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Now consider a time series that consists of samples of a sine wave.
Such a time series could be obtained, for instance,
by measuring the position of a forced pendulum as a function of
time when transients have died out.
If we reconstruct a twodimensional phase space with a delay
, where is the period of the sine, we obtain
a circle. That is, if we choose the sample frequency in such a way
that it is incommensurable with the frequency of the sine, and if
we have an infinitely long time series. The dimension of this
circle is one. Now, define a homogeneous probability measure on
the circle so that the distance, measured along the circle, between
two random chosen points is uniformly distributed. By applying a
coordinate transformation, we obtain for the probability density
function of the Euclidean distance (divided by the diameter
of the circle):

(4.10) 
Then, an analytic expression for the correlation integral is:

(4.11) 
and

(4.12) 
In figure 4.1
the correlation integral and the correlation dimension
are plotted as a function of .
We observe that we will overestimate the dimension.
In figure 5.9
the correlation integrals were plotted which were
calculated using a time series consisting of 10000 points.
Compare this figure with figure 4.1!
It would be interesting to know what happens if
we would choose a higher embedding dimension, another lag, a sum of sines,
introduce noise, or if we
would use methods of dimension estimation from Chapter 3 or 5.
Fortunately, in these cases the problem gets very complicated.
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