Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient

Abstract

The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient’s value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.

Passive tracer advection in the equatorial Pacific region: statistics, correlations and a model of fractional Brownian motion

“…Both measures have a transparent definition: in our setup, the FPT is the time when a given longitude is crossed by a parcel and the MSD is calculated as the time evolution of the squared distances from the initial location. In the second case, a refined statistics is also adopted by computing the time-averaged mean square displacement (TAMSD) (Lubel- ski et al, 2008;Kepten et al, 2015;Sikora et al, 2017;Maraj et al, 2021). In this case, the whole trajectory of a parcel is evaluated by introducing the time window parameter w, which changes from 12 h to 315 d (2 × 315 time steps in the record).…”

Passive tracer advection in the equatorial Pacific region: statistics, correlations, and a model of fractional Brownian motion

“…Both measures have a transparent definition: in our setup, FPT is the time when a given longitude is crossed by a parcel, and MSD is calculated as the time evolution of the squared distances from the initial location. In the second case, a refined statistics is also adopted by computing the Time Averaged Mean Square Displacement (TAMSD) (Lubelski et al, 2008;Kepten et al, 2015;Sikora et al, 100 2017; Maraj et al, 2021). In this case, a whole trajectory of a parcel is evaluated by introducing a time window parameter w, which changes from 12 h to 315 days (2 × 315 time steps in the record).…”