# Preprocessing for Lessening the Influence of Eye Artifacts in EEG Analysis
## 4 Artifact Processing Methods
### Rejection of ICA Components  
Following the idea of ICA in the EEG analysis context, we will identify components or artifacts in specific components for their removal or attenuation. The trial subdivision of the signals is considered irrelevant regarding ICA. Nevertheless, the resulting \(\mathbf{s}(t)\) maintains the trial substructure so that each IC is defined as \(s_{n}(t)=[s_{n,1}(t),s_{n,2}(t),\ldots,s_{n,m}(t)]\) where \(m\) is the number of trials.  
In the literature, clean EEG data are obtained by simply removing components, which were previously selected via visual inspection of the data [9; 11], automatically by template matching [13] or by EOG cross-correlation measures [14].  
In this paper, manual IC identification is not done but the automatic identification of components and their artifactual excerpts is considered by measuring the cross-correlation coefficient with the EOG channel based on the fact that the EOG channel contains primarily ocular motion.  
Possible small delays between the recorded signals are considered though these should not be significant [3]. Thus, it is expected to find the most prominent peak around \(\tau=0\); for most cases, the peak was found within \(\pm 20\) ms from that point, which, at the considered sampling rate, means a slack of less than 7 samples. The cross-correlation function is normalized by the variances of the signals involved in the calculation. Note that only negative correlation values are valid, since the polarity of eye artifact potentials are inverted in the scalp electrodes [1]. Hence, the minimum cross-correlation coefficient for each component was chosen.  
The negative cross-correlations coefficients \(\rho\) between the corresponding EOG trial and each trial of each IC are computed as follows:  
\[\rho(n,m)=\frac{\min_{|\tau|<\mathcal{T}}\left(\sum\limits_{i=-\inf}^{\inf} EOG_{m}(i)s_{n,m}(i-\tau)\right)}{\sqrt{\sigma_{EOG_{m}}^{2}*\sigma_{s_{n,m}}^{2}}} \tag{2}\]  
This produces a \(n\times m\) matrix of negative cross-correlation coefficients \(\mathbf{C}\) for each participant:  
\[\mathbf{C}=\left[\begin{array}{cccc}\rho(1,1)&\rho(1,2)&\ldots&\rho(1,m)\\ \rho(2,1)&\rho(2,2)&\ldots&\rho(2,m)\\ \vdots&\vdots&\ddots&\vdots\\ \rho(n,1)&\rho(n,2)&\ldots&\rho(n,m)\end{array}\right] \tag{3}\]where \(n\) is the number of different ICs and \(m\) is the number of trials.  
An illustration of the coefficients of \(\mathbf{C}\) is plotted in Figure 3 where every dashed line corresponds to a different trial for a selected subject.  
Then, the modulus of the elements of \(\mathbf{C}\) are computed and the rows summed (which represent the trials) according to:  
\[\mathbf{cc}=\left[\begin{array}{c}\sum\limits_{m=1}^{5}\left|\rho(1,m)\right| \\ \sum\limits_{m=1}^{5}\left|\rho(2,m)\right|\\ \vdots\\ \sum\limits_{m=1}^{5}\left|\rho(n,m)\right|\end{array}\right]^{T} \tag{4}\]  
Figure 4 shows an example of the vector \(\mathbf{cc}\) of cumulative cross-correlation coefficients for the same subject as in Figure 3. The two components with the highest values in \(\mathbf{cc}\) are selected as artifactual ones and marked for further processing, i.e., complete removal (CR). This choice was adopted since it was observed that, in the subjects analyzed, two of the components showed a significantly larger cumulative cross-correlation coefficient than the others; see Figure 4 (note that an analysis of this fact with a larger set of subjects would be beneficial for the definition of this or a different approach to select such components).  
A scheme of the process described in this section is drawn in Figure 5.  
Though processing can be concreted in the complete removal (CR) of selected components, their partial rejection (PR) can also be considered as it will be described in the next section, for the later analysis of reconstructed EEG channels.  
Figure 3: Illustration of the cross-correlation coefficients obtained between every IC and EOG in all trials for subject \(S_{12}\) (negative values are reversed).