Accuracy of the box-counting algorithm for noisy fractals

Abstract

The box-counting (BC) algorithm is applied to calculate fractal dimensions of four fractal sets. The sets are contaminated with an additive noise with amplitude $\gamma = 10^{-5} \div 10^{-1}$. The accuracy of calculated numerical values of the fractal dimensions is analyzed as a function of $\gamma$ for different sizes of the data sample ($n_{tot}$). In particular, it has been found that a tiny ($10^{-5}$) addition of noise generates much larger (three orders of magnitude) error of the calculated fractal exponents. Natural saturation of the error for larger noise values prohibits the power-like scaling. Moreover, the noise effect cannot be cured by taking larger data samples.