Scale-free to Pareto–Tsallis transitions in the distributions of waiting times : weather, sea-level, currency trading and automotive datasets

Abstract

We report a series of detailed statistical analyses on the distributions of waiting times pertaining to a diverse set of complex systems, including terrestrial and space weather, sea-level variations, currency trading (for both fiat and cryptocurrencies), and synthetic automotive data. Given a generic time series, we define a waiting time as the shortest time interval needed to find an entry of value of at least (Formula presented) (Formula presented), with (Formula presented) (Formula presented) a given threshold, after a certain entry of value (Formula presented) (Formula presented) was observed. Going through the entire time series we obtain the complete set of waiting times for a specific value of (Formula presented) (Formula presented) and can determine their distribution. This distribution can be seen as a dynamic fingerprint of the process to which the time series pertains and is particularly useful to directly compare the dynamics of otherwise very different systems. To this end, we show that the aforementioned distributions have a prominent scale-free character for small values of (Formula presented) (Formula presented) for all datasets under scrutiny, while for large values of (Formula presented) (Formula presented) the observed distributions of waiting times converge to a Pareto–Tsallis distribution. We identify the threshold values (Formula presented) (Formula presented) at which this transition occurs using the goodness-of-fit indicators, and further substantiate these results by analyzing the behavior of the generalized Kullback–Leibler divergence. Our results are robust across all of the considered datasets.