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Practical problems with correlation dimension

Pavel Pokorny Pavel.Pokorny at vscht.REMOVEME.cz
Wed Jan 21 09:30:11 EST 2004

In sci.nonlinear Karl <karlknoblich at yahoo.de> wrote:
> Hallo!

> I want to calculate the correlation dimension of a time serie.

> What I have done
> I calculated the correlation integral C(r) (number of point having a
> distance smaller than r) for different embedding dimensions. Taking
> the slopes of the curve of log C(r) against log r for the different
> embedding dimensions and plotting them against the embedding dimension
> should result in a limes of the slopes: the correlation dimension.

> My problem
> Which slope shall I take?

> In examples I saw in text books there is a nice limit of the slopes
> with higher embedding dimensions. In my data I do not know which slope
> I should take because the slope of the curve varies. If I take the
> slope at a certain value of log r I can not get a limes.

> My curves (log C(r) against log r) can be seen in
> http://karlknoblich.4t.com/korrdim.jpg

> What to do? Does anybody knows such data and how to handle it?

> Hope somebody can help!

> Karl

There is no guarantee that the limit exists.
There may be different slopes on different scales with different widths.
There is a huge difference between real life data and solution of 
low dim mathematical models without noise (as seen in books).

Pavel Pokorny
Math Dept, Prague Institute of Chemical Technology

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