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Singular values

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 matrix7.3. 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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Next: Discussion Up: APPLICATIONS Previous: The respiration   Contents
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