Nonlinear Functional Principal Component Analysis Using Neural Networks

TL;DR

This paper introduces a neural network-based nonlinear FPCA method for functional data analysis, improving upon classical linear FPCA especially when data exhibits nonlinear structures.

Contribution

The paper develops a novel nonlinear FPCA approach using neural networks, extending traditional linear FPCA to handle complex nonlinear functional data.

Findings

Effective curve reconstruction demonstrated in simulations

Superior performance on real-world datasets

Universal approximation property of the proposed networks

Abstract

Functional principal component analysis (FPCA) is an important technique for dimension reduction in functional data analysis (FDA). Classical FPCA method is based on the Karhunen-Lo\`{e}ve expansion, which assumes a linear structure of the observed functional data. However, the assumption may not always be satisfied, and the FPCA method can become inefficient when the data deviates from the linear assumption. In this paper, we propose a novel FPCA method that is suitable for data with a nonlinear structure by neural network approach. We construct networks that can be applied to functional data and explore the corresponding universal approximation property. The main use of our proposed nonlinear FPCA method is curve reconstruction. We conduct a simulation study to evaluate the performance of our method. The proposed method is also applied to two real-world data sets to further demonstrate its superiority.