7.3.3 Nonparametric approaches to the description of piecewise stationary structure of EEG

Earlier attempts

Study (Michael & Houchin 1979) is an example of one of the first nonparametric approaches to EEG segmentation. The authors also used the technique of running window, but compared the referent and tested EEG intervals not by the parameters of the autoregressive model but rather by the autocorrelation function. The integral index of the relative amplitude and shape discrepancy between the normalized autocorrelation functions of the referent and tested EEG intervals served as a nonparametric test of their difference (Michael & Houchin 1979). The later modification of this technique, which used the calculation of the normalized sum of the squares of differences of five autocorrelation coefficients as a measure of spectral dissimilarity between the referent and tested windows, performed satisfactory with clinical EEG recordings (Creutzfeldt et al. 1985).

Indices of spectral expansion also belong to the nonparametric estimations of time series. The Fast Fourier Transform (FFT) was one of the techniques employed for the fixed-interval EEG segmentation discussed above. As we noted, the main disadvantage of this approach to segmentation was the lack of adaptability of segment boundaries to the actual piecewise stationary structure. It seems natural to apply the FFT to a running time window and a referent window and then compare the obtained spectral estimations, in analogy with the adaptive segmentation procedure employing autoregressive modelling. A very high variance of the single spectral estimations (Jenkins & Watts 1972) is a serious obstacle on this way. Nevertheless, the only work applied this approach (Skrylev 1984) did demonstrate that it is quite efficient. In this study, the author used the maximal ratio between the narrow-band spectral power estimations as a measure of EEG spectral difference in two jointly running windows (Skrylev 1984), which made the method sufficiently sensitive to the EEG transition processes. However, the lack of the analytical justification of the threshold conditions, which is characteristic also for most of the adaptive segmentation techniques, still remained. In study (Omel'chenko et al. 1988) the use of an empirical statistical test for the assessment of inhomogeneity of spectral estimations of two EEG intervals made possible the justification of the choice of the threshold for detection of spectral differences. However, this work was not developed in the direction of EEG segmentation.

Though the first attempts to apply the nonparametric approach for EEG segmentation were rather successful, its further development was restricted by the apparent condition that, in each specific case, a statistical EEG characteristic most responsible for the EEG segmental structure (expected value, variance, other statistical moments etc.) is unknown a priori. Moreover, the development of a specific technique of quasi-stationary segmentation for each of these statistics is necessary; therefore, the task of nonparametric EEG segmentation would consist in exhaustion of a rather large number of possible solutions.

Our approach to nonparametric segmentation of the EEG

A new technology of the nonparametric EEG segmentation was developed on the basis of the theory of detecting the sharp changes or change-points in time series with a clear-cut piecewise stationary structure (Brodsky & Darkhovsky 1993). The change-points determined in such a way in a continuous EEG recording are the markers of the boundaries between stationary segments of the signal. The algorithm was based on the method of detection of multiple change-points in a stochastic process, which is described in Chapter 3 and, in more detail, in Appendix. Using this method in the EEG segmentation technique, after its "tuning" in a numerical experiment on the EEG-like simulated signals, made it possible not only reliably detect the intersegmental boundaries, but also to estimate, for the first time, the confidence intervals of their positions within the tested EEG record (Shishkin et al. 1997; Kaplan et al. 1997c; Brodsky et al. 1998).

A starting point for the development of this technology was provided by the piecewise-stationary metaphor of the EEG, i.e., the assumption that any EEG recording is "pasted" from relatively homogenous (in statistical sense) pieces (segments). The transition intervals between such pieces are supposed to be of minor length, and therefore can be roughly treated as a point (a change-point). Each homogenous segment of the signal is assumed to correspond to a period of stable activity of some brain system; thus the transitions from one segment to another show the time moments of the switching of the neuronal network activity (Kaplan et al. 1995; Brodsky et al. 1999). One may also expect that the change-points can reveal the consecutive operations of neuronal networks, with different time scales according to parameters of the algorithm of change-point detection (Kaplan et al. 1997b). The covert dynamics of the operational activity of brain systems, which cannot be disclosed by usual methods of EEG analysis, now can be studied.

