A third criterion is defined from the spectral error measure of the periodogram. A periodogram of the mth frame is obtained by Fourier transforming of the correlation function of the EEG signal:
(4.11)
where
(4.12)
The test window sample autocorrelation for the measurement of both d 1(m) and d 3(m) can be updated through the following recursive equation (Eq. 4.13) over the test windows of size N:
and thereby computational complexity can be reduced in practise. A fourth criterion corresponds to the error energy in autoregressive (AR) ‐based modelling of the signals. The prediction error in the AR model of the mth frame is simply defined as:
where p is the prediction order and ak (m), k = 1, 2, …, p, are the prediction coefficients. For certain p the coefficients can be found directly (for example by using Durbin's method) in such a way to minimize the error (residual) between the actual and predicted signal energy. In this approach it is assumed that the frames of length N overlap by one sample. The prediction coefficients estimated for the (m − 1)th frame are then used to predict the first sample in the mth frame, which we denote it as
where ∇ n (.) denotes the gradient with respect to n, approximated by a first‐order difference operation. Figure 4.2 shows the residual and the gradient defined in Eq. (4.15), wherein the prediction order has been selected as 12.
Finally, a fifth criterion d 5(m) may be defined by using the AR‐based spectrum of the signals in the same way as short‐time Fourier transform (STFT) for d 3(m). The above AR model is a univariate model, i.e. it models a single‐channel EEG. A similar criterion may be defined when multichannel EEGs are considered [14]. In such cases a multivariate AR (MVAR) model is analyzed. The MVAR can also be used for characterization and quantification of the signal propagation within the brain and is discussed in Chapter 8 of this book.
Although the above criteria can be effectively used for segmentation of EEG signals, better systems may be defined for the detection of certain abnormalities. In order to do that, the features, which best describe the behaviour of the signals, need to be identified and used. Therefore, the segmentation problem becomes a classification problem for which different classifiers can be used.
4.5 Signal Transforms and Joint Time–Frequency Analysis
If the signals are statistically stationary it is straightforward to characterize them in either time‐ or frequency‐domains. The frequency‐domain representation of a finite‐length signal can be found by using linear transforms such as the (discrete) Fourier transform (DFT), cosine transform (DCT) or other semi‐optimal transform, which have kernels independent of the signal. However, the results of these transforms can be degraded by spectral smearing due to the short‐term time‐domain windowing of the signals and fixed transform kernels. An optimal transform such as Karhunen–Loéve transform (KLT) requires complete statistical information, which may not be available in practise.
Figure 4.2 (a) An EEG seizure signal including preictal, ictal, and postictal segments, (b) the error signal and (c) the approximate gradient of the signal, which exhibits a peak at the boundary between the segments. The number of prediction coefficients p = 12.
Parametric spectrum estimation methods such as those based on AR or autoregressive moving average (ARMA) modelling can outperform the DFT in accurately representing the frequency‐domain characteristics of a signal, but they may suffer from poor estimation of the model parameters mainly due to the limited length of the measured signals. For example, in order to model the EEGs using an AR model, accurate values for the prediction order and coefficients are necessary. A high prediction order may result in splitting the true peaks in the frequency spectrum and low prediction order results in combining peaks in close proximity in the frequency domain.
For an AR model of the signal x(n) the error or driving signal is considered to be zero‐mean white noise. Therefore, by applying z‐transform to Eq. (4.14), dropping the block index m, and replacing z by ejω we have:
(4.16)
where, E(ω) = Kω (Constant), is the power spectrum of the white noise and Xp (ω) is used to denote the signal power spectrum. Hence,
(4.17)
and the parameters Kω , ak , k = 1, …, p, are the exact values. In practical AR modelling these would be estimated from the finite‐length measurement, thereby degrading the estimate of the spectrum. Figure 4.3 provides a comparison of the spectrum of an EEG segment of approximately 1550 samples of a single‐channel EEG using both DFT analysis and AR modelling.
The fluctuations in the signal DFT as shown in Figure 4.3b is a consequence of the statistical inconsistency of periodogram‐like power spectral estimation techniques. The result from the AR technique (Figure 4.3c) overcomes this problem provided
