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Two-sample and independence tests with the kernel-based MMD and HSIC haveshown remarkable results on i.i.d. data and stationary random processes.However, these statistics are not directly applicable to non-stationary randomprocesses, a prevalent form of data in many scientific disciplines. In thiswork, we extend the application of MMD and HSIC to non-stationary settings byassuming access to independent realisations of the underlying random process.These realisations - in the form of non-stationary time-series measured on thesame temporal grid - can then be viewed as i.i.d. samples from a multivariateprobability distribution, to which MMD and HSIC can be applied. We further showhow to choose suitable kernels over these high-dimensional spaces by maximisingthe estimated test power with respect to the kernel hyper-parameters. Inexperiments on synthetic data, we demonstrate superior performance of ourproposed approaches in terms of test power when compared to currentstate-of-the-art functional or multivariate two-sample and independence tests.Finally, we employ our methods on a real socio-economic dataset as an exampleapplication.

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