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Abstract
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The aim of the definition of inverse structure functions is to study the fractals and multifractals behaviors of complex systems. This method claims that has many advantages in comparison with the normal structure function. In spite of some findings related to a few number of empirical studies, there is also a debate among scientists about the applicability of this method. Although, there is no extensive investigation on the effects of the linear/nonlinear correlations and non Gaussianity on the inverse structure function. The main goal is to more identify the dynamical characteristics of the systems, and to better predict the system’s future. In this thesis, we study this issue. At first, we investigate the standard and inverse structure functions, in details and then by using programming with Matlab, we calculate the inverse structure function for three fractal and multifractal stochastic processes: a fractional Brownian motion, an alpha-stable series, and a multiplicative process. We observe that the inverse structure function obeys a power-law behavior. On the other hand, for a fractal Brownian motion, the scaling exponent of such power-law functionality is a linear function of the order of the inverse structure function. However, for both multifractal series, this functionality is nonlinear. This indicates that the method of the inverse structure function is able to detect the fractality/multifractality characteristics.
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