Asymmetric Anomalous Diffusion: an Efficient Way to Detect Memory in Time Series

Abstract

We study time series concerning rare events. The occurrence of a rare event is depicted as a jump of constant intensity always occurring in the same direction, thereby generating an asymmetric diffusion process. We consider the case where the waiting time distribution is an inverse power law with index $μ$. We focus our attention on $μ<3$, and we evaluate the scaling $δ$ of the resulting diffusion process. We prove that $δ$ gets its maximum value, $δ=1$, corresponding to the ballistic motion, at $μ=2$. We study the resulting diffusion process by means of joint use of the continuous time random walk and of the generalized central limit theorem, as well as adopting numerical treatment. We show that rendering asymmetric the diffusion process yelds the significant benefit of enhancing the value of the scaling parameter $δ$. Furthermore, this scaling parameter becomes sensitive to the power index $μ$ in the whole region $1<μ<3$. Finally, we show our method in action on real data concerning human heartbeat sequences.