  Operator-theoretic analysis of nonlinear dynamical systems has attracted much
attention in a variety of engineering and scientific fields, endowed with
practical estimation methods using data such as dynamic mode decomposition. In
this paper, we address a lifted representation of nonlinear dynamical systems
with random noise based on transfer operators, and develop a novel Krylov
subspace method for estimating the operators using finite data, with
consideration of the unboundedness of operators. For this purpose, we first
consider Perron-Frobenius operators with kernel-mean embeddings for such
systems. We then extend the Arnoldi method, which is the most classical type of
Kryov subspace method, so that it can be applied to the current case.
Meanwhile, the Arnoldi method requires the assumption that the operator is
bounded, which is not necessarily satisfied for transfer operators on nonlinear
systems. We accordingly develop the shift-invert Arnoldi method for
Perron-Frobenius operators to avoid this problem. Also, we describe an approach
of evaluating predictive accuracy by estimated operators on the basis of the
maximum mean discrepancy, which is applicable, for example, to anomaly
detection in complex systems. The empirical performance of our methods is
investigated using synthetic and real-world healthcare data.
