The generalized Langevin equation (GLE) overcomes the limiting Markov approximation of the Langevin equation by an incorporated memory kernel and can be used to model various stochastic processes in many fields of science ranging from climate modeling over neuroscience to finance. Generally, Bayesian estimation facilitates the determination of both suitable model parameters and their credibility for a measured time series in a straightforward way. In this work we develop a realization of this estimation technique for the GLE in the case of white noise. We assume piecewise constant drift and diffusion functions and represent the characteristics of the data set by only a few coefficients, which leads to a numerically efficient procedure. The kernel function is an arbitrary time-discrete function with a fixed length $K$. We show how to determine a reasonable value of $K$ based on the data. We illustrate the abilities of both the method and the model by an example from turbulence.
Download