Multiscaling comparative analysis of time series and geophysical phenomena

Different methods are used to determine the scaling exponents associated with a time series describing a complex dynamical process, such as those observed in geophysical systems. Many of these methods are based on the numerical evaluation of the variance of a diffusion process whose step increments are generated by the data. An alternative method focuses on the direct evaluation of the scaling coefficient of the Shannon entropy of the same diffusion distribution. The combined use of these methods can efficiently distinguish between fractal Gaussian and L\'{e}vy-walk time series and help to discern between alternative underling complex dynamics

Multiscaling Comparative Analysis of Time Series and Geophysical Phenomena

Different methods are used to determine the scaling exponents associated with a time series describing a complex dynamical process, such as those observed in geophysical systems. Many of these methods are based on the numerical evaluation of the variance of a diffusion process whose step increments are generated by the data. An alternative method focuses on the direct evaluation of the scaling coefficient of the Shannon entropy of the same diffusion distribution. The combined use of these methods can efficiently distinguish between fractal Gaussian and Lévy-walk time series and help to discern between alternative underling complex dynamics. © 2005 Wiley Periodicals, Inc. Complexity 10: 51–56, 2005 Key Words: scaling exponents; geophysical systems; scaling coefficient; Lévy-walk time series; fractal Gaussian time series The evaluation of the scaling exponents is of fundamental importance to describe a number of complex systems [1–3]. The mathematical definition of scaling is as follows [4]. The function �(r 1, r 2, …) is termed scaling invariant, if it fulfills the property: ��r 1, r 2,··· � � � a �� � b r 1, � c r 2,···�. (1) Thus, if we scale all coordinates {r i} by means of an appropriate choice of the exponents a, b, c,..., then we always recover the same function. This scaling invariance is th