Estimation of drift and diffusion functions from time series data: A maximum likelihood framework

TL;DR

This paper introduces a maximum likelihood framework for estimating drift and diffusion functions from time series data, improving efficiency and accuracy in modeling stochastic dynamics of complex systems.

Contribution

It develops a Bayesian likelihood-based estimation framework with approximations that enhance efficiency and address applicability and accuracy issues in stochastic differential equation estimation.

Findings

Framework provides confidence regions for estimated functions

Approximations significantly improve estimation efficiency

Addresses key problems in parameter applicability and accuracy

Abstract

Complex systems are characterized by a huge number of degrees of freedom often interacting in a non-linear manner. In many cases macroscopic states, however, can be characterized by a small number of order parameters that obey stochastic dynamics in time. Recently techniques for the estimation of the corresponding stochastic differential equations from measured data have been introduced. This contribution develops a framework for the estimation of the functions and their respective (Bayesian posterior) confidence regions based on likelihood estimators. In succession approximations are introduced that significantly improve the efficiency of the estimation procedure. While being consistent with standard approaches to the problem this contribution solves important problems concerning the applicability and the accuracy of estimated parameters.