3.1 Optimization strategy of DTW algorithm for automatic English teaching

The traditional English teaching method has many problems, such as large number of students, too much content and limited class time, which limits the quality and efficiency of English teaching ^[16]. DTW algorithm is a common technique in time series pattern matching, especially in speech recognition, which can compare the similarity between two time series ^[17]. Therefore, the study attempts to construct an English intelligent teaching platform based on the DTW algorithm, dynamically adjusting teaching strategies to meet the personalized needs of students. The DTW algorithm first defines two time series. The definition method is shown in Eq. (1).

In Eq. (1), $A$ and $B$ are two different time series. $i$ and $j$ are the lengths of $A$ and $B$, respectively. $a$ and $b$ represent the time points in $A$ and $B$, respectively. The distance between the two sequences $A$ and $B$ is displayed in Eq. (2).

In Eq. (2), $D$ is the Euclidean distance calculation Eq.. Compared to Euclidean distance, DTW distance can more accurately measure the similarity between time series. The superiority of DTW is shown in Fig. 1.

Fig. 1. Euclidean distance and DTW distance.

In Fig. 1, compared to Euclidean distance, DTW distance has compression and stretching effects on time. It can dynamically plan and find the optimal path ^[18]. The DTW distance calculation is as follows. Firstly, the distance matrix between each point in the sequence is calculated. Next, the path of a matrix from the bottom left corner to the top right corner is determined to minimize the sum of elements along the path. The next grid point is shown in Eq. (3).

In Eq. (3), $D(n,m)$ is the distance between $a_{n} $ and $b_{m} $ for the $(n,m)$-th element. The operational principle of the DTW algorithm is to extend and shorten two time series to calculate their similarity. It decomposes the problem into several sub problems through dynamic programming. The optimal solution of the sub-problem is solved to obtain the optimal solution of the original problem. Next, the cumulative distance operation of DTW is performed, as displayed in Eq. (4).

In Eq. (4), $G$ is the cumulative distance. Fig. 2 is a schematic diagram of the cumulative distance calculation process for DTW.

Fig. 2. Schematic representation of the calculation steps of the DTW algorithm.

Fig. 2 shows the cumulative distance calculation matrix for DTW. The first column of the grid starts from bottom to top. In addition to calculating the distance between the corresponding points, the distance between adjacent grids below should also be added to achieve distance accumulation. The conventional DTW algorithm has high computational complexity, long computation time, sensitivity to noise and outliers. It is also prone to mismatches. To this end, improvements are made on the basis of the DTW algorithm. K-means clustering algorithm is introduced to impose distance and conditional constraints on DTW, aiming to improve the performance and the DTW. The improved DTW search path is dynamically bent and split. The expression is shown in Eq. (5).

In Eq. (5), $X_{a} $ and $X_{b} $ are the inflection points after path optimization. Next, the lengths of $i$ and $j$ are re-restricted. The limiting condition is shown in Eq. (6).

In Eq. (6), the length constraint specifies that when calculating the cumulative distance between two sequences, the lengths of the two sequences must be equal. If the lengths of two sequences are not equal, then in the process of calculating the cumulative distance, there will be a mismatch of points at the corresponding positions. This makes the calculated cumulative distance inaccurate. At this point, multi-scale DTW is introduced, which calculates the DTW distance at different scales and then integrates the results of each scale to obtain the final similarity. This method enables the DTW algorithm to perform well at different scales. It can handle sequences with unequal length. The optimized DTW is shown in Fig. 3.

Fig. 3. Pathway changes before and after the global constraint improvement.

Fig. 3 shows the path changes of the optimized DTW algorithm. Specifically, the multi-scale DTW algorithm divides the time series into multiple subsequences and calculates the DTW distance for each subsequence. These subsequences can be of fixed length or dynamically adjusted based on the length of the time series. According to this approach, the multi-scale DTW algorithm can better handle length mismatch problems. The dynamic programming can be applied to find the optimal path. In each calculation step, the similarity of the entire time series is considered and the optimal matching point pair is selected. In summary, the study introduces clustering algorithms and optimizes DTW with multi-scale and global constraints, ultimately forming the K-means-PDTW algorithm. It provides the main technology for the intelligent platform design of English automatic teaching.

