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.