Learning Transfer Operators by Kernel Density Estimation

Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The usual description, which we will call the Ulam-Galerkin method, is in terms of projection onto basis functions that are characteristic functions supported over a fine grid of rectangles. In these terms, the usual Ulam-Galerkin approach can be understood as density estimation by the histogram method. Here we show that the problem can be recast in statistical density estimation formalism. This recasting of the classical problem, is a perspective that allows for an explicit and rigorous analysis of bias and variance, and therefore toward a discussion of the mean square error. Keywords: Transfer Operators; Frobenius-Perron operator; probability density estimation; Ulam-Galerkin method;Kernel Density Estimation.