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 realisations 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) = ae bH in case of fBm, and A(H) = a + bH c 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.

Analytical representation of Gaussian processes in the A−T plane

“…(34) simplifies then to just ln n = n−1 n log n 4. For n = 2 14 , these values are 1.49991 and 0.14285, respectively, in perfect agreement with Fig. 3 in [14].…”

“…11(d)]. Hints that fully developed chaos behaves this way were noted in case of the Lorenz system [14,17]. With increasing r, a gradual decrease in T occurs, with wells in periodic windows [ Fig.…”

“…The Abbe value of a time series {x i } n i=1 is defined as half the ratio of the mean-square successive difference to the variance [14,[26][27][28][29]:…”

Correlation between the Hurst exponent and the maximal Lyapunov exponent: Examining some low-dimensional conservative maps

“…In a long run, those from the Lorenz system resemble a white noise. On the other hand, the time series of the Chirikov map more resemble those of a fractional Brownian motion [32]. Therefore, the correlations between the maximal Lyapunov exponents (mLEs) and H are to be examined herein.…”

Scaling Properties of Atmospheric Wind Speed in Mesoscale Range

“…H is an important parameter in the description of fluid turbulence since it is related to the auto-correlation function of the velocity time series: the rate at which C( ) decreases with lag . Moreover, for self-similar (monofractal) processes, H is directly related to fractal dimension, D, by the relation: [49]. When H = 0, the average fluctuations δu exhibit a scale dependence.…”