How to detect and fit “fractal” curves, containing power-law exponents? Part 1

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) behavior. This paper consists of two parts. In this first part and according to the self-similarity principle (SSP), the authors of this study completely distinguish between independent “fractal” (power-law) behavior and “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. The 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.