An Innovative Study of Traditional Music Pedagogy Based on Time Series Analysis

The so-called “traditional music teaching” mainly refers to the traditional music teaching method that takes the mastery of single knowledge, skill and convenient assessment of external knowledge as the teaching goal. Traditional music is conducted in its own oral transmission, teacher-disciple face-to-face order, resulting in music activities in most colleges and universities are also centered on this single-introduction, textbook-based, music mastery-centered teaching method. “Traditional music teaching” focuses on the completion of a piece of music and reveals a cult of mechanical imitation. For example, in music activities, teachers will listen and imitate mainly, from the complete whole piece of music to listen, to listen in small sections, and then listen to the complete piece of music to appreciate, and then imitate in this order after appreciation. They seldom consider the learners’ subjectivity and originality, and only ask them to passively complete the music in a complete and coherent way as the ultimate goal.

Moving average [24] (MA) model and autoregressive sliding average [25] (ARMA) model, which lays the foundation for the improvement of time series models. With the further research of related scholars, the time series fitting model application and construction methods are gradually diversified.

Time series fitting models are mostly used to do forecasting. For example, the product ARIMA is used to fit and predict the passenger flow in and out of each station of Guangzhou Railway Transportation, and a high prediction accuracy has been achieved. In order to improve the prediction accuracy, some scholars use segmented fitting to make predictions. For example, the segmented curve fitting method is used to excavate the trend characteristics of passenger flow time series, and it is found that the automatic segmentation strategy is used in segmented fitting, which can avoid the thought. The subjectivity of determining the segmentation point, achieving the best optimization, and further improving the efficiency and accuracy of forecast analysis.

In the research of time series fitting model, the autoregressive model, moving average model and the evolution model of autoregressive model are mainly used to analyze and study the time series in various fields, and the overall fitting of the time series fitting model is mainly used to do forecasting. In order to further improve the fitting accuracy and discover the stage characteristics of the data, the research hotspot evolves from overall fitting to segmented fitting. In the field of education informatics, the application of time series fitting models to education data analysis requires further research.

In the field of music teaching, time series can be used to analyze the innovative history and evolution of traditional music teaching. By analyzing time series data on historical teaching methods and effects, future teaching trends and innovations are predicted. Time series analysis is used to track these changes and analyze the impact of the evolution of policies, teaching materials, and teaching methods on music education.

The online learning behavior analysis model is shown in Figure 1, in which the online learning behavior in the online learning space is divided into four categories in the structural model of the online learning space: independent learning behavior, system interaction behavior, resource interaction behavior and social interaction behavior. The data mining technology is refined into three analysis methods: correlation analysis, classification analysis, and cluster analysis, with different analysis methods corresponding to different forms of analysis results, so as to explore the hidden objective laws through the analysis of online learning behaviors, which can directly influence the stakeholders (administrators, teachers, and learners), and provide scientific guidance for educational strategies.

Figure 1.

Online learning behavior analysis model based on data mining technology

In this study, after rigorous screening of course level, course nature, and course type, full-time undergraduate students in the public course “Traditional Music Appreciation” of an art college in M province were selected as the research subject, and the online learning behavior data generated by them were subjected to case data analysis. The course is a national boutique open course, a provincial blended first-class course, and a public course in the teacher education segment of the college, with a total of 150 second-year college full-time learners majoring in vocal performance within the course. The online teaching of the course “Appreciation of Traditional Music” is mainly carried out on the Super Star Learning Channel platform, where a total of 62 learning resources such as illustrations and videos have been uploaded, with in-class discussion forums and post-course Q&A forums, and a total of 6 post-course assignments, 25 chapter quizzes and 1 final test have been released.

The learning content of the course is more related to computer operation and focuses on the skill enhancement of the learners. The data source of learners’ online learning behavior in this study is based on the exportable learning activity data of the online course learning platform, and the content analysis of the learning outcome data on the platform is used as a supplement. First, Excel is used to perform basic statistics and organization of the raw data set of online learning behaviors obtained from the background learning logs of the online course learning platform, such as learners’ task completion, resource learning, discussion, and homework detection. Second, the learning behavior data obtained from the analysis of discussion forums, assignments, etc. All data is added, checked, and counted again to improve the accuracy of the analyzed data and confirm that all the data required for the study has been collected completely.

