Time-series thresholding and the definition of avalanche size

Abstract

Avalanches whose sizes and durations are distributed as power laws appear in many contexts, from physics to geophysics and biology. Here we show that there is a hidden peril in thresholding continuous times series - from either empirical or synthetic data - for the identification of avalanches. In particular, we consider two possible alternative definitions of avalanche size used, e.g., in the empirical determination of avalanche exponents in the analysis of neural-activity data. By performing analytical and computational studies of an Ornstein-Uhlenbeck process (taken as a guiding example) we show that (1) if relatively large threshold values are employed to determine the beginning and ending of avalanches and (2) if - as sometimes done in the literature - avalanche sizes are defined as the total area (above zero) of the avalanche, then true asymptotic scaling behavior is not seen, instead the observations are dominated by transient effects. This problem - that we have detected in some recent works - leads to misinterpretations of the resulting scaling regimes.