  In the paper, we consider the problem of robust approximation of transfer
Koopman and Perron-Frobenius (P-F) operators from noisy time series data. In
most applications, the time-series data obtained from simulation or experiment
is corrupted with either measurement or process noise or both. The existing
results show the applicability of algorithms developed for the finite
dimensional approximation of deterministic system to a random uncertain case.
However, these results hold true only in asymptotic and under the assumption of
infinite data set. In practice the data set is finite, and hence it is
important to develop algorithms that explicitly account for the presence of
uncertainty in data-set. We propose a robust optimization-based framework for
the robust approximation of the transfer operators, where the uncertainty in
data-set is treated as deterministic norm bounded uncertainty. The robust
optimization leads to a min-max type optimization problem for the approximation
of transfer operators. This robust optimization problem is shown to be
equivalent to regularized least square problem. This equivalence between robust
optimization problem and regularized least square problem allows us to comment
on various interesting properties of the obtained solution using robust
optimization. In particular, the robust optimization formulation captures
inherent tradeoffs between the quality of approximation and complexity of
approximation. These tradeoffs are necessary to balance for the proposed
application of transfer operators, for the design of optimal predictor.
Simulation results demonstrate that our proposed robust approximation algorithm
performs better than the Extended Dynamic Mode Decomposition (EDMD) and DMD
algorithms for a system with process and measurement noise.
