Choice of First Embedding Dimension and Sampling Time

By ``first embedding dimension'' we mean here $n$, the dimension of the space with which we begin the procedure described in the last section. We did not specify $n$ yet, but simply said that it should be greater than or equal to $2m+1$, in order to guarantee that embedding is possible. Broomhead and King in [7] 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).

$\parbox{12cm}{ Figure 7. Example for a band-limited power spectrum which allows to determine the band-limit frequency $\omega^*$. }$

If the deterministic contribution is only significant for frequencies which are smaller than some ``band-limit'' frequency $\omega^*$ (which corresponds to the time-interval $\tau^*=\frac{2\pi}{\omega^*}$) then one can choose $n\tau=\tau^*$. This can be justified as follows: On the one hand one wants to make $n\tau$ large, in order to have it $\geq(2m+1)\tau$ (see section 3.1), on the other hand choosing $n$ so large that $n\tau>\tau^*$ 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 $n\tau=\tau^*$. Although this justification for the choice of $n$ is rather handwaving and only valid for band-limited data, numerical experiments in [7] 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 [14]). 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 $\tau$ while keeping the ``window length'' $n\tau$ constant then one gets vectors with more and more components and thus more and more singular values $\sigma_i$. Doing this one will reach a point where the number $d$ of those singular values which are not noise-dominated does not go up further: decreasing $\tau$ then results essentially in increasing the number of singular values in the noise floor. This means that the corresponding value of $\tau$ is small enough to match the characteristic time of the system and we can take $\tau$ as the sampling time. A similar approach to the choice of sampling time is described by Schuster (chapter 5.3 in [14]): He considers some fixed $n$ and determines $\tau$ as the decay time of the autocorrelation function $C(t)$ of the time series which can be computed as follows:

$$\quad C(t)=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N-1}v_nv_{n+1} \quad,$$ (47)

where $t$ is the time between two successive measurements of $v$. Then we get $\tau$ from

$$\quad C(\tau)\approx\frac{1}{2}C(0) \quad.$$ (48)

Since the power spectrum is proportional to the Fourier transform of the autocorrelation function¹⁵ both approaches ([7] and [14], chapter 5.3) should give comparable estimates for $\tau$.

Footnotes

... function¹⁵

This is the Wiener-Khinchine theorem.