Independence Testing for Temporal Data

Temporal data are increasingly prevalent in modern data science. A fundamental question is whether two time-series are related or not. Existing approaches often have limitations, such as relying on parametric assumptions, detecting only linear associations, and requiring multiple tests and corrections. While many non-parametric and universally consistent dependence measures have recently been proposed, directly applying them to temporal data can inflate the p-value and result in invalid test. To address these challenges, this paper introduces the temporal dependence statistic with block permutation to test independence between temporal data. Under proper assumptions, the proposed procedure is asymptotically valid and universally consistent for testing independence between stationary time-series, and capable of estimating the optimal dependence lag that maximizes the dependence. Notably, it is compatible with a rich family of distance and kernel based dependence measures, eliminates the need for multiple testing, and demonstrates superior power in multivariate, low sample size, and nonlinear settings. An analysis of neural connectivity with fMRI data reveals various temporal dependence among signals within the visual network and default mode network.