# A test of weak separability for multi-way functional data, with application to brain connectivity studies

## TL;DR

This paper introduces a formal test for weak separability in multi-way functional data, facilitating the analysis of complex brain connectivity patterns with applications to magnetoencephalography signals.

## Contribution

It proposes a novel statistical test for weak separability in multi-way functional data, linking tensor methods to brain connectivity analysis.

## Key findings

- The test effectively distinguishes weakly separable structures.
- Application to MEG data reveals meaningful brain connectivity insights.
- The asymptotic distribution of the test statistic is derived.

## Abstract

This paper concerns the modeling of multi-way functional data where double or multiple indices are involved. We introduce a concept of weak separability. The weakly separable structure supports the use of factorization methods that decompose the signal into its spatial and temporal components. The analysis reveals interesting connections to the usual strongly separable covariance structure, and provides insights into tensor methods for multi-way functional data. We propose a formal test for the weak separability hypothesis, where the asymptotic null distribution of the test statistic is a chi-square type mixture. The method is applied to study brain functional connectivity derived from source localized magnetoencephalography signals during motor tasks.