Preliminaries

A univariate EEG signal $\mathbf{x}\in {\mathbb{R}}^t$ is a sequence of $t$ real-valued observations sampled at uniform time intervals from a single EEG channel (i.e., electrode):

$$\mathbf{x}=[x_1,x_2,\dots ,x_t{]}^{\top }$$

Here, $t$ denotes the sequence length of the signal.

A multivariate EEG time series $X\in {\mathbb{R}}^{d\times t}$ consists of $d$ univariate signals recorded concurrently across $d$ EEG channels:

$$X=\left[\begin{array}{c}{\mathbf{x}}^{(1)}\\ {\mathbf{x}}^{(2)}\\ ⋮\\ {\mathbf{x}}^{(d)}\end{array}\right]$$

Each ${\mathbf{x}}^{(i)}\in {\mathbb{R}}^t$ represents a univariate signal from channel $i$, and $t$ is the shared sequence length.

An EEG time series classification (TSC) dataset $\mathcal{D}$ comprises $n$ multivariate EEG samples and their corresponding class labels:

$$\begin{array}{rl}\mathcal{X}=\{X_1,X_2,\dots ,X_n\}& ,X_i\in {\mathbb{R}}^{d\times t_i}\\ \mathbf{y}=[y_1,y_2,\dots ,y_n{]}^{\top }\in {\mathbb{N}}^n& ,y_i\in \mathcal{C}=\{1,2,\dots ,c\}\end{array}$$

Here, $d$ is the number of EEG channels (constant across all samples), $t_i$ is the sequence length for sample $i$, and $\mathcal{C}$ is the set of target classes.

A time series classification model is a function

$$\mathcal{M}:{\mathbb{R}}^{d\times t}\to [0,1{]}^c$$

that maps a multivariate time series $X\in {\mathbb{R}}^{d\times t}$ to a class probability vector

$$\hat{\mathbf{y}}=\mathcal{M}(X)=[{\hat{y}}_1,{\hat{y}}_2,\dots ,{\hat{y}}_c{]}^{\top },\sum _{i=1}^c{\hat{y}}_i=1$$

Each ${\hat{y}}_i$ represents the predicted probability of class $i$. The predicted label is given by

$$\hat{y}=\arg \underset{i\in \mathcal{C}}{\max }{\hat{y}}_i$$