Accuracy of the box-counting algorithm for noisy fractals

TL;DR

This paper investigates how additive noise impacts the accuracy of the box-counting algorithm in calculating fractal dimensions, revealing that even minimal noise significantly degrades precision regardless of sample size.

Contribution

It provides a detailed analysis of noise effects on box-counting fractal dimension estimates, highlighting limitations in accuracy and scalability.

Findings

Tiny noise causes large errors in fractal dimension estimates.

Error saturation occurs at higher noise levels, preventing improved accuracy.

Increasing data sample size does not mitigate noise-induced errors.

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.