Noise-Driven Return Statistics: Scaling and Truncation in Stochastic Storage Processes

Tomás Aquino,Gavan Mcgrath,Diogo Bolster,Antoine Aubeneau,Suresh Rao

doi:10.1038/s41598-017-00451-x

Tomás Aquino, Gavan Mcgrath + Show 3 more

Open Access

https://doi.org/10.1038/s41598-017-00451-x

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Journal: Scientific Reports	Publication Date: Mar 22, 2017
Citations: 7	License type: open-access

Affiliation: University of Notre Dame

Abstract

In countless systems, subjected to variable forcing, a key question arises: how much time will a state variable spend away from a given threshold? When forcing is treated as a stochastic process, this can be addressed with first return time distributions. While many studies suggest exponential, double exponential or power laws as empirical forms, we contend that truncated power laws are natural candidates. To this end, we consider a minimal stochastic mass balance model and identify a parsimonious mechanism for the emergence of truncated power law return times. We derive boundary-independent scaling and truncation properties, which are consistent with numerical simulations, and discuss the implications and applicability of our findings.

Highlights

Many natural and engineered systems may be conceptualized as perturbed by external noise
Here we focus on first return times, our framework may be adapted to describe first passage times where the initial and target levels differ, and to discriminate between excursions above and below the target
We provide a mechanistic explanation for the emergence of truncated power law behavior in return times for systems described by a noise-driven mass balance

Summary

Storage Processes

Tomás Aquino 1, Antoine Aubeneau 2, Gavan McGrath[3], Diogo Bolster1 & Suresh Rao[2]. Truncated power laws (TPLs) represent a natural choice to model such systems, as they strike a balance between finite moments arising from the truncation of the process and heavy tailing characteristics. They are well founded with sound theoretical bases[18]. Characteristics of the processes involved, rather than representing useful fitting distributions We demonstrate their emergence rigorously and theoretically in a parsimonious model: a system with a fixed loss rate and stochastic input modeled as shot noise, which is an idealized representation of many of the natural and engineered systems mentioned above. Our theoretical analysis is consistent with results of numerical simulations presented here

Conceptual Model

First return time densities

Mean first return times

Discussion and Conclusions

Author Contributions

Additional Information