Influence of dictionary's structure on statistical properties of MP decomposition

In order to separate the problem of algorithm's properties from the possible signal's characteristics, we analyze MP decompositions of 50 different realizations of white noise. Figure 7c presents histogram of frequency centers of waveforms fitted to noise by MP decomposition over a dyadic dictionary. We observe high peaks in the middle of the frequency range, then in the quarters etc. In case of atoms fitted to EEG (7a) observed structure is superimposed on spectral characteristics of the signal. The structure observed in the two lower plots on the left side of the Fig. 7 comes from the fact that for each realisation of the signal the same time-frequency grid was used. In consequence, in the plot consisting of the several realisations, the privileged positions connected with dyadic dictionary are observed.

Figure 7: Statistical properties of MP decomposition of 50 epochs of sleep EEG (a, b) and white noise (c-f) over dyadic (a, c, e) and stochastic (b, d, f) dictionaries. Histograms of frequency centers of atoms fitted by MP decomposition over dyadic dictionary to EEG (a) and noise (c) reveal additional structure, absent in corresponding decompositions performed over stochastic dictionaries b and d, respectively. Maximum in the middle of frequency range in panel d results from convention of assigning half of Nyquist frequency to Diras'c delta. In the top panel centers of atoms fitted to white noise are given in the time-frequency plane for dyadic (left, e) and stochastic (right, f) dictionaries.

Figure 7e presents centers of atoms from discussed decompositions in dyadic dictionary in time (horizontal) vs frequency (vertical) coordinates. Due to a wide band of decomposed signals, full structure of the dictionary can be observed. Higher density of atoms in certain regions of dyadic dictionary (equation 8) ``attracts'' atoms chosen for decomposition.