Improvement of statistical properties of decomposition

Plots on the right side of Figure 7 (b, d, f) are constructed from a statistically unbiased MP representation obtained by decomposition in stochastic dictionaries. Histogram of frequencies in Figure 7d (decomposition of white noise realizations) fluctuates around flat characteristics¹, while 7b gives a shape related to spectral characteristics of decomposed EEG.²

Figure 7f presents centers of atoms, fitted to white noise realizations, in the time-frequency plane. Unlike the corresponding picture in 7e no particular structure is present, apart from the line in the half of Nyquist frequency, discussed in the footnote.

Figure 8: Histograms of frequencies of sleep spindles detected in the same EEG recording as in Figure 2, decomposed in stochastic dictionaries. Nine central derivations are presented.

Figure 9: Amplitudes of sleep spindles (vertical) plotted versus their frequencies (horizontal) detected in the same EEG recording as in Figure 3 (as well as 2 and 8), based upon MP decomposition over stochastic dictionaries.

Figures 8 and 9 present improvement of distributions, shown in Figures 2 and 3, achieved by decomposition of the same data over stochastic rather than dyadic dictionaries. General trend, discussed in section ``Frequencies in frontal an parietal derivations'' is still clearly visible, while evenly distributed local maxima of spindle's density disappeared, which proves that they were indeed artifacts of the algorithm.

Footnotes

... characteristics¹

peak in the middle stems from a convention of assigning half of Nyquist frequency to Dirac's delta (compare also Table 1)

... EEG.²

Histogram in Figure 7b is not an estimate of the average signal's spectrum, since frequency centers of fitted atoms were counted regardless of their amplitudes and frequency widths