## 3 Results  
### Parameter Tuning and Spectral Modeling  
To tune the rank of decomposition \(Q\), we evaluate the MOSM-GP performance for modeling the selected output channels from the data posterior distribution at specific temporal locations. Moreover, the mean absolute error (MAE) quantifies the difference between the target and predicted outputs as a function of the rank. Figure 3 presents the mean MAE across the GP outputs against the number of spectral components defined for the MOSM kernel. As a first insight, the MAE values evidence that the MOSM-GP effectively reconstruct EEG recordings. Nonetheless, the error increases over six spectral components, due to a large number of kernel parameters to be tuned, which in turn increases the computational complexity without providing relevant information to the probabilistic model. On the contrary, a single component lacks the complexity to account for the brain activity changes. Therefore, a rank of decomposition between three and five benefits the most the MAE performance, implying a balance between model complexity and generalization capability. Consequently, for the remaining of the work, we selected three spectral mixtures as the optimal Q for testing the MOSM-GP scheme. For the purpose of visualization, Figure 4 exemplifies the MOSM-GP output for channels FP1 and AF3 using \(Q=3\). As seen, the posterior MOSM distribution suitably models EEG data at all time locations with bounded deviations. An analysis of the spectral information quantified by the MOSM-GP is carried out using Equation (6) that decomposes the spectral content shared by two channels into \(Q\) terms. Figure 5 plots the component-wise spectral distribution between channels FC2 and FC6 in a trial from left- (Figure 5a) and right-hand movement (Figure 5b). Attained spectra prove that each component automatically fits a particular frequency band. Moreover, the component magnitude \(\alpha_{ij}^{(q)}\) highlights the dominant component from each trial as the most discriminative frequency band. As expected, such frequencies lay around 15 Hz to 35 Hz for MI experiments, being associated with an activation of alpha and beta rhythms. Later, Equation (8) is used to compute the cross-spectrum visualizing the computed covariances by the MOSM kernel for every channel pair. Since there is valuable information on the analysis of the cross-covariances between channel pairs, a complete trial visualization of the quantified spectral information is presented in Figure 6a,b for left- and right-hand movement trials, respectively. Each one  
Figure 3: Mean absolute error (MAE) values for the data reconstruction of electroencephalography (EEG) channels after training a MOSM-Gaussian Process (GP) compared to the original recordings. of the horizontal axis sections, corresponds to a particular channel \(i\) and its MOSM PSD against all the channels. The vertical axis represents the frequency bin at which the connectivity is assessed. Then, lighter green colors are related with strong spectral densities shared between \(i,j\) channels, while darker blue colors are related with lower interdependencies. Despite most of the strong spectral density being constrained to the \([15,40]\) Hz band, there is clear evidence that not all the channels are synchronized at this frequency when performing MI activity. Particularly for Figure 5(a), there are channels sections that seems to be uncorrelated in the complete spectrum for this particular task. For example, channels Cz, C2, and CP6 seem to have low frequency dependencies shared with the rest of the EEG array. On the other hand, channel CP2 presents strong connections with most of the channels. Furthermore, for the opposite MI task, there exist variations on the captured frequency relationships among the EEG array. C2 and C4 result as the most highly correlated the other channels when performing the MI task. Figure 4: Channels FP1 and AF3 from EEG recordings adjusted by an MOSM with three spectral mixtures and its corresponding model posterior along time.