Measuring Fractal Dimension ?
Peter Billam
peter at www.pjb.com.au
Mon Jun 15 01:06:49 CEST 2009
>> In message <slrnh33j2b.4bu.peter at box8.pjb.com.au>, 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 REMOVETHIS.cybersource.com.au> 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
box-counting-algorithm:
http://en.wikipedia.org/wiki/Box-counting_dimension
Also easy to compute is the Correlation Dimension (D2):
http://en.wikipedia.org/wiki/Correlation_dimension
Between the two, but much slower, is the Information Dimension (D1)
http://en.wikipedia.org/wiki/Information_dimension
which most closely corresponds to physical entropy.
Multifractals are very common in nature
(like stock exchanges, if that counts as nature :-))
http://en.wikipedia.org/wiki/Multifractal_analysis
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 www.pjb.com.au www.pjb.com.au/comp/contact.html
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