Accuracy analysis of the box-counting algorithm

TL;DR

This paper investigates the accuracy of the box-counting algorithm for calculating fractal exponents, revealing that standard deviation underestimates true error and providing a formula for more realistic error estimates.

Contribution

It introduces a new understanding of the error scaling in the box-counting method and offers a formula for more accurate error estimation.

Findings

Standard deviation underestimates actual error

Error scales with the number of data points as a power law

Errors are larger for higher-dimensional and fractal functions

Abstract

Accuracy of the box-counting algorithm for numerical computation of the fractal exponents is investigated. To this end several sample mathematical fractal sets are analyzed. It is shown that the standard deviation obtained for the fit of the fractal scaling in the log-log plot strongly underestimates the actual error. The real computational error was found to have power scaling with respect to the number of data points in the sample ($n_{tot}$). For fractals embedded in two-dimensional space the error is larger than for those embedded in one-dimensional space. For fractal functions the error is even larger. Obtained formula can give more realistic estimates for the computed generalized fractal exponents' accuracy.