Preprocessing noisy functional data: a multivariate perspective

TL;DR

This paper presents a multivariate approach to preprocessing noisy functional data, revealing underlying factor structures and improving principal component estimation, while also addressing noise hypothesis testing.

Contribution

It introduces a factor model perspective for functional data, enhancing estimation accuracy and challenging the common iid noise assumption.

Findings

Latent signals can be extracted via factor models.

Principal components are better estimated with a multivariate approach.

The iid noise assumption is often invalid in practice.

Abstract

We consider functional data which are measured on a discrete set of observation points. Often such data are measured with additional noise. We explore in this paper the factor structure underlying this type of data. We show that the latent signal can be attributed to the common components of a corresponding factor model and can be estimated accordingly, by borrowing methods from factor model literature. We also show that principal components, which play a key role in functional data analysis, can be accurately estimated after taking such a multivariate instead of a `functional' perspective. In addition to the estimation problem, we also address testing of the null-hypothesis of iid noise. While this assumption is largely prevailing in the literature, we believe that it is often unrealistic and not supported by a residual analysis.