3.2 Design of an English automatic teaching intelligent platform based on K-means-PDTW algorithm

At present, there are some problems in English teaching, such as asymmetrical teaching test and untimely feedback, which makes it difficult for students to improve their English level due to the lack of sufficient support and guidance in the process of English learning ^[19,20]. Therefore, based on the K-means-PDTW algorithm, an intelligent platform for English automatic teaching is designed. Firstly, the weight of the learning path is calculated. The matrix calculation is shown in Eq. (7).

In Eq. (7), $\alpha $ and $\beta $ are the sets of knowledge points and edges in English teaching videos, respectively. $\chi $ and $\lambda $ are different sets of knowledge points. Next, the initial pheromones of the learning path are recorded, as shown in Eq. (8).

In Eq. (8), $\tau $ is the initial pheromone for learning, reflecting the frequency and familiarity of students learning knowledge points. Simultaneously, learning pheromones is related to the learner's path length, time required to complete learning tasks, and learning outcomes. In teaching, due to the correlation between knowledge points, the initial pheromone of learning can be updated. The expression is shown in Eq. (9).

In Eq. (9), $\rho $ is the volatilization factor for learning initial pheromones. Then, to calculate the learning effectiveness of the knowledge points, heuristic information is used for representation, as shown in Eq. (10).

In Eq. (10), $c$, $k$, and $p$ represent block processing, known decision, and exploratory decision, respectively. They are used to understand the learning situation of learners. Finally, the learning path for English teaching is planned based on the initial pheromone and heuristic information of students. The learning path planning framework based on the K-means-PDTW is displayed in Fig. 4.

Fig. 4. Learning pathway planning framework for the K-means-PDTW.

In the learning path framework, the differences in the learning path are analyzed after inputting temporal data, exercise test data, and learning interaction behavior data. The expression is shown in Eq. (11).

In Eq. (11), $\varphi $ is the attenuation coefficient. Then, clustering algorithms are used to classify the learning abilities and knowledge reserves of different students. Their initial pheromone and initial pheromone update data are collected. At the same time, through automatic English teaching, the knowledge point mastery and the interaction between knowledge points are integrated to form heuristic information data. The calculation method for the knowledge point mastery is shown in Eq. (12).

In Eq. (12), $x$ denotes the student quantity. $y$ is the number of exercises. The corresponding knowledge matrix of the test question can be obtained, as shown in Eq. (13)

In Eq. (13), $l$ denotes the knowledge points in the exercise. Then, the heuristic information data and initial pheromone update data are simultaneously input into the transfer matrix to form the final learning path planning. The transition matrix is shown in Eq. (14).

In Eq. (14), $\sigma $ and $k$ are the influence coefficients of $\tau $ and $\eta _{\chi \lambda } $ on transfer, respectively. The learning effect of students is shown in Eq. (15).

In Eq. (15), $u$ and $T$ represent student identity and course duration, respectively. The classification of learning methods for the English automatic teaching intelligent platform based on the K-means-PDTW is displayed in Fig. 5.

Fig. 5. Classification of learning methods of English automatic teaching intelligent platform.

Fig. 5 shows the classification of learning methods for the designed English automatic teaching intelligent platform. The designed platform can automatically process, analyze, and judge input English language data, which enables the platform to quickly and accurately process a large amount of English language data. To meet the different learning needs and preferences of users, in the design of an English automatic teaching intelligent platform based on K-means-PDTW algorithm, English automatic teaching is divided into different learning methods such as graphic and text learning, audio learning, video learning, etc. Displaying English language knowledge and skills such as vocabulary, grammar, and reading comprehension through graphic and textual means can provide more intuitive and easily understandable learning content. This is suitable for users who need to master basic language knowledge and skills. The audio learning method allows users to engage in listening and speaking exercises anytime and anywhere, making it convenient for users to learn English anytime, anywhere. The provided English listening and speaking exercises help users improve their English listening and speaking abilities. The video format provides comprehensive learning content such as English speaking, writing, and speeches, presenting more realistic and vivid language scenes and learning experiences. This is suitable for users who need to improve their practical English application skills.

