7.5 Some general theoretical considerations

7.5.1 Unsolved problems and new prospects

Deviation of the signal from the piecewise stationary model

It is necessary to note that a change-point detection method which we applied to a real EEG signal was developed on the basis of the piecewise stationarity model of the signal. The performance of the method in the cases of gradual transitions from one segment to another, which are often observed in the EEG recordings, could not be completely adequate. But how much this discrepancy between the real signal and the model influenced the results?

Indeed, when the transitions from one segment to another were gradual, they usually were broken by the estimated change-point approximately in the middle, so that they were partly included into the estimated adjacent segments (Fig. 7.4, 7.7, 7.9). The other known methods of the EEG segmentation are also subject to this shortcoming, in part because many of them also utilize the piecewise stationarity model (Barlow 1985). In our work, however, the transitions themselves were usually well detected even if they were substantially long (see Fig. 7.4, 7.7, 7.9), making possible an effective analysis based on the describing the transitions as the points on time axis. Particularly, the interchannel synchronization of the EEG change-points was considerably above the stochastic level, excepting the cases of the highest interelectrode distance. It must be kept in mind, however, that in the cases of higher deviations from the piecewise stationarity model than those observed in our data the signal may lose the structure available for the analysis of any kind, and it will be just senseless to characterize it with change-points.

Nonstationarity of the synchronization level and of other characteristics estimated with change-points

One of the problems related to the nonstationarity may appear to be fundamentally unsolvable in the application of the change-point methodology to highly nonstationary signals like the EEG. In the calculation of any averaged estimate of synchronization between signals, not only the traditional correlation/coherency but also change-point based, the possibility of the nonstationarity of synchronization level itself is ignored. No evidence exists that at the different time intervals the coupling between signals (or signal components) cannot possess different mean values. More generally, any characteristic estimated for a large time interval, even if segmentation approach is applied, can be a subject to this problem. The same task as discussed above in the formulation of the problem of EEG segmentation appears here: any value computed for some time interval may be quite far from the real levels of the estimated characteristic at any part of this interval. This problem, in a special case of estimating the probabilities of transitions between different classes of EEG stationary segments, was noted in the paper of Jansen and Cheng (Jansen & Cheng 1988). It seems to be reasonable to keep always in mind that if the signal of weakly understood nature is nonstationary (like the EEG), any of its characteristic may appear to be nonstationary; this means that any averaged estimates of such signals must be used with caution.

New prospects in the signal analysis opened by the effective change-point detection

The revealing of the instants of changes in the EEG or other complex signal makes more precise the routine statistical analysis, which usually require stationarity of the signal interval in question. We will not consider here the possibilities of the EEG analysis being opened by viewing it as a sequence of stationary intervals, since they were extensively discussed by other authors in the works related to the EEG segmentation (e.g., Bodenstein & Praetorius 1977; Sanderson et al. 1980; Creutzfeldt et al. 1985; Barlow 1985; Jansen & Cheng 1988; Cerutti et al. 1996; Pardey et al. 1996). In our work, probably for the first time, the emphasis was made on the other aspects of the utilizing the information about the signal structure given by the time instants of changes.

The data presented in this Chapter demonstrate that the effective detection of the change-points opens the way for a number of new methods of the analysis of complex signals such as the EEG. First, the segmentary time structure of the signal or (if the signal is composed of a number of components with more or less independent dynamics) its components can be accurately described. Secondly, if these descriptions are obtained for two or more simultaneously recorded signals or for the components of the same signal, their temporal relations can be estimated. These approaches are empirical in their essence and have no explicit mathematical justification, at least now, but at the current stage of the studies of the EEG structural characteristics they seem to be quite acceptable, because of the lack of the understanding of the EEG nature and the weak elaboration of more strict methodology of its structural analysis.

The separate detection of changes in different frequency bands and with different time scales was already used by other researchers (Deistler et al. 1986; Ferber 1987). In our work this was done with a more broad range of signal types, covering a substantial portion of the variety of patterns occurring in the EEG studies. It turned out that the detection algorithm described in this book easily worked with all of them, requiring only little "manual" efforts for tuning its parameters, or allowing unsupervised automatic data processing (Fig. 7.2-7.4, 7.6).

One of the possibilities of signal analysis offered by the change-point detection, namely the statistical estimation of the temporal relationship between the segmentary structures of different signals and of different components of the same complex signal, was not employed before our works at all. This is why we will focus our attention on it here.