7.3.2 Parametric segmentation of the EEG

In general terms, the procedure of adaptive segmentation could be based on the estimation of the extent of similarity of an initial fixed interval of EEG with an EEG interval of the same duration viewed through the time window running along the EEG recording. The similarity index will drop sharply when the window runs over a segment boundary, giving a formal indication of the transition to the following segment. The autoregressive methods, which predict the EEG amplitude at a given moment by analysing a series of amplitudes at prior moments, seems to be adequate for this task. The discordance between predicted and real EEG amplitude could be a sufficient indication of a local nonstationarity (Bodenstein & Praetorius 1977; Jansen 1991).

Parametric EEG segmentation based on autoregressive models

The methods of predicting time series are based on the assumption that their stochastic nature is substantially confined by certain dynamic rules. In this case, if mathematical models could be fitted to these regularities, the EEG amplitude will be predicted with a certain accuracy for a number of successive samples. Beyond the stationary segment to which the model parameters were fitted the prediction error will sharply increase, thus signalling the termination of the foregoing segment and the beginning of the next one. For the initial portion of this next segment, new model parameters can be computed, and then search for the next boundary can be continued. Thus, the parameters of the mathematical model of the EEG become the key element in search for segment-to-segment transitions, and a correct choice of the EEG model is very important.

In the framework of this idea, the coefficients of Kalman filter were first used for the model EEG description. A decision about the boundary were made if a sharp change in at least one of 10 filter coefficients was observed (Duquesnoy 1976, cit. by Barlow 1985). More recently, the most advanced technique for the EEG simulation, linear extrapolation, was applied for the EEG segmentation. This technique was developed by N. Wiener as early as 1942 as a supplement for autoregression analysis (cit. by Bodenstein & Praetorius 1977) and applied for the EEG analysis in the late 1960s (for a review see Kaipio & Karjalainen 1997). In the framework of the autoregression model, the EEG amplitude at a given moment can be predicted, with some error, as a sum of several previous amplitude values taken with certain coefficients. The principle procedures of the EEG adaptive segmentation based on the autoregressive models of a rather low order were first developed by Bodenstein and Praetorius (1977) and then in various modifications were successfully used by other authors (Bodunov 1985; Aufrichtig et al. 1991; Jansen 1991; Sanderson et al. 1980; Barlow & Creutzfeld 1981; Creutzfeldt et al. 1985; see also Barlow 1985 for a review of earlier works). According to different authors, the number of segment types lied in the range 6 to 50, and the duration of a stationary segment varied, in general, from 1--2 to 20 s (Bodunov 1985; Barlow & Creutzfeld 1981; Creutzfeldt et al. 1985). Use of the multiple regression analysis employing computation of the contribution of each of the several model parameters made the segmentation procedure more correct. With this technique, the authors managed to detect the EEG segments associated with some mental operations. They reported a similar duration range (2--10 s) for the majority of stationary EEG segments (Inouye et al. 1995).

Although the algorithms of many of the EEG segmentation methods based on the regression analysis were thoroughly elaborated, almost all of them operate with the empirically chosen threshold criteria. This makes it difficult to compare the results of segmentation not only from different subjects but even from different EEG channels in the same subject. In addition to the inevitable empirical predetermination, the threshold criterion for EEG segmentation in these techniques has a more serious disadvantage, i.e., the tuning of the threshold cannot be refined in accord with the changing parameters of the EEG process. The autoregressive model with the time-varying parameters tested in speech recognition (Hall et al. 1983) seems to be an appropriate solution for this problem. Some attempts have been made to apply this approach to the EEG (Amir & Gath 1989; Gath et al. 1992). However, in the lack of a priori knowledge about the law of the variations of model parameters it was necessary to construct an additional model, which should result, in the general case, in accumulation of even greater error.

Time scales in EEG segmentation

The methods of EEG adaptive segmentation based of autoregressive modelling used the same technique of running comparison of the EEG parameters in the referent and tested intervals, which made it possible to view the EEG structure only through a fixed time window. This approach determined a single time scale for EEG heterogeneities and, thus, prevented the insight into the total EEG structure, just like only neighbouring mountain peaks can be seen in the view-finder of a camera, while the mountain chain relief, as a whole, escapes from the visual field. It is quite possible, however, that the EEG contains larger transformations which are superimposed on the local segment structure and corresponds to a segment description of the EEG signal on a larger time scale.

Close to the solution of this problem was the study (Deistler et al. 1986), where a type of the regressive EEG modelling was also used, like in the works discussed above. The method described in this paper was quite sensitive to find the time moment of the beginning of action of neurotropic drugs. The authors analysed the EEG power in alpha band (8--12 Hz) on the assumption that its dynamics in a stationary interval can be approximated by a simple linear regression of y = at + b type, where y is the power in alpha band computed in a short time window with number t. In this case, the problem of finding a boundary between two quasi-stationary segments was reduced to a well-developed statistical procedure of comparison between coefficients a and b for two linear regressions at both sides of the presumed boundary. The point of the maximal statistically significant difference between two regressions indicated the joint point between the largest EEG segments. The authors emphasised the ability of their method to find only the most pronounced change if there is a number of change-points in the EEG recording, which was important for the specific application area of the method (Deistler et al. 1986). The structural analysis of the EEG in more general terms was not the objective of their study, and this was probably the reason why they did not pay attention to the potential of the method in this area.

From our point of view, the change-point obtained just as they described could be placed at the macroscopic level of the EEG structural description. If a similar procedure was performed further for each of the two detected segments separately, the segments corresponding to more detailed structure of the EEG could be obtained. By repetitions of such a procedure a description of the microscopic level of the EEG segment structure could be provided. Thus, there were prospects for the description of the structural EEG organization as a hierarchy of segmental descriptions on different time scales (Kaplan 1998).

Inherent contradiction of the parametric segmentation

In principle, the parametric methods of adaptive segmentation makes it possible to describe adequately the piecewise stationary structure of the EEG signal. However, all these methods designed for the analysis of nonstationary processes are based on a procedure which may be applied only to stationary processes, namely on fitting a mathematical model (usually the autoregressive one). It is evident that accurate fitting of a model can be achieved only on a stationary interval. The longer the interval, the finer characteristics of the process can be represented by the model. But the longer the analyzed interval of the real EEG, the more probable the incidence of heterogeneities within it (see, for example, McEwen & Anderson 1975). If the model is constructed on a very short interval, it will be very rough and the results of segmentation based on the parameters of this model cannot be expected to be of high quality (Brodsky et al. 1998; Brodsky et al. 1999).

Thus, the parametric methods of search for quasi-stationary EEG segments carry a rather strong contradiction: segmentation into stationary fragments is impossible without construction of an adequate mathematical model, but such a model cannot be constructed without previous segmentation. Moreover, since the EEG is a highly composite and substantially nonlinear process (Steriade et al. 1990; Nunez 1995), the development of a rigorous linear mathematical model adequately representing the EEG intrinsic nature is hardly possible (Kaipio & Karjalainen 1997). The parameters of even the well-fitted EEG models (e.g., Kaipio & Karjalainen 1997; Wright & Liley 1995) thus cannot follow the essence of the processes underlying the EEG (Lopes da Silva 1981; Jansen 1991) and inevitably make the procedure of EEG segmentation substantially rough. This is why the development of nonparametric EEG segmentation methods is undoubtedly of interest. Application of such methods do not require previous testing for stationarity, since they are not associated with fitting mathematical models to a process but rather are based on the analysis of its individual statistical characteristics.