4.1 Performance analysis of improved DTW algorithm in automatic teaching intelligent platform

The clustering algorithm was used to improve the DTW algorithm. An intelligent platform for English automatic teaching was constructed. Firstly, the proposed K-means PDTW algorithm was validated for performance. The experiment was performed in MATLAB 2017b and simulated using Simulink. Table 1 displayed the experimental environment settings.

Simulink was used as the overall implementation platform in the experiment. The operating platform is Windows10. Using MATLAB as the experimental environment, the memory of the PC side memory is 4GB, and the CPU frequency is 2.4TGHz. Data was stored in MySQL database, and the visual tool in the experiment was Pajck32 5.09. The DAP dataset in UCI (University of California) was tested. The classification result before and after combining SA-GA was statistically analyzed. The DAP dataset is a commonly used classification experiment dataset, called the ship dataset, which is a type of multivariate analysis dataset. The dataset contains 200 samples, divided into 4 categories with 50 data in each category. Each contains 5 attributes. Next, to analyze the superiority of the K-means-PDTW algorithm in the application of English automatic teaching platforms, SSE and DB indicators were selected. The research algorithm was compared with Hierarchical Clustering, OPTICS, and HC-PDTW algorithms. The analysis results were shown in Fig. 6.

Fig. 6. Comparison of SSE and DB metrics.

Fig. 6(a) shows the comparison results of SSE indicators for each algorithm. SSE is the squared error sum in the predicted values of the regression model. A small value of SSE indicates good performance of the algorithm model. In Fig. 6(a), the SSE of the four algorithms rapidly decreased after reaching four clusters. Regardless of the number of clusters, the proposed algorithm was always at the lowest value. When the number of clusters was 8, the SSE value of the research algorithm was 13.3, which was 2.3, 2.8, and 6.2 units lower than the Hierarchical Clustering, OPTICS, and HC-PDTW algorithms, respectively. Fig. 6(b) shows the comparison results of DB metrics for each algorithm. The DB indicator represents the power attenuation degree of sound. A low DB value indicates that the algorithm performs well in recognizing and analyzing audio in the course. In Fig. 6(b), the DB value of the research algorithm remained below 5 after the cluster group exceeded 6. It was also the lowest value. The average DB value of the research algorithm was 4.4, which was 4.2, 6.7, and 7.3 lower than the average DB values of the Hierarchical Clustering, OPTICS, and HC-PDTW algorithms, respectively. Overall, the algorithm used in this study had a small calculation error and the lowest power attenuation of sound. Next, the clustering effects of the four algorithms were analyzed. The results were shown in Fig. 7.

Fig. 7. Clustering comparison of different algorithms.

Fig. 7 displays the clustering analysis comparison of different algorithms on the learning level of students. Fig. 7 (c) displays the clustering performance. In the figure, there was an unclear boundary between data objects in the OPTICS algorithm. There were intersections between data points from different datasets. The clustering effect was not ideal. The HC-PDTW algorithm and Hierarchical Clustering algorithm improved the unclear dataset boundaries in the OPTICS algorithm, but the data points were still relatively scattered. The proposed algorithm effectively improved the weaknesses of the other three algorithms. There was no crossing between data points. The clustering effect of data points was significant. After calculation, the Mahalanobis distances of the three samples in the initial, intermediate, and advanced stages of the research method were 1.58, 1.03, and 0.81, respectively. Compared to the OPTICS algorithm, the Mahalanobis distances were 0.72, 0.73, and 0.47 less. Compared to the HC-PDTW algorithm, the Mahalanobis distances were 0.45, 0.43, and 0.55 less. Compared to the Hierarchical Clustering algorithm, the Mahalanobis distances were 0.34, 0.43, and 0.54 less, respectively. From this, the clustering effect of the research method is the best. Finally, the accuracy of different algorithms under different numbers of learners is analyzed. The accuracy analysis results were shown in Fig. 8.

