# Toward Improving the Reliability of Discrete Movement Recognition of sEMG Signals
## 4 Proposed Methods
### GMM Classifier with a Reject Option  
GMM is a parameter-based classifier, and it makes an assumption about the samples. Our previous study shows that GMM has an excellent performance in sEMG signal-based movement recognition [5] for high accuracy and fast responses. Therefore, the GMM classifier is used as a baseline classifier in this study.  
Theoretically, GMM can be seen as the weighted sum of several Gaussian components that best approximate the input. It can be modeled as follows.  
\[p(\mathbf{x}|\Theta)=\sum_{i=1}^{N}w_{i}N(\mathbf{x}|\mathbf{\mu}_{i},\mathbf{ \Sigma}_{i}) \tag{7}\]  
\[\sum_{i=1}^{N}w_{i}=1 \tag{8}\]  
where \(w_{i}\) represents the prior probability of the ith component; and \(N(\mathbf{x}|\mathbf{\mu}_{i},\mathbf{\Sigma}_{i})\) is the Gaussian distribution of the ith component defined by mean vector \(\mathbf{\mu}_{i}\) and covariance matrix \(\mathbf{\Sigma}_{i}\). Then, given an observation \(\mathbf{x}\) with dimension \(d\), the conditional probability density of \(\mathbf{x}\) obtained from the cth category can be written as follows.  
\[p(\mathbf{x}|\Theta_{c})=\sum_{i=1}^{N}\pi_{i}\frac{1}{\left(2\pi\right)^{d/2 }\left|\mathbf{\Sigma}_{i}\right|^{\frac{1}{2}}}\exp\left(-\frac{1}{2}(\mathbf{x}- \mathbf{\mu}_{i})^{T}\mathbf{\Sigma}_{i}{}^{-1}(\mathbf{x}-\mathbf{\mu}_{i})\right) \tag{9}\]  
where \(\Theta_{c}=\{\mathbf{\pi},\mathbf{\mu},\mathbf{\Sigma}\}\) represents the parameter set of the Gaussian mixture model of the cth category, obtained from model training using the Expectation-Maximization (EM) algorithm. According to [PERSON] theory, the probability of \(\mathbf{x}\) belonging to class \(c\) is proportional to the conditional probability density if each class's prior probability is the same. Therefore, the class label of \(\mathbf{x}\) can be determined by finding the category of the maximum conditional probability as long as the conditional probability densities of \(\mathbf{x}\) from each class are obtained.  
The test sample \(\mathbf{X}\) of sEMG signals collected from contracting muscles for each discrete movement consists of many observations \(\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\), where \(N\) is the number of observations. Therefore, the prediction on the discrete movement by using (10) is a series of class labels. To extend the GMM to have a rejection option (GMM-R), we introduce a majority voting (MV) scheme to the series of class labels obtained in this study. We tabulate these class labels into a 2-column table, where the first column contains the unique values of class labels, and the second column contains the percentage of each class label. The GMM-R makes its decision based on the maximum percentage value. Suppose the percentage of each class label is \(G_{1},\ldots,G_{C}\), and  
\[\hat{g}=\underset{c\in\text{1:C}}{\text{argmax}}\text{G}_{c} \tag{10}\]  
Then given a predefined threshold \(\delta_{g}\), the decision of GMM-R can be expressed as follows.  
\[g(\mathbf{X})=\left\{\begin{array}{ll}\hat{g},\text{G}_{\delta}>\delta_{g} \\ \text{\text{\text@underline{\text{\text{\text{\text{\text{\text{\text{\text{ \text{\text{\where 1 represents that the prediction is in doubt, and a further judgment should be made. In the GMM classifier, the number of mixture components plays a vital role in its performance. We have conducted a preliminary study on the influence of the number of mixture components on the performance of GMM [41], and it was found that when the number is three, a trade-off between classification accuracy and computational complexity can be obtained [5]. Therefore, the number of mixture components will be three in the following study.