Statistical Inference for inter-arrival times of extreme events in bursty time series

TL;DR

This paper introduces a model for inter-arrival times of extreme events in bursty time series, showing they follow a Mittag-Leffler distribution and form a fractional Poisson process, with methods for parameter estimation and empirical validation.

Contribution

It proposes a new model for high-threshold exceedance times in bursty systems, extending the Poisson process to fractional cases with practical estimation techniques.

Findings

Inter-arrival times follow a Mittag-Leffler distribution.

The model captures the bursty, fractal-like nature of event timings.

Application to real data demonstrates improved prediction of extreme events.

Abstract

In many complex systems studied in statistical physics, inter-arrival times between events such as solar flares, trades and neuron voltages follow a heavy-tailed distribution. The set of event times is fractal-like, being dense in some time windows and empty in others, a phenomenon which has been dubbed "bursty". A new model for the inter-exceedance times of events above high thresholds is proposed. For high thresholds and infinite-mean waiting times, it is shown that the times between threshold crossings are Mittag-Leffler distributed, and thus form a "fractional Poisson Process" which generalizes the standard Poisson Process of threshold exceedances. Graphical means of estimating model parameters and assessing model fit are provided. The inference method is applied to an empirical bursty time series, and it is shown how the memory of the Mittag-Leffler distribution affects prediction of the time until the next extreme event."