Corrective Function Method for the Fractal Analysis

Abstract

One of the main numerical characteristics used in numerous methods of fractal analysis is the corresponding fractal dimensions. The accuracy of estimating these dimensions in the vast majority of cases is quite small, which cannot satisfy, first of all, researchers-practitioners. The method of the corrective function is put forward, which makes it possible to compensate for the ever-existing nonlinearity of the dependence between the true value of the fractal dimension and its estimation, performed using the selected method of monofractal analysis of signals and processes for a known number of samples of the discrete data vector of the investigated signal. The main idea of the method is to build and apply a special correction function using a set of model fractal signals with previously known values of the fractal dimension. The mathematical bases of the new method are outlined. Features of the practical application of the corrective function method are considered on the example of the evaluation of regularization, boxing, variation and Hurst fractal dimensions. For them, the minimum values of the number of samples of the discrete data vector of the investigated signal, at which these dimensions can still be estimated, are defined. Using a set of model monofractal and multifractal signals on the example of the dynamical fractal analysis method, the effectiveness of the created method of the corrective function is shown. It is proven that due to the application of the correction function method, the maximum deviation of the estimated fractal dimension from the true known value for the specified dimensions is reduced from 25 – 55% to 5 – 7%.