One difference between a chaotic signal from a strange attractor and a signal from a noisy random process is that points on the chaotic attractor are spatially organized. One measure of this spatial organization is the correlation integral ,
where n is the total number of points in the time series. This correlation function can be written more formally by making use of the Heaviside function H(z),
where H(z) = 1 for positive z, and 0 otherwise.
Typically, the vector 
used in the correlation integral
is a point in the embedded phase space constructed from
a single time series according to
For a limited range of 
it is found that
that is, the correlation integral is proportional to some power of
[30]. This power 
is called the correlation
dimension , and is a simple measure of the (possibly
fractal) size of the attractor.
The correlation integral gives us an effective procedure for assigning a
fractal dimension to a strange set. This fractal
dimension is a simple way to distinguish a random signal from a signal generated
by a strange (possibly chaotic) set. In principle, a random
process has an ``infinite'' correlation dimension. Intuitively, this is because
an orbit of a random process is not expected to have any spatial structure.
In contrast, the correlation
dimension for a closed curve (a periodic orbit) is 1, and for a two-dimensional
surface (such as quasiperiodic motion on a torus) is 2. A strange
(fractal) set can have a correlation dimension that is not an integer.
For instance, the strange set arising at the end of the period doubling
cascade found in the quadratic map has a correlation dimension of
, indicating that the dimension of this strange attractor is
somewhere between that of a finite collection of points ( 
) and a curve
( 
).
There exist some technical issues associated with the calculation of
a correlation dimension from a time series that have to do with the choice
of the embedding dimension m and the delay time r. These issues are dealt
with more fully in references [1] and [23]. There now
exist several computer codes in the public domain that have,
to a large extent, automated the calculation of from a single
time series 
. Such a time series could come from a simulation or
from an experiment. See, for example, the BINGO code by Albano
[31], or the efficient algorithm discussed by Grassberger [32]. Thus,
the correlation dimension is now a standard tool in a nonlinear dynamicist's
toolbox that helps one distinguish between noise and low-dimensional
chaos.
