Two Sample Test for Eigendecompositions of Functional Data

TL;DR

This paper introduces a new statistical test for comparing eigendecompositions of functional data, revealing trial-to-trial variability in neural activation patterns beyond sampling noise.

Contribution

The authors develop a novel eigendecomposition comparison test for functional data, extending it to paired samples, with demonstrated superior power over existing methods.

Findings

The test effectively detects differences in latent activation patterns.

Simulation studies show the test's superior power.

Applied to neural data, it uncovers meaningful trial-to-trial variability.

Abstract

Neuron-level firing data is believed to be governed by latent activation patterns during task completion. Analysing repeated trials of a task allows us to study these patterns, typically by averaging in-vivo neural spikes across trials. However, estimates of underlying latent activation patterns show trial-to-trial variability. Our aim is to determine whether this variation arises from observed data differences or changes in the latent activation patterns themselves. The latter would imply that current approaches overlook meaningful activation changes, necessitating adjustments in dimension reduction and downstream analysis. We propose a test that compares the eigendecompositions of two samples of functional data based on the covariance matrix of scores derived from a functional principal component analysis of the pooled data. Initially developed for independent samples, we later extend the test to paired samples, as necessary for our data. Simulation studies demonstrate its superior power compared to leading methods across various scenarios. In an experiment with 157 trials, we analyse all pairwise comparisons using a permutation approach to test the null hypothesis of shared latent activation patterns across trials. Our findings reveal trial-to-trial variation in latent activation patterns that cannot be attributed to sampling noise.