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.

Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories

“…Since each of these models describes different physical causes for anomalous diffusion, identifying the best-fitting stochastic model is an important step in unraveling the physical origin of an experimentally observed anomalous diffusion. ,− , Similarly determining specific parameters attributed to each model, such as the anomalous diffusion exponent α and coefficient K α , can help quantify and/or differentiate between trajectories or systems. , Typically this task is tackled through the use of statistical observables, aiming at quantifying the expected differences between the models. ,− However, the stochastic nature of these models in combination with the often noisy and limited experimental data can severely hinder this process and may lead to conflicting results from different observables. For example, it has been shown that noisy data can lead to a mistaken identification as subdiffusion. , …”

“…The aforementioned methods cover only a fraction of possibilities. Other techniques not further specified here include the use of the velocity autocorrelation, the first passage statistics, the codifference, or the autocovariance . The applications of these statistical techniques, however, struggle when data is sparse and often require an ensemble of trajectories. , …”

“…Other techniques not further specified here include the use of the velocity autocorrelation, 47 the first passage statistics, 75 the codifference, 76 or the autocovariance. 77 The applications of these statistical techniques, however, struggle when data is sparse and often require an ensemble of trajectories. 74,79 As an alternative approach to classify SPT data, Thapa et al demonstrated that Bayesian inference may be used to determine the best-fitting model and its parameters directly from the position data of an SPT experiment using their mathematical description.…”

Preface: characterisation of physical processes from anomalous diffusion data

“…Transient anomalous diffusion is also obtained in a tempered fractionally integrated process [33]. Maraj et al [34] introduces the empirical anomaly measure as a means to measure the distance between the anomalous diffusion process and normal diffusion. Limit properties of Lévy walks are shown to be useful in the recognition and verification of Lévy walk-type motion, as well as the parameter estimation in maximum likelihood methods [35].…”

Preface: Characterisation of Physical Processes from Anomalous Diffusion Data

“…Ref. [33] introduces the empirical anomaly measure as a means to measure the distance between the anomalous diffusion process and normal diffusion. Limit properties of Lévy walks are shown to be useful in the recognition and verification of Lévy walk-type motion, as well as the parameter estimation in maximum likelihood methods [34].…”