Next: Singular System Analysis
Up: Analysis of RealWorld Data
Previous: Topological Considerations
Contents
One of the many quantities which are usually used to characterize a
strange attractor is the correlation dimension. It was introduced by
Grassberger and Procaccia as a measure of strangeness which is easier to
handle (especially numerically) than the measures used until then
[5,6].
For instance, if the dynamical laws of a system are not known then
the Hausdorff dimension is usually computed using boxcounting algorithms
(see for example [14], chapter 3.3).
These are very timeconsuming, especially for higherdimensional
systems, because the number of calculations grows exponentially with the
dimension [5,6,8]. The problems we encounter for
our special case of analyzing a dynamical system where the only
information about the system is given in the form of a time series are
even larger, since we have to go through the embedding process first.
Grassberger and Procaccia suggested to use, instead of the Hausdorff
dimension, the correlation dimension , which can be computed
directly from the time series without greater difficulties. We will see
that calculation of will provide us with a method to find a proper
embedding dimension ([8] and [14], chapter 5.3), as
well.
To determine the correlation dimension we first calculate the
correlation integral:

(25) 
where is the correlation length and the Heaviside function. For
sufficiently large and sufficiently small the logarithm of
as a function of the logarithm of will have a linear region, the
scaling region, and the slope in this region is the correlation
dimension :

(26) 
Because is closely related to the Hausdorffdimension^{11} which itself is a generalization of the intuitive concept of
dimensionality [17],
it seems sensible to apply a procedure similar to the one
presented in the previous section [8]: We calculate for
a sequence of embedding dimensions and infer the respective
's. We expect that if the embedding dimension is too small then
equals . (In this case
is not the true correlation dimension of the attractor but only an
artefact, due to the embedding dimension being too small.) So increasing
yields an increasing . But for sufficiently large
will be smaller than and equal to the true correlation
dimension. A further increase should not change the value of any
more.
Thus we have a method to compute the proper embedding dimension: we simply
use the correlation dimension as the embeding dimension.
Footnotes
 ... Hausdorffdimension^{11}

For the relationship of to the Hausdorff and information
dimensions see [5,6]; Schuster [14], chapter
5.3, shows that these
three types of dimensions can be organized within the framework of
generalized dimensions : , for example. It seems
to be true that in most cases
all three dimensions do not differ very much and often they are even the
same, e.g. when the attractor is covered uniformly by the
[6].
Next: Singular System Analysis
Up: Analysis of RealWorld Data
Previous: Topological Considerations
Contents
Martin_Engel
20000525