The next aim of the EEG segmentation is estimation of the characteristic features of brain operational activity. The emphasis here should be made on the method capacity to work with various EEG types, which may differ substantially by their spectral components. If this task were solved, the EEGs recorded under different mental loading, during different functional states (e.g., during sleep or awake states), under different medication, etc., could be compared quantitatively by rate (frequency) of change-point occurence. Variations of the change-point rate across brain sites could also be of special interest, because they may help to compare the degree of the involvement of these sites in brain operations (Kaplan 1998; Kaplan 1999).

The high interconnectivity of different brain sites offers another field of the application of change-point detection with respect to spatial domain. It seems to be highly probable that the order of appearance of different segments in EEG channels deriving the electrical signal from spatially different regions of the brain cortex is co-ordinated across the channels in a degree dependent on the functional co-operation of these regions. It seems possible that investigation of the coincidence of change-points would enable, for the first time, the direct estimation of the coupling of inherent elementary operations going on in different brain areas, instead of routine phase-frequency synchrony in the terms of correlation and coherency (Kaplan et al. 1995; Kaplan 1998). A qualitative description (see below) of this type of synchrony, which we call the operational synchrony, provides the means for a radically new insight into the co-operation of brain structures.

It was shown during the adaptation of the change-point detection methods for EEG segmentation, that the following type of diagnostic sequence is appropriate for this purpose: y_τ(t) = x(t)x(t+τ), where x(t) is the routine EEG, τ = 0,1,... is a fixed time lag (Brodsky et al. 1998). The main reason to choose this diagnostic sequence was that the nonstationarity of the EEG signal generally results from variations in its spectrum or, what is the same - in its correlation function. The EEG could therefore be viewed as "pasted" from a large number of random stationary (by the correlation function) processes. As it was described in Chapter 1 (Section 1.4), these diagnostic sequences should be used for the reduction of the problem of detecting changes in correlation function to the problem of detecting changes in expected value. All the experimental results described below were obtained with a particular variant of this diagnostic sequence (referred to as basic diagnostic sequence): y₀(t)=x²(t).

While developing the algorithm of change-point detection in the EEG, we also intended to make possible to assess all the levels of the hierarchy of segmental description of the EEG, with different time scales (see above Subsection 7.3.2). The estimation of change-points were made by stages. The most "powerful" change-point was detected first, providing a boundary between two large segments of the recording; then the procedure was performed for these segments, which were more homogenous than the total recording, and the change-points of second level could be found if existed. The division into smaller segments was proceeded this way, until all the resulted segments were found to be homogenous (without change-points) or shorter than a certain threshold corresponding to the minimal length of sequence required for consistent statistical estimates. In more detail the algorithm is described in Chapters 3, in this Chapter in Subsection 7.4.1, and in Appendix (see also Brodsky et al. 1998; Shishkin et al. 1997; Kaplan et al. 1997c; Brodsky et al. 1999).

The threshold for change-point detection in our method is a function of "false alarm" probability (the probability to detect a change-point which in fact not exists). The latter parameter is set in an explicit form (Brodsky & Darkhovsky 1993; Brodsky et al. 1998; Shishkin et al. 1997), which is especially important because, irrespective of a specific technology of detection, the results of change-point detection could be obtained only as probabilistic estimates. The use of "false alarm" probability not only enables a wide range of the adaptation of detection procedure to specific research tasks, but also makes possible the work with different time scales. By increasing the "false alarm" level, for example, one may tune the procedure for most prominent intersegmentary transitions and work with the macroscopic segmentary structure of the EEG. The lower false alarm level results in revealing more detailed, microscopic segmentary structure. The repetitive processing of the same signal with different thresholds will yield the outlines of the hierarchy of EEG segmental description (see above subsection 7.3.2).

The application of the method in neurophysiological studies demonstrated its sufficiently high sensitivity in estimation of the dynamics of structural changes in EEG related to cognitive processing (Kaplan et al. 1997c; Kaplan et al. 1998).