Generalized Hurst Hypothesis: Description of Time-Series in Communication Systems

Abstract

In this paper, we focus on the generalization of the Hurst empirical law and suggest a set of reduced parameters for quantitative description of long-time series. These series are usually considered as a specific response of a complex system (economic, geophysical, electromagnetic and other systems), where successive fixations of external factors become impossible. We consider applying generalized Hurst laws to obtain a new set of reduced parameters in data associated with communication systems. We analyze three hypotheses. The first one contains one power-law exponent. The second one incorporates two power-law exponents, which are in many cases complex-conjugated. The third hypothesis has three power-law exponents, two of which are complex-conjugated as well. These hypotheses describe with acceptable accuracy (relative error does not exceed 2%) a wide set of trendless sequences (TLS) associated with radiometric measurements. Generalized Hurst laws operate with R/S curves not only in the asymptotic region, but in the entire domain. The fitting parameters can be used as the reduced parameters for the description of the given data. The paper demonstrates that this general approach can also be applied to other TLS.

New Empirical Laws in Geosciences: A Successful Proposal

“…Apart from the technological advances that have allowed its implementation, NMR values allow estimating some properties of the subsurface, such as porosity and permeability. The most relevant formulation is the one that allows an estimation of medium permeability from the free porosity Ø mr obtained using NMR, k nmr = C• p mr •T 2 1 , where C is an adjustment factor, T 1 is the NMR relaxation time, and p is an exponent with two possible values. These two values come from the fact that the relationship k nmr (Ø,T 1 ) has two distinct sources: (1) a theoretical-empirical origin [28], which concludes that p = 1; and (2) a purely empirical character [29], which concludes that the best fit is obtained for p ≈ 4.…”

“…However, there are many cases in practice in which the empirical data do not adequately fit these types of functions but correspond to two different behaviours dependent on the variation of the independent variable. This study presents a new type of single variable function, named the function of double asymptote, which allows the acceptance of new empirical laws, unlike other more complex processes such as the one shown in Godoy et al [1], and those showing a single asymptote are usually studied using the empirical Hurst's law [2]. A function of double asymptote of a single variable is one that presents an infinite approximation to a straight line (horizontal, vertical, or oblique), i.e., an asymptotic behaviour, at two different values, and these functions have been called y 2A (x).…”

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

“…The authors of the publication determine the Wavelet-based method as an accurate method. Although the R/S method is not the most accurate, nowadays it is actively applied in the analysis of fractal processes in various scientific fields due to its easy of use [16][17][18][19][20][21][22].…”