Empirical anomaly measure for finite-variance processes

Abstract

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.