EEG Signal Processing and Machine Learning. Saeid Sanei

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a proper prediction order for the AR estimates. EEG signals are often statistically nonstationary, particularly where there is an abnormal event captured within the signals. In these cases the frequency‐domain components are integrated over the observation interval and do not show the characteristics of the signals accurately. A TF approach is the solution to the problem.

      In the case of multichannel EEGs, where the geometrical positions of the electrodes reflect the spatial dimension, space–time–frequency (STF) analysis through multiway processing methods has also become popular [15]. The main concepts in this area together with the parallel factor analysis (PARAFAC) algorithm will be reviewed in Chapter 17 aimed at detection of movement‐related potentials or eye‐blinking artefacts from the EEG signals.

      The STFT is defined as the discrete‐time Fourier transform evaluated over a sliding window. The STFT can be performed as:

      (4.18)

Figure 4.3 Single‐channel EEG spectrum. (a) A segment of an EEG signal with a dominant alpha rhythm. (b) Spectrum of the signal in (a) using DFT. (c) Spectrum of the signal in (a) using a 12‐order AR model.

where the discrete‐time index n refers to the position of the window w(n). Analogous with the periodogram a spectrogram is defined as:

      (4.19)

      Based on the uncertainty principle, i.e. , where and are respectively time and frequency‐domain variances, perfect resolution cannot be achieved in both time‐ and frequency‐domains. Windows are typically chosen to eliminate discontinuities at block edges and to retain positivity in the power spectrum estimate. The choice also impacts upon the spectral resolution of the resulting technique, which, put simply, corresponds to the minimum frequency separation required to resolve two equal amplitude‐frequency components.

Figure 4.4 shows the TF representation of an EEG segment during the evolution from preictal to ictal and to postictal stages. In Figure 4.4 the effect of time resolution has been illustrated using a Hanning windows of different durations of one and two seconds. Importantly, in Figure 4.4 the drift in frequency during the ictal period is observed clearly.

      4.5.1 Wavelet Transform

      The wavelet transform (WT) is another alternative for TF analysis. There is already a well established literature detailing the WT such as [16, 17]. Unlike the STFT, the TF kernel for the WT‐based method can better localize the signal components in TF space. This efficiently exploits the dependency between time and frequency components. Therefore, the main objective of introducing the WT by Morlet [16] was likely to have a coherence time proportional to the sampling period. To proceed, consider the context of a continuous time signal.

      4.5.1.1 Continuous Wavelet Transform

      The Morlet–Grossmann definition of the continuous WT for a 1D signal f(t) is:

(4.20)

      where (.)^* denotes the complex conjugate, is the analyzing wavelet, a (>0) is the scale parameter (inversely proportional to frequency) and b is the position parameter. The transform is linear and is invariant under translations and dilations, i.e.:

      (4.21)

      and

      (4.22)

The last property makes the WT very suitable for analyzing hierarchical structures. It is similar to a mathematical microscope with properties that do not depend on the magnification. Consider a function W(a,b) which is the WT of a given function f(t). It has been shown [18, 19] that f(t) can be recovered according to:

Figure 4.4 TF representation of an epileptic waveform in (a) for different time resolutions using the Hanning window of (b) 1 ms, and (c) 2 ms duration.

      (4.23)

      where

      (4.24)

      Although often it is considered that ψ(t) = ϕ(t), other alternatives for ϕ(t) may enhance certain features for some specific applications [20]. The reconstruction of f(t) is subject to having C_ϕ defined (admissibility condition). The case ψ(t) = ϕ(t) implies , i.e. the mean of the wavelet function is zero.

      4.5.1.2 Examples of Continuous Wavelets

      Different waveforms/wavelets/kernels have been defined for the continuous WTs. The most popular ones are given below.

      Morlet's wavelet is a complex waveform defined as:

      (4.25)

      This wavelet may be decomposed into its constituent real and imaginary parts as:

      (4.26)

      (4.27)

where b₀ is a constant, and it is considered that b₀ > 0 to satisfy the admissibility condition. Figure

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