Better than DFA? A Bayesian Method for Estimating the Hurst Exponent in Behavioral Sciences

TL;DR

This paper compares DFA and a Bayesian Hurst-Kolmogorov method for estimating the Hurst exponent, demonstrating the Bayesian approach's superior accuracy and reliability, especially with short or empirical time series.

Contribution

It introduces and validates the Bayesian Hurst-Kolmogorov method as a more reliable alternative to DFA for estimating the Hurst exponent in behavioral sciences.

Findings

HK method outperforms DFA on synthetic data

HK provides more stable estimates regardless of series length

Empirical data analysis supports HK's advantages

Abstract

Detrended Fluctuation Analysis (DFA) is the most popular fractal analytical technique used to evaluate the strength of long-range correlations in empirical time series in terms of the Hurst exponent, $H$. Specifically, DFA quantifies the linear regression slope in log-log coordinates representing the relationship between the time series' variability and the number of timescales over which this variability is computed. We compared the performance of two methods of fractal analysis -- the current gold standard, DFA, and a Bayesian method that is not currently well-known in behavioral sciences: the Hurst-Kolmogorov (HK) method -- in estimating the Hurst exponent of synthetic and empirical time series. Simulations demonstrate that the HK method consistently outperforms DFA in three important ways. The HK method: (i) accurately assesses long-range correlations when the measurement time series is short, (ii) shows minimal dispersion about the central tendency, and (iii) yields a point estimate that does not depend on the length of the measurement time series or its underlying Hurst exponent. Comparing the two methods using empirical time series from multiple settings further supports these findings. We conclude that applying DFA to synthetic time series and empirical time series during brief trials is unreliable and encourage the systematic application of the HK method to assess the Hurst exponent of empirical time series in behavioral sciences.