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### Choice of First Embedding Dimension and Sampling Time

By first embedding dimension'' we mean here , the dimension of the space with which we begin the procedure described in the last section. We did not specify yet, but simply said that it should be greater than or equal to , in order to guarantee that embedding is possible. Broomhead and King in  suggested to compute the power spectrum, which shows to which degree the frequencies contribute. Typically, the power spectrum consists of a noise floor (i.e. all frequencies contribute equally in the case of white noise) to which the amplitudes due to the deterministic contribution are added (see Fig. 7). If the deterministic contribution is only significant for frequencies which are smaller than some band-limit'' frequency (which corresponds to the time-interval ) then one can choose . This can be justified as follows: On the one hand one wants to make large, in order to have it (see section 3.1), on the other hand choosing so large that seems not to be sensible, since it is obviously easier to work with a lower-dimensional embedding space. So the only consistent a priori estimate seems to be . Although this justification for the choice of is rather handwaving and only valid for band-limited data, numerical experiments in  show that in many cases it gives good results. One reason for this is that most dynamical systems which have been investigated until now usually have rather low-dimensional attractors due to the dissipative properties of the systems. This can be true even if the system is moving in a phase space as high-dimensional as in the case of the Belousov-Zhabotinski reaction (see chapter 1.1 in ). According to Broomhead and King the choice of the sampling time can be based on physical considerations as well. Many systems have a characteristic time-scale: The observables of the system do not change significantly in times smaller than this. If one decreases while keeping the window length'' constant then one gets vectors with more and more components and thus more and more singular values . Doing this one will reach a point where the number of those singular values which are not noise-dominated does not go up further: decreasing then results essentially in increasing the number of singular values in the noise floor. This means that the corresponding value of is small enough to match the characteristic time of the system and we can take as the sampling time. A similar approach to the choice of sampling time is described by Schuster (chapter 5.3 in ): He considers some fixed and determines as the decay time of the autocorrelation function of the time series which can be computed as follows: (47)

where is the time between two successive measurements of . Then we get from (48)

Since the power spectrum is proportional to the Fourier transform of the autocorrelation function15 both approaches ( and , chapter 5.3) should give comparable estimates for .

#### Footnotes

... function15
This is the Wiener-Khinchine theorem.    Next: On the Required Number Up: Singular System Analysis Previous: Singular Value Decomposition as   Contents
Martin_Engel 2000-05-25