Analytical Determination of Fractal Structure in Stochastic Time Series

TL;DR

This paper introduces a Bayesian framework for objectively assessing fractal structures in time series by analyzing their diffusion scaling properties, providing a new criterion and a superior maximum likelihood estimator for the Hurst exponent.

Contribution

It presents the Bayesian Assessment of Scaling, an analytical method that offers objective inference on fractal structures and a new estimator for the Hurst exponent, outperforming existing methods.

Findings

The Bayesian criterion accurately characterizes the evidence for fractal structures.

The derived maximum likelihood estimator of H outperforms previous estimators.

The method is simple to compute and applicable to various time series.

Abstract

Current methods for determining whether a time series exhibits fractal structure (FS) rely on subjective assessments on estimators of the Hurst exponent (H). Here, I introduce the Bayesian Assessment of Scaling, an analytical framework for drawing objective and accurate inferences on the FS of time series. The technique exploits the scaling property of the diffusion associated to a time series. The resulting criterion is simple to compute and represents an accurate characterization of the evidence supporting different hypotheses on the scaling regime of a time series. Additionally, a closed-form Maximum Likelihood estimator of H is derived from the criterion, and this estimator outperforms the best available estimators.