Kernel Two-Sample and Independence Tests for Non-Stationary Random Processes

TL;DR

This paper extends kernel-based two-sample and independence tests to non-stationary processes by leveraging multiple realizations, enabling effective testing on complex time-series data with improved power over existing methods.

Contribution

It introduces a novel approach to apply MMD and HSIC to non-stationary data using multiple realizations, with optimized kernel selection for enhanced test performance.

Findings

Superior test power on synthetic data

Effective application to real socio-economic data

Outperforms existing state-of-the-art tests

Abstract

Two-sample and independence tests with the kernel-based MMD and HSIC have shown remarkable results on i.i.d. data and stationary random processes. However, these statistics are not directly applicable to non-stationary random processes, a prevalent form of data in many scientific disciplines. In this work, we extend the application of MMD and HSIC to non-stationary settings by assuming access to independent realisations of the underlying random process. These realisations - in the form of non-stationary time-series measured on the same temporal grid - can then be viewed as i.i.d. samples from a multivariate probability distribution, to which MMD and HSIC can be applied. We further show how to choose suitable kernels over these high-dimensional spaces by maximising the estimated test power with respect to the kernel hyper-parameters. In experiments on synthetic data, we demonstrate superior performance of our proposed approaches in terms of test power when compared to current state-of-the-art functional or multivariate two-sample and independence tests. Finally, we employ our methods on a real socio-economic dataset as an example application.