Precision in Brief: The Bayesian Hurst–Kolmogorov Method for the Assessment of Long-Range Temporal Correlations in Short Behavioral Time Series

Abstract

Various fields within biological and psychological inquiry recognize the significance of exploring long-range temporal correlations to study phenomena. However, these fields face challenges during this transition, primarily stemming from the impracticality of acquiring the considerably longer time series demanded by canonical methods. The Bayesian Hurst–Kolmogorov (HK) method estimates the Hurst exponents of time series—quantifying the strength of long-range temporal correlations or “fractality”—more accurately than the canonical detrended fluctuation analysis (DFA), especially when the time series is short. Therefore, the systematic application of the HK method has been encouraged to assess the strength of long-range temporal correlations in empirical time series in behavioral sciences. However, the Bayesian foundation of the HK method fuels reservations about its performance when artifacts corrupt time series. Here, we compare the HK method’s and DFA’s performance in estimating the Hurst exponents of synthetic long-range correlated time series in the presence of additive white Gaussian noise, fractional Gaussian noise, short-range correlations, and various periodic and non-periodic trends. These artifacts can affect the accuracy and variability of the Hurst exponent and, therefore, the interpretation and generalizability of behavioral research findings. We show that the HK method outperforms DFA in most contexts—while both processes break down for anti-persistent time series, the HK method continues to provide reasonably accurate H values for persistent time series as short as N=64 samples. Not only can the HK method detect long-range temporal correlations accurately, show minimal dispersion around the central tendency, and not be affected by the time series length, but it is also more immune to artifacts than DFA. This information becomes particularly valuable in favor of choosing the HK method over DFA, especially when acquiring a longer time series proves challenging due to methodological constraints, such as in studies involving psychological phenomena that rely on self-reports. Moreover, it holds significance when the researcher foreknows that the empirical time series may be susceptible to contamination from these processes.

Keywords

• detrended fluctuation analysis
• fractal fluctuation
• fractional
• Hurst–Kolmogorov process
• long-range correlation
• physiology