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Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods

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  • Caccia, David C.
  • Percival, Donald
  • Cannon, Michael J.
  • Raymond, Gary
  • Bassingthwaighte, James B.

Abstract

Precise reference signals are required to evaluate methods for characterizing a fractal time series. Here we use fGp (fractional Gaussian process) to generate exact fractional Gaussian noise (fGn) reference signals for one-dimensional time series. The average autocorrelation of multiple realizations of fGn converges to the theoretically expected autocorrelation. Two methods, commonly used to generate fractal time series, an approximate spectral synthesis (SSM) method and the successive random addition (SRA) method, do not give the correct correlation structures and should be abandoned. Time series from fGp were used to test how well several versions of rescaled range analysis (R/S) and dispersional analysis (Disp) estimate the Hurst coefficient (0 < H < 1.0). Disp is unbiased for H < 0.9 and series length N ⩾ 1024, but underestimates H when H > 0.9 R/S-detrended overestimates H for time series with H < 0.7 and underestimates H for H > 0.7. Estimates of H(Ĥ) from all versions of Disp usually have lower bias and variance than those from R/S. All versions of dispersional analysis, Disp, now tested on fGp, are better than we previously thought and are recommended for evaluating time series as long-memory processes.

Suggested Citation

  • Caccia, David C. & Percival, Donald & Cannon, Michael J. & Raymond, Gary & Bassingthwaighte, James B., 1997. "Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 246(3), pages 609-632.
  • Handle: RePEc:eee:phsmap:v:246:y:1997:i:3:p:609-632
    DOI: 10.1016/S0378-4371(97)00363-4
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    References listed on IDEAS

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    1. Chandra, Ramesh & Rohrbach, Kermit & Willinger, G Lee, 1995. "A Comparison of the Power of Parametric and Nonparametric Tests of Location Shift for Event Studies," The Financial Review, Eastern Finance Association, vol. 30(4), pages 685-710, November.
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    Cited by:

    1. Janusz Miśkiewicz & Zenon Trela & Zbigniew Burdach & Waldemar Karcz & Wanda Balińska-Miśkiewicz, 2020. "Long range correlations of the ion current in SV channels. Met3PbCl influence study," PLOS ONE, Public Library of Science, vol. 15(3), pages 1-15, March.
    2. Peter F. Craigmile, 2003. "Simulating a class of stationary Gaussian processes using the Davies–Harte algorithm, with application to long memory processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(5), pages 505-511, September.
    3. Hartmann, András & Mukli, Péter & Nagy, Zoltán & Kocsis, László & Hermán, Péter & Eke, András, 2013. "Real-time fractal signal processing in the time domain," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(1), pages 89-102.
    4. Fu, Yang & Zheng, Zeyu & Xiao, Rui & Shi, Haibo, 2017. "Comparison of two fractal interpolation methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 563-571.
    5. McGaughey, Donald R & Aitken, George J.M, 2000. "Statistical analysis of successive random additions for generating fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 277(1), pages 25-34.
    6. Biermé, Hermine & Meerschaert, Mark M. & Scheffler, Hans-Peter, 2007. "Operator scaling stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 312-332, March.
    7. Turvey, Calum G., 2007. "A note on scaled variance ratio estimation of the Hurst exponent with application to agricultural commodity prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 377(1), pages 155-165.
    8. Mante, Claude, 2007. "Application of resampling and linear spline methods to spectral and dispersional analyses of long-memory processes," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4308-4323, May.
    9. Almurad, Zainy M.H. & Delignières, Didier, 2016. "Evenly spacing in Detrended Fluctuation Analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 451(C), pages 63-69.
    10. Alvarez-Ramirez, Jose & Echeverria, Juan C. & Rodriguez, Eduardo, 2008. "Performance of a high-dimensional R/S method for Hurst exponent estimation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(26), pages 6452-6462.
    11. McGaughey, Donald R. & Aitken, G.J.M., 2002. "Generating two-dimensional fractional Brownian motion using the fractional Gaussian process (FGp) algorithm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 311(3), pages 369-380.

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