Can Self-Similarity Processes Be Reflected by the Power-Law Dependencies?

Abstract

This work was greatly influenced by the opinions of one of the authors (JS), who demonstrated in a recent book that it is important to distinguish between “fractal models” and “fractal” (power-law) behaviors. According to the self-similarity principle (SSP), the authors of this study completely distinguish between independent “fractal” (power-law) behavior and the “fractal models”, which result from the solution of equations incorporating non-integer differentiation/integration operators. It is feasible to demonstrate how many random curves resemble one another and how they can be predicted by functions with real and complex-conjugated power-law exponents. Bellman’s inequality can be used to demonstrate that the generalized geometric mean, not the arithmetic mean, which is typically recognized as the fundamental criterion in the signal processing field, corresponds to the global fitting minimum. To highlight the efficiency of the proposed algorithms, they are applied to two sets of data: one without a clearly expressed power-law behavior, the other containing clear power-law dependence.