Testing of Multifractional Brownian Motion.

Michał Balcerek,Krzysztof Burnecki

doi:10.3390/e22121403

Michał Balcerek, Krzysztof Burnecki

Open Access

https://doi.org/10.3390/e22121403

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Abstract

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index . In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena.

Highlights

We introduce a statistical test on multifractional Brownian motion (MFBM) based on its autocovariance function (ACVF) which is presented as a quadratic form
We consider ensemble and time average mean-squared displacement (MSD) (EATAMSD) which is an average of Time average MSD (TAMSD): EATAMSD(τ ) =
We follow an approach based on the ACVF which was introduced by Balcerek and Burnecki [38]

Summary

Introduction

Massive advances in single-particle tracking (SPT), partially based on superresolution microscopy of fluorescently tagged tracers, or fluorescence correlation spectroscopy allow experimentalists to obtain insight into the motion of submicron tracer particles or even single molecules in complex environments, such as living biological cells, down to nanometer precision and at submillisecond time resolution [1,2]. Entropy 2020, 22, 1403 diffusion refers to the power-law behavior (1) with a fixed α, an increasing number of systems are reported in which the local scaling exponent of the MSD (1) is an explicit function of time, α(t) Such transient behavior has, for instance, been observed for green fluorescent proteins in cells or for the motion of lipid molecules in protein-crowded bilayer membranes [25,32]. Ryvkina [45] uses such covariance functions to define Gaussian processes to extend FBM and MFBM to a class of fractional Brownian motions with a variable Hurst parameter parameterized by a set of all measurable functions with values in (1/2, 1), and different from MFBMs. from a biological data point of view, such a range for H values is not practical since it only corresponds to a superdiffusive (long-range dependent) case.

Model and Methods

Mean-Squared Displacement

Three Cases of the Hurst Exponent Function

Results

Three Power Case Studies

Discussion and Conclusions