M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond and J. B. Bassingthwaighte (1997), `Evaluating Scaled Windowed Variance Methods for Estimating the Hurst Coefficient of Time Series,' Physica A, 241, no. 3-4, pp. 606-26.
Three scaled windowed variance methods (standard, linear regression detrended, and bridge detrended) for evaluating the Hurst coefficient (H) are evaluated. The Hurst coefficient, with 0 < H < 1, characterizes self-similar decay in the time series autocorrelation function. The scaled windowed variance methods estimate H for fractional Brownian motion (fBm) signals which are cumulative sums of fractional Gaussian noise (fGn) signals. For all three methods both the bias and standard deviation of estimates are less than 0.05 for series having 512 points or more. Estimates for short series (less than 256 points) are unreliable. To have a 95% probability of distinguishing between two signals with true H differing by 0.1, more than 32,768 points are needed. All three methods proved more reliable (based on bias and variance of estimates) than Hurst's rescaled range analysis, periodogram analysis, and autocorrelation analysis, and as reliable as dispersional analysis. These latter methods can only be applied to fGn or differences of fBm, while the scaled windowed variance methods must be applied to fBm or cumulative sums of fGn.
Fractals; Fractional Gaussian noise; Fractional Brownian motion; Autocorrelation; Covariance; Long memory process; Dispersional analysis
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