Analytical representation of Gaussian processes in the $\mathcal{A}-\mathcal{T}$ plane

TL;DR

This paper derives exact and approximate formulas for the paths of fractional Brownian motion and Gaussian noise in the - plane, using the fraction of turning points and the Abbe value, applicable to Gaussian processes and ARMA models.

Contribution

It provides the first closed-form expressions and accurate approximations for  and  in Gaussian processes, including fBm, fGn, and ARMA, with applications to real-world data.

Findings

Exact formulas for  in fBm using special functions

Accurate exponential approximations for  and 

Regions of - plane for ARMA processes

Abstract

Closed-form expressions, parametrized by the Hurst exponent $H$ and the length $n$ of a time series, are derived for paths of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) in the $\mathcal{A}-\mathcal{T}$ plane, composed of the fraction of turning points $\mathcal{T}$ and the Abbe value $\mathcal{A}$. The exact formula for $\mathcal{A}_{\rm fBm}$ is expressed via Riemann $\zeta$ and Hurwitz $\zeta$ functions. A very accurate approximation, yielding a simple exponential form, is obtained. Finite-size effects, introduced by the deviation of fGn's variance from unity, and asymptotic cases are discussed. Expressions for $\mathcal{T}$ for fBm, fGn, and differentiated fGn are also presented. The same methodology, valid for any Gaussian process, is applied to autoregressive moving average processes, for which regions of availability of the $\mathcal{A}-\mathcal{T}$ plane are derived and given in analytic form. Locations in the $\mathcal{A}-\mathcal{T}$ plane of some real-world examples as well as generated data are discussed for illustration.