Operator Fitting for Parameter Estimation of Stochastic Differential Equations
For researchers modeling stochastic dynamical systems, this offers a potentially more efficient parameter estimation approach, though tested only on simple systems and incremental in nature.
The paper proposes a method for parameter estimation in stochastic differential equations by matching a finite-dimensional Koopman operator approximation with an extended dynamic mode decomposition approximation, achieving objective evaluation cost independent of sample size for some systems. Tests on two simple systems show competitive performance compared to benchmark techniques.
Estimation of parameters is a crucial part of model development. When models are deterministic, one can minimise the fitting error; for stochastic systems one must be more careful. Broadly parameterisation methods for stochastic dynamical systems fit into maximum likelihood estimation- and method of moment-inspired techniques. We propose a method where one matches a finite dimensional approximation of the Koopman operator with the implied Koopman operator as generated by an extended dynamic mode decomposition approximation. One advantage of this approach is that the objective evaluation cost can be independent the number of samples for some dynamical systems. We test our approach on two simple systems in the form of stochastic differential equations, compare to benchmark techniques, and consider limited eigen-expansions of the operators being approximated. Other small variations on the technique are also considered, and we discuss the advantages to our formulation.