## 2 Material and Methods
### Cross-Spectral Estimation from Kernel Mixtures  
The communication between neuron cells is the basis of every neuronal processing task. The electrical impulses result in every possible cognitive or physiological condition, such as behavior, sensation, thoughts, and emotions [27]. Due to equally measuring normal and abnormal BEA, EEG is considered a well-suited neuroimaging technique for diagnosis, treatment, and clinical procedures across several neurological pathologies [1]. Equation (1) presents the mathematical representation of the BEA from an EEG of \(C\) channels holding \(T\) time instants [28]. \[\chi=\{\mathbf{t}\in\mathbb{R}^{T},\mathbf{X}\in\mathbb{R}^{C\times T}\}, \tag{1}\]  
with \(\mathbf{t}\) as the time sample positions of the recordings, and \(\mathbf{X}=\{\mathbf{x}_{i}\}_{i=1}^{C}\) holding the brain electrical responses measured by the EEG array at channel \(i\). To quantify the spectral content between channels, the introduced cross-spectrum estimation relies on specific covariance functions. The [PERSON]'s theorem states that a family of integrable functions \(\{\kappa_{ij}(\tau)\}_{i,j=1}^{C}\) are the covariance functions of a weakly-stationary stochastic process if and only if they admit the following representation:  
\[\kappa_{ij}(\tau)=\int_{\mathbb{R}^{n}}e^{\omega^{\gamma}\tau}S_{ij}(\omega)d\omega, \tag{2}\]  
being \(\imath\) the imaginary unit, each \(S_{ij}(\omega)\) an integrate complex-valued function \(S_{ij}:\mathbb{R}\rightarrow\mathbb{C}\) that is also positive definite, and \(i,j\) the indices of two EEG channels. This relationship between covariance functions \(\kappa_{ij}\) in the time domain with argument \(\tau\in\mathbb{R}\) and their corresponding spectral density \(S_{ij}\) with arguments \(\omega\) in the Fourier domain allows designing a desired spectral density and obtaining a covariance function [24]. Now, a family \(\mathbf{S}=\{S_{ij}\}_{i,j=1}^{C}\in\mathbb{R}^{C\times C}\) of positive-definite complex-valued functions can be used as cross-spectral densities for multi-output data [24]. These functions are designed by including specific parameters that allow physical interpretation of the obtained covariance kernel regarding the inputdata. Moreover, complex-valued and positive-definite matrices can be decomposed in the form \(\mathbf{S}(\omega)=\mathbf{R}^{H}(\omega)\mathbf{R}(\omega)\) where \(\mathbf{R}(\omega)\in\mathbb{R}^{Q\times C}\), \(Q\) represents the rank of decomposition, and \((\cdot)^{H}\) denotes the Hermitian operator. Since Fourier transforms and multiplications of squared exponential (SE) functions are also SE, the autocovariance function \(R_{i}(\omega)\) of \(i\)-th channel is modeled as the complex-valued SE in Equation (3). \[R_{i}(\omega)=w_{i}\exp\left(-\frac{1}{4}\frac{(\omega-\mu_{i})^{2}}{\sigma_{i }^{2}}\right)\exp\left(-\iota(\theta_{i}\omega+\phi_{i})\right), \tag{3}\]  
with \(w_{i},\phi_{i},\mu_{i},\theta_{i}\in\mathbb{R}\) and \(\sigma_{i}\in\mathbb{R}^{+}\). With such a choice of functions, the cross-spectral density between channels \(i\) and \(j\) is given by Equation (4) with covariance \(\sigma_{ij}\in\mathbb{R}^{+}\), mean \(\mu_{ij}\in\mathbb{R}\), magnitude \(w_{ij}\in\mathbb{R}\), delay \(\theta_{ij}\in\mathbb{R}\), and phase \(\phi_{ij}\in\mathbb{R}\). \[S_{ij}(\omega)=w_{ij}\exp\left(-\frac{1}{2}\frac{(\omega-\mu_{ij})^{2}}{ \sigma_{ij}}+\iota(\theta_{ij}\omega+\phi_{ij})\right). \tag{4}\]  
Finally, in order to guarantee that the model is restricted to real-valued stochastic processes, the spectral density is reassigned to become symmetric with respect to \(\omega\) by \(S_{ij}(\omega)\rightarrow\frac{1}{2}(S_{ij}(\omega)+S_{ij}(-\omega))\). Then, the inverse Fourier transform of the resulting cross-spectral density becomes the corresponding temporal domain real-valued kernel; the kernel and the symmetric version of the spectral density are presented in Equations (5) and (6), respectively.