Study the accuracy of the correlation fractal dimension estimation

Fractals are widely used in different fields of science. The main characteristic of a fractal set is its fractal dimensions. The presented work addresses the accuracy of estimation of the correlation fractal dimension D-2 and corresponding confidence interval. For example, in geosciences D-2 is applied for characterizing surfaces or the spatial distribution of fracture systems. Two qualitatively different problems are studied. The first problem is related to the analysis of the fractal sets assigned on regular grids. The fractals consisting of a family of irregular points are considered in the second problem. The Monte Carlo method is proposed to estimate the confidence intervals of correlation fractal dimension for the first problem. This approach is based on the statistical generation of the fractal sets realizations with a given D-2. Numerical testing using synthetic models showed the reliability of this method. A comparison with PBM and BCA bootstrap methods is performed. The paper also investigates the accuracy of the D-2 and the corresponding confidence intervals estimate for the second studied problem. An algorithm employing the jackknife method is suggested. Its accuracy has been studied and proved numerically.