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

Area of Science:

• Time Series Analysis
• Statistical Modeling
• Stochastic Processes

Background:

• Accurate characterization of fractal time series is crucial for various scientific disciplines.
• Existing methods for generating fractal time series, such as spectral synthesis (SSM) and successive random addition (SRA), often fail to produce correct correlation structures.
• The Hurst coefficient (H) is a key parameter for quantifying fractal properties and long-range dependence.

Purpose of the Study:

• To generate exact fractional Gaussian noise (fGn) reference signals using the fractional Gaussian process (fGp) for evaluating fractal time series analysis methods.
• To compare the performance of rescaled range analysis (R/S) and dispersional analysis (Disp) in estimating the Hurst coefficient (H).
• To identify reliable methods for analyzing time series exhibiting long-memory properties.

Main Methods:

• Generation of exact fGn reference time series using fGp.
• Evaluation of two common fractal time series generation methods (SSM and SRA) for their correlation accuracy.
• Application and comparison of various rescaled range analysis (R/S) and dispersional analysis (Disp) techniques to fGp-generated series.
• Assessment of bias and variance in Hurst coefficient (H) estimations.

Main Results:

• fGp successfully generates fGn reference signals with accurate autocorrelation properties.
• SSM and SRA methods were found to be inadequate for generating fractal time series with correct correlations.
• Dispersional analysis (Disp) generally provides less biased and lower variance estimates of the Hurst coefficient (H) compared to R/S methods.
• Disp shows unbiased performance for H < 0.9 and series length N ≥ 1024, though it underestimates H for H > 0.9.
• R/S-detrended methods exhibit overestimation of H for H < 0.7 and underestimation for H > 0.7.

Conclusions:

• The fractional Gaussian process (fGp) is a reliable tool for generating precise reference signals for fractal time series analysis.
• Commonly used spectral synthesis (SSM) and successive random addition (SRA) methods should be abandoned due to inaccurate correlation structures.
• Dispersional analysis (Disp) is a recommended and robust method for evaluating time series with long-memory properties, offering superior performance over R/S methods.
• Further research may be needed to refine Disp for cases with very high Hurst coefficients (H > 0.9).