Optimizing a Bayesian Method for Estimating the Hurst Exponent in Behavioral Sciences
The Bayesian Hurst–Kolmogorov (HK) method estimates the Hurst exponent of a time series more accurately than the age-old Detrended Fluctuation Analysis (DFA), especially when the time series is short. However, this advantage comes at the cost of computation time. The computation time increases expon...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/6/421 |
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| Summary: | The Bayesian Hurst–Kolmogorov (HK) method estimates the Hurst exponent of a time series more accurately than the age-old Detrended Fluctuation Analysis (DFA), especially when the time series is short. However, this advantage comes at the cost of computation time. The computation time increases exponentially with the time series length <i>N</i>, easily exceeding several hours for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1024</mn></mrow></semantics></math></inline-formula>, limiting the utility of the HK method in real-time paradigms, such as biofeedback and brain–computer interfaces. To address this issue, we have provided data on the estimation accuracy of the Hurst exponent <i>H</i> for synthetic time series as a function of a priori known values of <i>H</i>, the time series length, and the simulated sample size from the posterior distribution <i>n</i>—a critical step in the Bayesian estimation method. The simulated sample from the posterior distribution as small as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>25</mn></mrow></semantics></math></inline-formula> suffices to estimate <i>H</i> with reasonable accuracy for a time series as short as 256. Using a larger simulated sample from the posterior distribution—that is, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>50</mn></mrow></semantics></math></inline-formula>—provides only a marginal gain in accuracy, which might not be worth trading off with computational efficiency. Results from empirical time series on stride-to-stride intervals in humans walking and running on a treadmill and overground corroborate these findings—specifically, allowing reproduction of the rank order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>H</mi><mo>^</mo></mover></semantics></math></inline-formula> for time series containing as few as 32 data points. We recommend balancing the simulated sample size from the posterior distribution of <i>H</i> with the user’s computational resources, advocating for a minimum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>50</mn></mrow></semantics></math></inline-formula>. Larger sample sizes can be considered based on time and resource constraints when employing the HK process to estimate the Hurst exponent. The present results allow the reader to make judgments to optimize the Bayesian estimation of the Hurst exponent for real-time applications. |
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| ISSN: | 2075-1680 |