Estimation of Kramers-Moyal coefficients at low sampling rates.

An optimization procedure for the estimation of Kramers-Moyal coefficients from stationary, one-dimensional, Markovian time series data is presented. The method takes advantage of a recently reported approach that allows one to calculate exact finite sampling interval effects by solving the adjoint Fokker-Planck equation. Therefore, it is well suited for the analysis of sparsely sampled time series. The optimization can be performed either by making a parametric ansatz for drift and diffusion functions or parameter free. We demonstrate the power of the method in several numerical examples with synthetic time series.