On the detection of superdiffusive behaviour in time series

TL;DR

The paper introduces a novel, versatile method for detecting superdiffusive behavior in various time series, effectively handling both stochastic and deterministic data without prior knowledge, and automatically removing linear drift effects.

Contribution

It presents a new detection technique for superdiffusion applicable to single realizations of data, outperforming traditional methods in versatility and automatic drift removal.

Findings

Effective in identifying superdiffusive behavior in diverse data types

Automatically removes linear drift without preprocessing

Outperforms standard estimation methods in numerical tests

Abstract

We present a new method for detecting superdiffusive behaviour and for determining rates of superdiffusion in time series data. Our method applies equally to stochastic and deterministic time series data (with no prior knowledge required of the nature of the data) and relies on one realisation (ie one sample path) of the process. Linear drift effects are automatically removed without any preprocessing. We show numerical results for time series constructed from i.i.d. $\alpha$-stable random variables and from deterministic weakly chaotic maps. We compare our method with the standard method of estimating the growth rate of the mean-square displacement as well as the $p$-variation method, maximum likelihood, quantile matching and linear regression of the empirical characteristic function.