Markov properties in presence of measurement noise

TL;DR

This paper investigates how measurement noise affects the Markov properties of stochastic processes, revealing that noise can artificially violate Markov assumptions and impact the reconstruction of stochastic differential equations from data.

Contribution

It demonstrates that measurement noise can spoil Markov properties, suggesting that small-scale limitations may be due to noise rather than intrinsic dynamics, aiding future data analysis techniques.

Findings

Measurement noise destroys Markov properties in data.

Small-scale limitations may be due to noise, not intrinsic process features.

Implications for reconstructing stochastic processes from noisy data.

Abstract

Recently, several powerful tools for the reconstruction of stochastic differential equations from measured data sets have been proposed [e.g. Siegert et al., Physics Letters A 243, 275 (1998); Hurn et al., Journal of Time Series Analysis 24, 45 (2003)]. Efficient application of the methods, however, generally requires Markov properties to be fulfilled. This constraint typically seems to be violated on small scales, which frequently is attributed to physical effects. On the other hand, measurement noise such as uncorrelated measurement and discretization errors has large impacts on the statistics of measurements on small scales. We demonstrate, that the presence of measurement noise, likewise, spoils Markov properties of an underlying Markov processes. This fact is promising for the further development of techniques for the reconstruction of stochastic processes from measured data, since limitations at small scales might stem from artificial noise sources rather than from intrinsic properties of the dynamics of the underlying process. Measurement noise, however, can be controlled much better than the intrinsic dynamics of the underlying process.