## 3 Feature Extraction
### Spatial Domain Feature Extraction  
In mathematical statistics, covariance is commonly used as a parameter to investigate the relationship between two random variables, reflecting the second-order statistical properties between variables. Currently, there are limited studies utilizing covariance features to study EMG signal recognition issues [24; 25]. The main reason is that EMG signals are one-dimensional time series, requiring the identification of an appropriate covariance matrix to explore the signal dynamics in high-dimensional space. Describing EMG information characteristics through the covariance matrix offers several advantages: it can integrate nonlinear feature information represented in the spatial domain; furthermore, the covariance matrix provides a method to filter out noise. By using spatial reconstruction to construct the covariance matrix from EMG signal sequences, potential signal dynamics information can be unearthed. An attempt is made to measure the similarity between covariance matrices using Euclidean distance, primarily achieved by vectorizing the matrices. Subsequently, through the use of classifiers, an effort is made to decode the underlying feature information and human kinematic information from higher dimensions, while restoring the spatial feature information between different channels of the EMG signals. In this study, by constructing the inter-channel covariance matrix of the EMG signals, referred to as the inter-channel correlation (ICC) matrix, we aim to delve deeper into the hidden spatial feature information and human movement information between channels of the EMG signals. As shown in Figure 2, the spatial correlation of EMG signals is embedded within adjacent or correlated channels. Figure 2: Example characteristics of EMG data. Cross correlation between channels of EMG data— note how many groups of channels have correlation. The ICC matrix is utilized to learn spatial representation from EMG signals. We constructed the ICC matrix in the following, let \(x_{t}\in\mathbb{R}^{c}\) denote the sEMG signal vector at a specific time point \(t\), with \(c\) denoting the number of recording channels. The ICC matrix is formally defined by \(\Sigma=E(x_{t}-E\{x_{t}\})(x_{t}-E\{x_{t}\})^{T}\), where \(E\{\cdot\}\) denotes the expected value. In ICC matrix we consider short-time sEMG signal, in the form of a matrix \(X_{i}=[x_{(t+T_{i})}\ldots x_{(t+T_{i}+T_{i}-1)}]\in\mathbb{R}^{c\times T_{n}}\), which corresponds to the \(i\)-th segments of movement started at time \(t=T_{i}\). Here, \(T_{i}\) denotes the number of sampled time points in each segment. \[C_{i}=\frac{1}{T_{n}-1}X_{i}*X_{i}^{T} \tag{2}\]  
where \(X\) is EMG signal pre-processed using the procedure is Section 4.2. It is with shape \((c\times n)\), where \(n,c\) are the signal's fragment length and the number of channels. The ICC matrix is known to be an unbiased estimator of the covariance matrix \(\Sigma\) provided that the number of observations \(T_{n}\) is much than the number of variables \(c\)[23]. The ICC matrix is based on the covariance matrix using Equation (2), revealing spatial correlations among different channels and partially presenting the distribution characteristics of EMG signal data within each channel. Given that covariance reflects the distribution and variability of data, the ICC matrix not only captures spatial correlations among different channels but also characterizes the distribution of EMG signal vectors within the same channel. The variable \(X_{i}\) represents the \(i\)-th signal segment containing a signal vector of length \(n\) and \(c\) channels, the covariance matrix between different channels of this signal segment is computed using Equation (2). Here, \(T_{n}\) denotes the time length \(n\) of the signal vector for the respective signal segment. Ultimately, the covariance matrix for signal segment \(i\) is a symmetric matrix with dimensions \((c\times c)\).