Measuring Fractal Dimension ?

Peter Billam peter at
Mon Jun 15 01:06:49 CEST 2009

>> In message <slrnh33j2b.4bu.peter at>, Peter Billam wrote:
>>> Are there any modules, packages, whatever, that will
>>> measure the fractal dimensions of a dataset, e.g. a time-series ?
> Lawrence D'Oliveiro wrote:
>> I don't think any countable set, even a countably-infinite set, can
>> have a fractal dimension. It's got to be uncountably infinite, and
>> therefore uncomputable.

You need a lot of data-points to get a trustworthy answer.
Of course edge-effects step in as you come up against the
spacing betwen the points; you'd have to weed those out.

On 2009-06-14, Steven D'Aprano <steve at> wrote:
> Strictly speaking, there's not one definition of "fractal dimension", there
> are a number of them. One of the more useful is the "Hausdorf dimension",

They can be seen as special cases of Renyi's generalised entropy;
the Hausdorf dimension (D0) is easy to compute because of the

Also easy to compute is the Correlation Dimension (D2):

Between the two, but much slower, is the Information Dimension (D1)
which most closely corresponds to physical entropy.

Multifractals are very common in nature
(like stock exchanges, if that counts as nature :-))
but there you really need _huge_ datasets to get useful answers ...

There have been lots of papers published (these are some refs I have:
  G. Meyer-Kress, "Application of dimension algorithms to experimental
  chaos," in "Directions in Chaos", Hao Bai-Lin ed., (World Scientific,
  Singapore, 1987) p. 122
  S. Ellner, "Estmating attractor dimensions for limited data: a new
  method, with error estimates" Physi. Lettr. A 113,128 (1988)
  P. Grassberger, "Estimating the fractal dimensions and entropies
  of strange attractors", in "Chaos", A.V. Holden, ed. (Princeton
  University Press, 1986, Chap 14)
  G. Meyer-Kress, ed. "Dimensions and Entropies in Chaotic Systems -
  Quantification of Complex Behaviour", vol 32 of Springer series
  in Synergetics (Springer Verlag, Berlin, 1986)
  N.B. Abraham, J.P. Gollub and H.L. Swinney, "Testing nonlinear
   dynamics," Physica 11D, 252 (1984)
) but I haven't chased these up and I don't think they contain
any working code. But the work has been done, so the code must
be there still, on some computer somwhere...

Regards,  Peter

Peter Billam

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