  Development of metrics for structural data-generating mechanisms is
fundamental in machine learning and the related fields. In this paper, we give
a general framework to construct metrics on random nonlinear dynamical systems,
defined with the Perron-Frobenius operators in vector-valued reproducing kernel
Hilbert spaces (vvRKHSs). We employ vvRKHSs to design mathematically manageable
metrics and also to introduce operator-valued kernels, which enables us to
handle randomness in systems. Our metric provides an extension of the existing
metrics for deterministic systems, and gives a specification of the kernel
maximal mean discrepancy of random processes. Moreover, by considering the
time-wise independence of random processes, we clarify a connection between our
metric and the independence criteria with kernels such as Hilbert-Schmidt
independence criteria. We empirically illustrate our metric with synthetic
data, and evaluate it in the context of the independence test for random
processes. We also evaluate the performance with real time seris datas via
clusering tasks.
