EEG Signal Processing and Machine Learning. Saeid Sanei
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Название: EEG Signal Processing and Machine Learning
Автор: Saeid Sanei
Издательство: John Wiley & Sons Limited
Жанр: Программы
isbn: 9781119386933
isbn:
(4.78)
After repeated sifting up to k times, d 1 becomes an IMF, i.e.:
(4.79)
C 1 = d 1k is considered as the first IMF of the signal x(t).
The iteration above can be stopped in different ways such as when the power (standard deviation) of the difference (between current and previous iteration) signal becomes less than a predefined threshold, or when the number of iterations reaches a reasonable number [27].
For calculation of the other IMFs:
(4.80)
The residue r 1 is then treated as the new signal and the same processing is applied to that. Therefore
(4.81)
The sifting process finally stops when the residue, rn , becomes a monotonic function from which no more IMFs can be extracted. From the above equations, it is induced that:
(4.82)
This results in decomposition of the data into n‐empirical modes [25, 26].
Ensemble EMD (EEMD) is a noise assisted data analysis method. EEMD consists of ‘sifting’ an ensemble of white noise‐added signal. EEMD can separate scales naturally without any a priori subjective criterion selection as in the intermittence test for the original EMD algorithm. Complete ensemble EMD with adaptive noise (CEEMDAN) is a variation of the EEMD algorithm that provides an exact reconstruction of the original signal and a better spectral separation of the IMFs.
4.7 Coherency, Multivariate Autoregressive Modelling, and Directed Transfer Function
In some applications such as in detection and classification of finger movement, it is very useful to find out how the associated movement signals propagate within the neural network of the brain. As will be shown in Chapter 16, there is a consistent movement of the source signals from the occipital to temporal regions. It is also clear that during the mental tasks different regions within the brain communicate with each other. The interaction and cross‐talk among the EEG channels may be the only clue to understanding this process. This requires recognition of the transient periods of synchrony between various regions in the brain. These phenomena are not easy to observe by visual inspection of the EEGs. Therefore, some signal processing techniques have to be used in order to infer such causal relationships. One time series is said to be causal to another if the information contained in that time series enables the prediction of the other time series.
The spatial statistics of scalp EEG are usually presented as coherence in individual frequency bands, these coherences result both from correlations among neocortical sources and volume conduction through the tissues of the head, i.e. brain, cerebrospinal fluid, skull, and scalp. Therefore, spectral coherence [28] is a common method for determining the synchrony in EEG activity. Coherency is given as:
(4.83)
Figure 4.10 Cross‐spectral coherence for a set of three electrode EEGs, one second before the right‐finger movement. Each block refers to one electrode. By careful inspection of the figure, it is observed that the same waveform is transferred from Cz to C3.
where
Granger causality (also called as Wiener–Granger causality) [30] is another measure, which attempts to extract and quantify the directionality from EEGs. Granger causality is based on bivariate AR estimates of the data. In a multichannel environment this causality is calculated from pair‐wise combinations of electrodes. This method has been used to evaluate the directionality of the source movement from the local field potential in the visual system of cats [31].
For multivariate data in a multichannel recording, however, application of the Granger causality is not computationally efficient [31, 32]. The directed transfer function (DTF) [33], as an extension of Granger causality, is obtained from multichannel data and can be used to detect and quantify the coupling directions. The advantage of the DTF over spectral coherence is that it can determine the directionality in the coupling when the frequency spectra of the two brain regions have overlapping spectra. The DTF has been adopted by some researchers for determining the directionality in the coupling [34, 35] since it has been demonstrated that [36] there is a directed flow of information or cross‐talk between the sensors around the sensory motor area before finger movement. The DTF is based on fitting the EEGs to an MVAR model. Assuming that x(n) is an M‐channel EEG signal, it can be modelled in vector form as:
where n is the discrete‐time index, p is the prediction order, v(n) is zero‐mean noise, and L k is generally an M × p matrix of prediction coefficients. A similar method to the Durbin algorithm for single‐channel signals, namely the Levinson–Wiggins–Robinson (LWR) algorithm is used to calculate the MVAR coefficients [14]. The Akaike information criterion (AIC) [37] is also used for the estimation of prediction order p. By multiplying both sides of the above equation (Eq. 4.84) by xT (n − k) and performing the statistical expectation the A set of Yule–Walker equation is obtained as [38]:
(4.85)
where R(q) = E[x(n)xT (n + q)] is the covariance matrix of x(n), and the cross‐correlations of the signal and noise are zero since they are assumed uncorrelated. Similarly, the noise autocorrelation is zero for non‐zero СКАЧАТЬ