Use the information dimension, not the Hausdorff

Details Use the information dimension, not the Hausdorff Use the information dimension, not the Hausdorff

2005-12-07

arxiv.org/abs/nlin/0512014v1

Nonlinear Sciences

Pattern Formation and Solitons

11 pages

Scientific paper

Multi-fractal patterns occur widely in nature. In developing new algorithms to determine multi-fractal spectra of experimental data I am lead to the conclusion that generalised dimensions $D_q$ of order $q\leq0$, including the Hausdorff dimension, are effectively \emph{irrelevant}. The reason is that these dimensions are extraordinarily sensitive to regions of low density in the multi-fractal data. Instead, one should concentrate attention on generalised dimensions $D_q$ for $q\geq 1$, and of these the information dimension $D_1$ seems the most robustly estimated from a finite amount of data.

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