Recurrence statistics for anomalous diffusion regime change detection

For many real-time series, specific behaviors are observed where the character of the time series changes over time. This temporal evolution may indicate that some properties of the data evolve or fluctuate. One can find such problems in many different applications including physical and biological experiments as well as in technical diagnostics. From the mathematical point of view, this complexity can be considered as a segmentation problem, i.e. extraction of the homogeneous parts from the original data. Most segmentation methods assume that a simple characteristic of the time series changes, for example the mean or the variance. However, many physical applications involve a more complex situation dealing with transient statistics. Here, a new technique of the critical change point detection is introduced for the case when the data consist of anomalous diffusion processes with transient anomalous diffusion exponents. The precise mathematical formulation of a new statistics based on recurrence statistics is provided. The proposed recurrence analysis counts the number of data points falling into the appropriate circle built from consecutive observations. This approach proves to be helpful in recognizing subdiffusive and superdiffusive regions, which characterize anomalous diffusion behaviors. The main characteristics of the recurrence statistics are presented and the application to the segmentation problem is described. The effectiveness of the proposed technique is validated for a family of classical anomalous diffusive models, namely fractional Brownian motion. Finally, the methodology is applied to biological data exhibiting anomalous diffusion behavior with transient anomalous diffusion exponents.