Zero-Crossing Statistics for Non-Markovian Time Series

TL;DR

This paper develops an analytical framework for calculating zero-crossing statistics in non-Markovian time series, extending classical results and providing new insights into complex stochastic processes.

Contribution

It introduces the Independent Interval Approximation to derive higher-order zero-crossing cumulants for non-Markovian processes, surpassing the limitations of Rice's formula.

Findings

Analytic expressions match well with simulations.

Extends zero-crossing analysis beyond Gaussian stationary processes.

Provides a new tool for analyzing complex stochastic signals.

Abstract

In applications spaning from image analysis and speech recognition, to energy dissipation in turbulence and time-to failure of fatigued materials, researchers and engineers want to calculate how often a stochastic observable crosses a specific level, such as zero. At first glance this problem looks simple, but it is in fact theoretically very challenging. And therefore, few exact results exist. One exception is the celebrated Rice formula that gives the mean number of zero-crossings in a fixed time interval of a zero-mean Gaussian stationary processes. In this study we use the so-called Independent Interval Approximation to go beyond Rice's result and derive analytic expressions for all higher-order zero-crossing cumulants and moments. Our results agrees well with simulations for the non-Markovian autoregressive model.