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Sample frequency and length of the time series

To avoid spurious dimension estimates due to autocorrelations, the sample frequency should not be too high. It is probably optimal to determine the delay using the mutual information criterion (see section 6.6) and then choose the sample frequency in such a way that the optimal lag is near 1. (Broomhead and King remark that it may be better to increase the sampling rate with noisy data [Broomhead and King, 1986].) However, with a finite signal length the resulting times series may become too short! Smith showed, using the example as described in section 4.4.1, that to obtain an estimate of the dimension $\hat{D}$ with an error of $(1-Q) \times 100$%, the number of (uniformly distributed and uncorrelated) points must be
\begin{displaymath}
P \geq {\left[ \frac{R(2-Q)}{2(1-Q)}\right] }^D
\end{displaymath} (6.1)

where $R$ denotes the range of the scaling region ($R = r_u / r_l$). To estimate $D$ with an error of 5% and with $R=4$, he obtained $P \geq 42^D$. It is often stated that one should have $P \geq 10^D$. In that case we can estimate $D$ with an error of 25%. If we require the scaling region to extend a decade and an error of 5%, we obtain $P \geq 105^D$ and for $P \geq 10^D$, the error is 100%. Furthermore, Eckmann and Ruelle showed [Ruelle, 1990] that if the scaling region extends a decade, the dimension estimated from the correlation integral cannot exceed $2 \log L$ where $L$ is the length of the time series. Increasing $L$ by interpolation does not help [Ruelle, 1990] and it is likely, as is oversampling, to produce a spurious scaling region at the small distance range with a slope of 1. Since we usually specify the number of distances to be computed, a similar bound - $\hat{D} \leq \log N$ - may be of use.


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Next: Theiler correction Up: PRACTICAL ASPECTS Previous: Filtering   Contents
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