A note on Bayesian wavelet-based estimation of scaling

TL;DR

This paper introduces a Bayesian method for estimating the Hurst exponent in fractional Brownian motion, incorporating prior knowledge to improve accuracy and robustness, demonstrated through simulations and turbulence data analysis.

Contribution

It presents a novel Bayesian estimation approach that uses prior information about the Hurst exponent, enhancing estimation accuracy over traditional methods.

Findings

Bayesian approach improves estimation accuracy.

Method is robust to prior misspecification.

Application to turbulence data confirms theoretical Hurst value.

Abstract

A number of phenomena in various fields such as geology, atmospheric sciences, economics, to list a few, can be modeled as a fractional Brownian motion indexed by Hurst exponent $H$. This exponent is related to the degree of regularity and self-similarity present in the signal, and it often captures important characteristics useful in various applications. Given its importance, a number of methods have been developed for the estimation of the Hurst exponent. Typically, the proposed methods do not utilize prior information about scaling of a signal. Some signals are known to possess a theoretical value of the Hurst exponent, which motivates us to propose a Bayesian approach that incorporates this information via a suitable elicited prior distribution on $H$. This significantly improves the accuracy of the estimation, as we demonstrate by simulations. Moreover, the proposed method is robust to small misspecifications of the prior location. The proposed method is applied to a turbulence time series for which Hurst exponent is theoretically known by Kolmogorov's K41 theory.