## 3 Experimental Results and Analysis
### Classification Results and Analysis Based on Support Vector Machine  
Formant patterns have a certain degree of discrimination for different vowels. Therefore, following the classification evaluation method based on the Support Vector Machines (SVM) algorithm described in reference [17], this study conducted a classification analysis of vowels produced by learners at different proficiency levels. This method helps correct pronunciation issues in tongue position and lip shape for Mandarin Chinese learners. SVM is a supervised learning algorithm primarily used for classification and regression tasks. Its core idea is to find an optimal hyperplane (or a hyperplane after non-linear mapping) that maximizes the margin between different classes of samples to achieve high generalization performance in classification [28; 29]. We optimized the SVM via grid search (five-fold cross-validation) with a penalty factor of C\(\in\){0.1,1,10} and an RBF kernel width of \(\sigma\in\){0.01,0.1,1}. The best performance (C = 10, \(\sigma\) = 0.1) was selected for all subsequent analyses. To benchmark our formant-based features (F1-F2), we extracted 256-dimensional embeddings using wav2 vec 2.0 [7] (mean-pooled frame-level features). SVM achieved comparable accuracy (92.1% vs. 91.8% with wav2 vec), confirming the efficacy of our approach for pronunciation assessment. The formant patterns for different monophthongs are shown in Figure 1. The formant trajectories in the figure demonstrate that different vowels exhibit distinctly different formant patterns. SVM is a learning method that performs optimal classification for two inseparable problems. SVM is widely applied in various fields such as image processing, speech signal processing, and speech recognition [30] because it considers the error as a constraint in the optimization problem and employs a structural risk minimization learning method. To separate monophthong samples in the speech of learners at different proficiency levels, the optimal solution involves minimizing the following objective function:  
\[\phi(\omega,\xi)=\frac{1}{2}(\omega,\omega)+c\left(\sum_{i=1}^{n}\xi_{i}\right) \tag{1}\]  
subject to the following constraints:  
\[y_{i}[(\omega\cdot x_{i})+b]-1+\xi_{i}\geq 0,i=1,2,\ldots,l \tag{2}\]  
where \(\xi\) is slack variables representing margin violations (Equation (1)); \(\omega\) is weight vector determining the decision boundary (Equation (1)); \(c\) is regularization parameter controlling the penalty for misclassified samples (Equation (1)); \(x_{i}\) is training samples; and \(y_{i}\in\){\(-\)1,1} is class labels. This formulation corresponds to the soft-margin Support Vector Machine (SVM) framework, where the goal is to maximize the margin between classes while allowing controlled misclassifications via slack variables \(\xi_{i}\). The parameter \(c\) balances the trade-off between maximizing the margin and minimizing classification errors. To solve its dual problem, first find the maximum value of a quadratic function:  
\[W(a)=\sum_{i=1}^{l}a_{i}-\frac{1}{2}\sum_{i,j=1}^{l}y_{i}y_{j}a_{i}a_{j}K\big{(} x_{i},x_{j}\big{)} \tag{3}\]  
The constraints are  
\[0\leq a_{i}\leq C,i=1,2,\ldots,l;\ \ \sum_{i=1}^{l}a_{i}y_{i}=0 \tag{4}\]  
In Equation (3), \(a_{i}\) is the Lagrange multiplier, \(K\big{(}x_{i},x_{j}\big{)}\) is the kernel function, and \(\big{(}x_{i},x_{j}\big{)}\) is the sample set. By using the radial basis function (RBF) as the inner product kernel function,  
\[K(x,x_{i})=exp\left\{-\frac{\left|x-x_{i}\right|^{2}}{\sigma^{2}}\right\} \tag{5}\]The output weights are automatically determined by the algorithm. In the equation, \(x_{i}\) is the sample and \(\sigma\) is a variable parameter. \[f(x)=\text{sign}\left\{\left[\sum_{i=1}^{l}a_{i}y_{i}K(x,x_{i})\right]+b\right\} \tag{6}\]  
Figure 2 below shows the classification results of each monophthong for learners at different levels based on the SVM. As shown in Figure 3, the overall classification performance for each vowel achieves an average accuracy of 92.08%.