Saeid Sanei - EEG Signal Processing and Machine Learning

where is the Fourier transform of the cross‐correlation coefficients between channel i and channel j of the EEGs. Figure 4.10shows an example of the cross‐spectral coherence around one second prior to finger movement. A measure of this coherency, such as an average over a frequency band, is capable of detecting zero time‐lag synchronization and fixed time non‐zero time‐lag synchronization, which may occur when there is a significant delay between the two neuronal population sites [29]. However, it does not provide any information on the directionality of the coupling between the two recording sites.

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:

(4.84)

where n is the discrete‐time index, p is the prediction order, v( n ) is zero‐mean noise, and L kis 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 x T( 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 ) x T( 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 shift since the noise samples are uncorrelated. The data segment is considered short enough for the signal to remain statistically stationary within that interval and long enough to enable accurate measurement of the prediction coefficients. Given the MVAR model coefficients, a multivariate spectrum can be achieved. Here it is assumed that the residual signal, v( n ), is white noise. Therefore,

(4.86)

where

(4.87)

and L(0) = I. Rearranging the above equation ( Eq. 4.87) and replacing noise by σ v 2 Iyields

(4.88)

which represents the model spectrum of the signals or the transfer matrix of the MVAR system. The DTF or causal relationship between channel i and channel j can be defined directly from the transform coefficients [32] given by:

(4.89)

Electrode i is causal to j at frequency f if:

(4.90)

A time‐varying DTF can also be generated (mainly to track the source signals) by calculating the DTF over short windows to achieve the short‐time DTF (SDTF) [32].

As an important feature in classification of left and right‐finger movements, or tracking the mental task related sources, SDTF plays an important role. Some results of using SDTF for detection and classification of finger movement have been given in the context of BCI.

The EEG signals are subject to noise and artefacts. Electrocardiograms (ECGs), electro‐oculograms (EOG) or eye blinks affect the EEG signals. Any multimodal recording such as EEG–functional magnetic resonance imaging (fMRI) significantly disturbs the EEG signals because of both magnetic fields and the change in the blood oxygen level and sensitivity of oxygen molecule to the magnetic field (balisto‐cardiogram). Artefact removal from the EEGs will be explained in the related chapters. The noise in the EEGs, however, may be estimated and mitigated using adaptive and non‐adaptive filtering techniques.

The EEG signals contain neuronal information below 100 Hz (in many applications the information lies below 30 Hz). Any frequency component above these frequencies can be simply removed by using lowpass filters. In the cases where the EEG data acquisition system is unable to cancel out the 50 Hz line frequency (due to a fault in grounding or imperfect balancing of the inputs to the differential amplifiers associated with the EEG system) a notch filter is used to remove it.

The nonlinearities in the recording system related to the frequency response of the amplifiers, if known, are compensated by using equalizing filters. However, the characteristics of the internal and external noises affecting the EEG signals are often unknown. The noise may be characterized if the signal and noise subspaces can be accurately separated. Using principal component analysis (PCA) or independent component analysis (ICA) we are able to decompose the multichannel EEG observations to their constituent components such as the neural activities and noise. Combining these two together, the estimated noise components can be extracted, characterized, and separated from the actual EEGs. These concepts are explained in the following sections and their applications to the artefact and noise removal will be brought in the later chapters.