Testing separability for continuous functional data

TL;DR

This paper introduces a new statistical test for assessing the separability of covariance structures in dependent functional time series, modeled in the space of continuous functions, with validated theoretical and practical results.

Contribution

The paper develops a novel test for covariance separability in continuous functional data, utilizing a supremum norm approach and a bootstrap method for dependent data.

Findings

Test effectively detects non-separability in simulations

Bootstrap method provides accurate critical values

Application demonstrates practical utility

Abstract

Analyzing the covariance structure of data is a fundamental task of statistics. While this task is simple for low-dimensional observations, it becomes challenging for more intricate objects, such as multivariate functions. Here, the covariance can be so complex that just saving a non-parametric estimate is impractical and structural assumptions are necessary to tame the model. One popular assumption for space-time data is separability of the covariance into purely spatial and temporal factors. In this paper, we present a new test for separability in the context of dependent functional time series. While most of the related work studies functional data in a Hilbert space of square integrable functions, we model the observations as objects in the space of continuous functions equipped with the supremum norm. We argue that this (mathematically challenging) setup enhances interpretability for users and is more in line with practical preprocessing. Our test statistic measures the maximal deviation between the estimated covariance kernel and a separable approximation. Critical values are obtained by a non-standard multiplier bootstrap for dependent data. We prove the statistical validity of our approach and demonstrate its practicability in a simulation study and a data example.