Testing Separability of Functional Time Series

TL;DR

This paper develops a statistical test to verify if a panel of functional time series has a separable covariance structure, which simplifies modeling and enhances computational efficiency in functional data analysis.

Contribution

The paper introduces a new significance test for separability in functional time series, applicable to autocovariances, with theoretical justification and practical validation.

Findings

Test controls size well and has high power

Applicable to autocovariance separability

Validated through simulations and real data applications

Abstract

We derive and study a significance test for determining if a panel of functional time series is separable. In the context of this paper, separability means that the covariance structure factors into the product of two functions, one depending only on time and the other depending only on the coordinates of the panel. Separability is a property which can dramatically improve computational efficiency by substantially reducing model complexity. It is especially useful for functional data as it implies that the functional principal components are the same for each member of the panel. However such an assumption must be verified before proceeding with further inference. Our approach is based on functional norm differences and provides a test with well controlled size and high power. We establish our procedure quite generally, allowing one to test separability of autocovariances as well. In addition to an asymptotic justification, our methodology is validated by a simulation study. It is applied to functional panels of particulate pollution and stock market data.