Analysis of fractional Gaussian noises using level crossing method

TL;DR

This paper uses level crossing analysis on fractional Gaussian noises to interpret the results and introduces a new empirical function relating the Hurst exponent to crossing frequency changes, with applications to financial data.

Contribution

It provides a novel interpretation framework for level crossing results using fractional Gaussian noises and establishes an empirical relation between the Hurst exponent and crossing frequency change.

Findings

Established an empirical function for Hurst exponent versus crossing change

Applied the method to financial series for practical insights

Demonstrated the interpretation of level crossing analysis in stochastic processes

Abstract

The so-called level crossing analysis has been used to investigate the empirical data set. But there is a lack of interpretation for what is reflected by the level crossing results. The fractional Gaussian noise as a well-defined stochastic series could be a suitable benchmark to make the level crossing findings more sense. In this article, we calculated the average frequency of upcrossing for a wide range of fractional Gaussian noises from logarithmic (zero Hurst exponent, H=0), to Gaussian, H=1, ($0<H<1$). By introducing the relative change of the total numbers of upcrossings for original data with respect to so-called shuffled one, $\mathcal{R}$, an empirical function for the Hurst exponent versus $\mathcal{R}$ has been established. Finally to make the concept more obvious, we applied this approach to some financial series.