On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points
Abstract
The long range dependence of the fractional Brownian motion (fBm), fractional Gaussian noise (fGn), and differentiated fGn (DfGn) is described by the Hurst exponent H. Considering the realizations of these three processes as time series, they might be described by their statistical features, such as half of the ratio of the mean square successive difference to the variance, A, and the number of turning points, T. This paper investigates the relationships between A and H, and between T and H. It is found numerically that the formulae A(H) = aebH in case of fBm, and A(H) = a + bHc for fGn and DfGn, describe well the A(H) relationship. When T(H) is considered, no simple formula is found, and it is empirically found that among polynomials, the fourth and second order description applies best. The most relevant finding is that when plotted in the space of (A , T) , the three process types form separate branches. Hence, it is examined whether A and T may serve as Hurst exponent indicators. Some real world data (stock market indices, sunspot numbers, chaotic time series) are analyzed for this purpose, and it is found that the H's estimated using the H(A) relations (expressed as inverted A(H) functions) are consistent with the H's extracted with the well known wavelet approach. This allows to efficiently estimate the Hurst exponent based on fast and easy to compute A and T, given that the process type: fBm, fGn or DfGn, is correctly classified beforehand. Finally, it is suggested that the A(H) relation for fGn and DfGn might be an exact (shifted) 3 / 2 power-law.
- Publication:
-
Physica A Statistical Mechanics and its Applications
- Pub Date:
- November 2016
- DOI:
- 10.1016/j.physa.2016.06.004
- arXiv:
- arXiv:1512.02928
- Bibcode:
- 2016PhyA..461..662T
- Keywords:
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- Time series;
- Hurst exponent;
- Abbe value;
- Turning points;
- Physics - Data Analysis;
- Statistics and Probability;
- Mathematics - Dynamical Systems;
- Nonlinear Sciences - Adaptation and Self-Organizing Systems
- E-Print:
- 20 pages in one-column format, 7 figures