Next: Theiler correction
Up: PRACTICAL ASPECTS
Previous: Filtering
Contents
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 with an error of
%, the number of
(uniformly distributed and uncorrelated) points must be

(6.1) 
where denotes the range of the scaling region ().
To estimate with an error of 5% and with , he obtained
.
It is often stated that one should have
. In that case we can estimate with an error of 25%.
If we require the scaling region
to extend a decade and an error of 5%, we obtain
and for , 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
where is the length of the time series.
Increasing
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 
 may be of use.
Next: Theiler correction
Up: PRACTICAL ASPECTS
Previous: Filtering
Contents
webmaster@rullf2.xs4all.nl