Geometric and projection effects in Kramers-Moyal analysis

TL;DR

This paper investigates how geometric projection effects influence the estimation of Kramers-Moyal coefficients in analyzing nonlinear stochastic time series, demonstrating the method's robustness and limitations through biological-inspired examples.

Contribution

It introduces a non-stochastic projection operator method to predict projection effects and compares these predictions with numerical simulations, enhancing understanding of Kramers-Moyal analysis.

Findings

Projection effects can significantly alter Kramers-Moyal coefficient estimates.

The method remains useful even when projections are non-Markovian.

Primary examples are close to Markovian, supporting the method's applicability.

Abstract

Kramers-Moyal coefficients provide a simple and easily visualized method with which to analyze stochastic time series, particularly nonlinear ones. One mechanism that can affect the estimation of the coefficients is geometric projection effects. For some biologically-inspired examples, these effects are predicted and explored with a non-stochastic projection operator method, and compared with direct numerical simulation of the systems' Langevin equations. General features and characteristics are identified, and the utility of the Kramers-Moyal method discussed. Projections of a system are in general non-Markovian, but here the Kramers-Moyal method remains useful, and in any case the primary examples considered are found to be close to Markovian.