Accuracy analysis of the box-counting algorithm

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.

💡 Research Summary

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The paper investigates the quantitative accuracy of the box‑counting method when it is used to compute fractal dimensions (including generalized dimensions). Although the technique is widely applied across many fields of physics, biology, economics, and beyond, the authors point out that the usual practice of quoting the standard error of the linear fit in a log‑log plot seriously underestimates the true computational error. To assess the real error, they perform systematic numerical experiments on six well‑known mathematical fractals: two one‑dimensional sets (the classic Cantor set and an asymmetric, multifractal Cantor variant), two two‑dimensional sets (the Sierpinski triangle and the Koch curve), and two instances of the Weierstrass‑Mandelbrot (WM) function, which are fractal curves defined as functions (i.e., for each value of the independent variable there is at most one point of the set).

For each fractal they generate point clouds with total numbers of points (n_{\text{tot}}) ranging from (10^{2}) up to (10^{5}). The box‑counting algorithm is applied, the scaling of the box count with box size (\varepsilon = 1/N) is plotted in log‑log coordinates, and a linear regression provides an estimate of the fractal exponent. The authors record both the regression’s standard error (the quantity most often reported as “uncertainty”) and the absolute deviation of the estimated exponent from its analytically known exact value (the “real error”).

The key finding is that the real error follows a clear inverse‑power law with respect to the number of data points, \