Anomalous diffusion phenomena are observed in many areas of interest. They manifest themselves in deviations from the laws of Brownian motion (BM), e.g. in the non-linear growth (mostly power-law) in time of the ensemble average mean squared displacement (MSD). When we analyze the real-life data in the context of anomalous diffusion, the primary problem is the proper identification of the type of the anomaly. In this paper, we introduce a new statistic, called empirical anomaly measure (EAM), that can be useful for this purpose. This statistic is the sum of the off-diagonal elements of the sample autocovariance matrix for the increments process. On the other hand, it can be represented as the convolution of the empirical autocovariance function with time lags. The idea of the EAM is intuitive. It measures dependence between the ensemble-averaged MSD of a given process from the ensemble-averaged MSD of the classical BM. Thus, it can be used to measure the distance between the anomalous diffusion process and normal diffusion. In this article, we prove the main probabilistic characteristics of the EAM statistic and construct the formal test for the recognition of the anomaly type. The advantage of the EAM is the fact that it can be applied to any data trajectories without the model specification. The only assumption is the stationarity of the increments process. The complementary summary of the paper constitutes of Monte Carlo simulations illustrating the effectiveness of the proposed test and properties of EAM for selected processes.
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.
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