How to analyze stochastic time series obeying a 2nd order differential equation

TL;DR

This paper presents a method to accurately analyze stochastic time series governed by second order differential equations, addressing errors from differencing and handling weak measurement noise.

Contribution

It introduces an approach that accounts for differencing errors and noise, enabling precise parameter estimation for second order stochastic systems.

Findings

The method corrects systematic errors caused by differencing.

It effectively handles weak measurement noise.

Provides accurate estimation of drift and diffusion functions.

Abstract

The stochastic properties of a Langevin-type Markov process can be extracted from a given time series by a Markov analysis. Also processes that obey a stochastically forced second order differential equation can be analyzed this way by employing a particular embedding approach: To obtain a Markovian process in 2N dimensions from a non Markovian signal in N dimensions, the system is described in a phase space that is extended by the temporal derivative of the signal. For a discrete time series, however, this derivative can only be calculated by a differencing scheme, which introduces an error. If the effects of this error are not accounted for, this leads to systematic errors in the estimation of the drift- and diffusion functions of the process. In this paper we will analyze these errors and we will propose an approach that correctly accounts for them. This approach allows an accurate parameter estimation and, additionally, is able to cope with weak measurement noise, which may be superimposed to a given time series.