## 2 Materials and Methods
### Improved Gaussian Mixture Model  
The Gaussian mixture model (GMM) is composed of k sub-Gaussian distribution models, which are the hidden variables of the mixture model, and the probability density function of the Gaussian mixture model is formed by the linear summation of these Gaussian distributions [21; 22]. A model is chosen randomly among the k Gaussian models according to the probability. The probability distribution of the Gaussian mixture model can be described as follows:  
\[p(x)=\sum_{k=1}^{K}\varphi_{i}N(x\mid\mu_{i},\sigma_{i}) \tag{11}\]  
where \(\varphi_{i}\) is the coefficient \((\varphi_{i}\geqslant 0,\quad\sum_{k=1}^{K}\varphi_{i}=1)\). \(N(x\mid\mu_{i})\) is the probability density function of the Gaussian distribution, and the associated parameters indicate that there are k Gaussian models, each with 3 parameters, namely, the mean \(\mu_{i}\), the variance \(\sigma_{i}\), and the generation probability \(\varphi_{i}\). The sample data are generated by their Gaussian probability density function \(N(x\mid\mu_{i})\) after model selection.  
\[N(x\mid\mu_{i},\sigma_{i})=\frac{1}{\sigma_{i}\sqrt{2\pi}}\exp\left(-\frac{(x- \mu_{i})^{2}}{2\sigma_{i}^{2}}\right) \tag{12}\]  
The maximum likelihood function method is used to train the Gaussian mixture model. The likelihood function can be expressed as follows:  
\[L=\prod_{i=1}^{L}p(x_{i}\mid\varphi) \tag{13}\]  
where L is the number of samples in the dataset, x is the data object (\(i=1,2,3\cdots\)N), and p(x) denotes the probability of data sample generation in the Gaussian mixture model. The mixed model with maximum likelihood is calculated as follows:  
\[\log L=\sum_{i=1}^{N}P(x_{i}\mid\varphi)=\sum_{i=1}^{N}\log\left(\sum_{k=1}^{ K}\alpha_{i}N(x\mid\varphi_{i})\right) \tag{14}\]  
The Gaussian mixture model cannot acquire the derivative, and the EM algorithm is used to solve the best parameters of the model according to the maximized likelihood function in the training phase of the GMM model; the mixture probability, \(\varphi_{k}\); the mean, \(\mu_{k}\); and the covariance, \(\sigma_{i}\), until the convergence of the model.  
However, the use of a single GMM as a classifier in movement quality assessment has certain drawbacks. First, it may oversimplify the complexity of motion data and affect the model performance [23; 24]. Second, the sensitivity to noisy data and statistics outliers is a weak feature of a single Gaussian mixture model, which reduces the model accuracy [25]. Meanwhile, promising results have been achieved in movement quality assessment by combining weak classifiers into a strong classifier in machine learning methods. Several studies have confirmed the validity of this method. For example, [PERSON] an automated FMS assessment method based on an AdaBoost.M1 classifier, which trains different weak classifiers for the FMS dataset collected by IMU and then combines these weak classifiers to form a powerful classifier for FMS [15]. In addition, [PERSON] evaluated the arm movements of stroke patients using a random forest model with randomly selected samples to form multiple classifiers, and then used a minority-majority approach to predict the classification labels through a voting method [16]. Therefore, we proposed an automated FMS assessment method based on the idea of combining three Gaussian mixture models into a strong classifier in this study.  
As shown in Figure 3, firstly, a Gaussian mixture model is trained separately for the movement features with different scores to obtain the Gaussian mixture model probability distributions of 1 point, 2 points, and 3 points (\(p_{1}(x)\), \(p_{2}(x)\), \(p_{3}(x)\), respectively). Next, the feature data with unknown scores are modeled using each of the 3 Gaussian mixture models, and finally, maximum likelihood estimation is performed to obtain the evaluation results. We evaluated the performance of the new classifier by comparing it with three basic classifiers, including the traditional Gaussian mixture model [26], Naive Bayes [27], and AdaBoost.M1 classifier [15].