The signal of an alpha rhythm was sampled using a 12-bit digital-to-analog converter (ADC) at a sampling frequency of 125 Hz. The obtained time series consisted of 8533 points (approximately 68 seconds in duration). Figure 7.1 shows a plot of (part of) the signal and its power spectrum7.1. The characteristic alpha rhythm is easily recognized near 10 Hz in the plot of the power spectrum (as is the hum caused by the 50 Hz mains supply wires!). Figure 7.4a shows a plot of the mutual information function. The first minimum is 4 samples away from the maximum. This value was used for the phase space reconstruction - see figure 7.5a for a phase portrait (at embedding dimension 2). There we see a cloud of points without any structure. This does not mean however, that there is no structure at all, since we are looking at a projection of the ``real'', possibly high-dimensional object. If we would connect the points with respect to the time evolution we would see many messed-up circles. We plotted the points only because it is between them that the distances are calculated in order to estimate the correlation dimension and entropy.
Figure 7.6 (figure 7.6b) and table 7.1 show the results from the MLDK2 program. The correlation integrals do not show a scaling region. Choosing scaling regions anyway results, as shown, in a non-saturating dimension curve. Under this condition, the entropy estimates are not useful. Some notes: we used for the entropy estimator to reduce its variance; the number (in the table) is caused by the fact that there is a (relatively) high probability of finding the same distance using a time series obtained from an ADC (this probability decreases as the number of bits of the ADC or the embedding dimension increases); the values for the test for the Takens estimator indicate floating underflow conditions7.2; the values of the sizes of the test for the Ellner estimator suggest valid scaling regions.
A recording of spike-waves was sampled at 200 Hz and consisted of only 2564 points (the seizure lasted approximately 13 seconds; see figure 7.2). The time delay for the phase space reconstruction was determined to be (see figure 7.4b and 7.5b). In the correlation integrals, obtained using the MLDK2 program (figure 7.7 (figure 7.7b) and table 7.2), we see ``bellies''; both these regions and the regions above the bellies look like scaling regions. The dimension seems to saturate at a reasonable low value. However, the bellies are caused by autocorrelation. They dissappear if we apply the Theiler correction (see figure 7.8 (figure 7.8b) and table 7.3) and the dimension curve does not saturate. One may argue that there is a scaling region at higher distance levels, but the dimension estimated in that region would also not saturate as a function of the embedding dimension.