  Evaluating whether data streams are drawn from the same distribution is at
the heart of various machine learning problems. This is particularly relevant
for data generated by dynamical systems since such systems are essential for
many real-world processes in biomedical, economic, or engineering systems.
While kernel two-sample tests are powerful for comparing independent and
identically distributed random variables, no established method exists for
comparing dynamical systems. The main problem is the inherently violated
independence assumption. We propose a two-sample test for dynamical systems by
addressing three core challenges: we (i) introduce a novel notion of mixing
that captures autocorrelations in a relevant metric, (ii) propose an efficient
way to estimate the speed of mixing relying purely on data, and (iii) integrate
these into established kernel two-sample tests. The result is a data-driven
method that is straightforward to use in practice and comes with sound
theoretical guarantees. In an example application to anomaly detection from
human walking data, we show that the test is readily applicable without any
human expert knowledge and feature engineering.
