Statistical test for fractional Brownian motion based on detrending moving average algorithm

TL;DR

This paper introduces a new statistical test for fractional Brownian motion using the detrending moving average method, applicable to anomalous diffusion systems, with validation through Monte Carlo simulations.

Contribution

It develops a novel statistical testing approach based on the detrending moving average statistic and its distribution, extendable to general Gaussian processes.

Findings

Test accurately distinguishes subdiffusive and superdiffusive behaviors

Distribution derived as a generalized chi-squared distribution

Validated through Monte Carlo simulations

Abstract

Motivated by contemporary and rich applications of anomalous diffusion processes we propose a new statistical test for fractional Brownian motion, which is one of the most popular models for anomalous diffusion systems. The test is based on detrending moving average statistic and its probability distribution. Using the theory of Gaussian quadratic forms we determined it as a generalized chi-squared distribution. The proposed test could be generalized for statistical testing of any centered non-degenerate Gaussian process. Finally, we examine the test via Monte Carlo simulations for two exemplary scenarios of subdiffusive and superdiffusive dynamics.