Fig. 8. Comparison of accuracy of different algorithms.

Fig. 8(a) displays the accuracy comparison results under different numbers of base learners. As the base learners increased, the complexity of the model also increased, which was beneficial for detecting the fitting situation of different algorithms. In Fig. 8(a), the accuracy of the research algorithm exceeded 0.8 when the base learners exceeded 2. When the base learners were 3, the maximum value reached 9.2. This was 0.21, 0.14, and 0.32 higher in accuracy than the Hierarchical Clustering, OPTICS, and HC-PDTW algorithms, respectively. Fig. 8(b) displays the accuracy comparison results under different classification point splitting numbers. The average accuracy of the research algorithm was 0.91, which was 0.21, 0.08, and 0.15 higher than the average accuracy of the Hierarchical Clustering, OPTICS, and HC-PDTW algorithms, respectively. In summary, the algorithm has low error rate, low sound attenuation rate, good clustering effect, and high data accuracy, providing key technologies for the intelligent platform construction of English automatic teaching.

4.2 Teaching effectiveness analysis of English automatic teaching intelligent platform

To further analyze the application effect of the English automatic teaching intelligent platform, the different sounds and signals that occur during the teaching process are modal decomposed to analyze the platform's recognition for noise and classroom sounds in complex teaching environments. The recognition result is shown in Fig. 9.

Fig. 9. Classroom different voice recognition results.

In Fig. 9, the explanation sound, coughing, door closing, and car honking were identified and classified. The results showed that the maximum correlation coefficient between the system platform and the recognition of explanatory sounds was 0.7-0.8. The center frequency was within 1500. The center frequency variance was within 200. Compared to other noises, its center frequency and center frequency variance were the smallest. The research system platform can effectively recognize different sounds. It has good stability in extracting frequency features of different audio. The correlation coefficient, center frequency, and center frequency variance of the whistle sound are the most dispersed, mainly due to the significant fluctuation in the frequency characteristics of the whistle sound. Next, the teaching duration of different learning stages were selected to analyze the degree of course goal achievement and teaching effectiveness of learners in each stage on the system platform. The analysis results were shown in Fig. 10.

Fig. 10. Teaching time of students at different stages of learning.

Fig. 10(a) shows a comparative analysis of the teaching duration in primary stage students. There was a significant fluctuation in the original duration when video knowledge points were different. When the knowledge points were 50, the system required a longer duration. However, when using a research system for teaching, as the number of teaching knowledge points increased, the teaching duration remained below 800 minutes, saving more than twice the original average duration. Fig. 10(b) shows a comparative analysis of teaching time for advanced scholars. The results indicated that students in the advanced stage required longer teaching time compared to students in the primary stage. However, compared to the original required teaching time, the teaching time using the research teaching system was reduced by an average of nearly 300 minutes. Finally, the teaching effectiveness of the English teaching system was analyzed from three aspects: English listening, English speaking, and English reading. The research system was compared with English teaching platforms based on Hierarchical Clustering and HC-PDTW algorithms. The results were shown in Fig. 11.

Fig. 11. Comparison of the English teaching effect of the different models.

Fig. 11 displays a comparison of confusion matrices for three models. Fig. 11(a) displays the teaching results of the Hierarchical Clustering model in English speaking, listening, and reading. Fig. 11(b) displays the teaching results of the HC-PDTW. Fig. 11(c) displays the teaching results of the research model. In English listening teaching, the teaching effectiveness of the research model was 21.3% and 12.2% higher than that of the Hierarchical Clustering and HC-PDTW models, respectively. In English oral teaching, the teaching effectiveness of the research model was 20.8% and 14.5% higher than that of the Hierarchical Clustering and HC-PDTW models, respectively. In English reading teaching, the teaching effectiveness of the research model was 22.1% and 13.5% higher than that of the Hierarchical Clustering and HC-PDTW models, respectively. In summary, from the three dimensions of speech signal recognition accuracy, teaching duration optimization, and English teaching effectiveness, the research model has more advantages.