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We could not identify strange attractors behind the biological
processes of the brain and of the respiratory system. We now
wonder if the singular value decomposition as introduced in
section 6.7 could make a difference.
Therefore we computed the normalized
singular spectra according to the
procedure as described in [Broomhead and King, 1986,Albano et al., 1987], as a function
of the time delay used in the construction of the ``embedding matrix''.
For every signal, the embedding matrix was constructed using 1000
10-dimensional vectors. We plotted the normalized singular values,
as defined by eq. (6.3), for
the studied signals in figure 7.10.
For all three signals, there is a group of two largest singular values and
a group of the remaining values, approximately half an order of magnitude smaller
(cf. [Albano et al., 1987]).
The first group most probably reflects the main frequency component
in the signals. (There are also two main singular values computed
using a sine wave; there are two directions needed to construct a
circle. The number of important singular values is also said to be
an upper limit for the dimension [Theiler, 1990].) The results hardly depend
on the time delay used.
If we would transform the phase space, we have to
decide whether to use only two principal components or all of them.
This means that the computed correlation dimensions using
the transformed space can either be less than 2, or non-saturating.
Figure 7.11
(figure 7.11b)
and table 7.5
show the results the MLDK2
program using
a rotated embedding matrix^{7.3}.
The dimension does indeed not saturate.
In [Albano et al., 1987], the dimension did saturate. That may be
caused by the fact that there a time delay was used which was a factor 16 smaller
than we used.

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