Framework for distinguishing anomalous diffusion models with constant and random parameters using statistical testing procedures

Anomalous diffusion refers to processes where the mean squared displacement grows non-linearly with time, following the relation E(X^2(t))~t^β, with β representing the anomalous exponent. This type of behavior, observed in complex systems like biological cells, often deviates from traditional diffusion models. Classical approaches, such as the fractional Brownian motion (FBM) and scaled Brownian motion (SBM), assume fixed exponents, which do not account for dynamics with varying anomalous parameters. To overcome this limitation, models like FBM with random exponents (FBMRE) and SBM with random exponents (SBMRE) have been developed. This work presents a universal procedure based on statistical testing to distinguish between anomalous diffusion models with constant and random anomalous exponents. This is done using time-averaged statistics and their ratio-based counterparts. In addition, a novel approach to optimizing time-lag selection using a divergence measure, specifically the Hellinger distance, is proposed. The methodology is widely applicable to distinguish constant from random anomalous diffusion, with its effectiveness depending on the choice of statistics, time lags, and process characteristics, as demonstrated through simulations (using a two-point distribution of the anomalous exponent) and analysis of real-world data.

anomalous diffusion

fractional Brownian motion

scaled Brownian motion

Monte Carlo simulations

statistical testing

real data analysis 

Computational Statistics

Symposium on Data Science and Statistics (SDSS) 2025