methods will be more overwhelmed by multivariate functional data. For that matter, this article proposes a functional nonlinear learning (FunNoL) method to sufficiently represent multivariate functional data in a lower-dimensional feature space. Furthermore, we merge a classification model for enriching the ability of representations in predicting curve labels. Hence, representations from FunNoL can be used for both curve reconstruction and classification. Additionally, we have endowed the proposed model with the ability to address the missing observation problem as well as to further denoise observations. The resulting representations are robust to observations that are locally disturbed by uncontrollable random noises. We apply the proposed FunNoL method to several real datasets and show that FunNoL can achieve better classifications than FPCA, especially in the multivariate functional data setting. Simulation studies have shown that FunNoL provides satisfactory curve classification and reconstruction regardless of data sparsity. Supplementary materials for this article are available online.

Keywords:

• Curve classification
• Feature mapping
• Functional data analysis
• Neural networks

Supplementary Materials

A supplementary document includes the additional application and simulation results, proofs of the generalization bound, and some technical details.

Acknowledgments

The authors would like to thank the editor, the associate editor, and one referee for many insightful comments. These comments are very helpful for us to improve our work.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

This research is supported by the Discovery grant (RGPIN-2023-04057) to J.Cao from the Natural Sciences and Engineering Research Council of Canada (NSERC).