In order to obtain a correct and complete dataset of learners’ online learning behavior, it is necessary to preprocess the original dataset. In the first step, the required data in the raw dataset are screened out, irrelevant data are deleted and processed, and the inconsistencies between the background exported data and the content analysis data in the raw dataset are deleted after checking again and deleting the erroneous items. In the second step, based on the constructed online learning behavior indicators under the blended learning mode, for the learning behaviors that cannot be directly represented by raw data, the raw data are obtained by formula conversion and other means. In the third step, observe whether there are missing or incomplete values in the processed dataset, and process and analyze them separately if there are any cases. In the fourth step, the personal information of the learners in the online learning behavior dataset is hidden, and the name part is replaced with an UID. In order to reduce the bias in the clustering analysis part of data analysis, the learners’ online learning behavior dataset is normalized in advance in the data preprocessing stage.

If time series x_t is related to both current and prior random error terms and to prior observations. It can be expressed as: (1)xt=ϕ0+ϕ1xt−1+⋯+ϕpxt−p+εt−θ1εt−1−⋯−θqεt−q$${x_t} = {\phi _0} + {\phi _1}{x_{t - 1}} + \cdots + {\phi _p}{x_{t - p}} + {\varepsilon _t} - {\theta _1}{\varepsilon _{t - 1}} - \cdots - {\theta _q}{\varepsilon _{t - q}}$$

Among them: (2){ ϕp≠0,θq≠0 E(εt)=0,Var(εt)=σε2,E(εtεs)=0,s≠t Exsεt=0,∀s<t$$\left\{ {\begin{array}{l} {{\phi _p} \ne 0,{\theta _q} \ne 0} \\ {E({\varepsilon _t}) = 0,Var({\varepsilon _t}) = \sigma _\varepsilon ^2,E({\varepsilon _t}{\varepsilon _s}) = 0,s \ne t} \\ {E{x_s}{\varepsilon _t} = 0,\forall s < t} \end{array}} \right.$$

Then model (1) is said to be an autoregressive moving average ARMA(p, q) model, and similarly the centered ARMA(p, q) model is expressed as: (3)xt=ϕ1xt−1+⋯+ϕpxt−p+εt−θ1εt−1−⋯−θqεt−q$${x_t} = {\phi _1}{x_{t - 1}} + \cdots + {\phi _p}{x_{t - p}} + {\varepsilon _t} - {\theta _1}{\varepsilon _{t - 1}} - \cdots - {\theta _q}{\varepsilon _{t - q}}$$

Introducing the delay operator B, model (3) is abbreviated as: (4)Φ(B)xi=Θ(B)εi$$\Phi (B){x_i} = \Theta (B){\varepsilon _i}$$

Eq:

Φ(B) = 1 − ϕ₁B − ⋯ − ϕ_pB^p is called the pnd order autoregressive coefficient.

Θ(B) = 1 − θ_iB − ⋯ − θ_qB^q is called the qth order moving average coefficient.

Obviously, when q = 0, the ARMA(p, q) model is transformed into a AR(p) model. When p = 0, ARMA(p, q) is transformed into a MA(q) model.

The observed and collected time series data of learners’ learning behavior need to be tested for smoothness before modeling. When the observed values of the time series are not smooth, it is necessary to carry out the necessary preprocessing to make it smooth. There are two main methods for smoothness testing: the first method mainly utilizes mathematical statistics methods, through the construction of test statistics for hypothesis testing. The second method mainly utilizes image features and makes judgments based on time series and autocorrelation diagrams. The first method is mainly quantitative analysis, while the second method is qualitative judgment, which is more subjective. Therefore, at present, the hypothesis testing method is usually utilized for testing, i.e., unit root test. Smoothing is mainly done by differencing or logarithmizing the series. However, too much differencing may eliminate some of the intrinsic laws of the series and affect the prediction effect. Therefore, the separation method can be utilized to separate the trend and random terms by curve fitting, wavelet decomposition, etc., and then build the model separately.

The determination of the model type is mainly through the values of the autocorrelation and partial autocorrelation functions of the sample data. Therefore, as long as the autocorrelation and partial autocorrelation function is judged to be truncated or trailing, the model type can be determined. There are two main ways to judge, one is the calculation method, if the partial autocorrelation function is p-step truncated tail: (5)ψkj={ ψj,1≤j≤p 0,p+1≤j≤k$${\psi _{kj}} = \left\{ {\begin{array}{l} {{\psi _j},1 \leq j \leq p} \\ {0,p + 1 \leq j \leq k} \end{array}} \right.$$

If the autocorrelation function is q-step truncated: (6)ρk={ 1,k=0 −θk+θ1θk+1+⋯+θq−kθq1+θ12+⋯+θq2 ,1≤k≤q 0,k>q$${\rho _k} = \left\{ {\begin{array}{l} {1,k = 0} \\ {\frac{{ - {\theta _k} + {\theta _1}{\theta _{k + 1}} + \cdots + {\theta _{q - k}}{\theta _q}}}{{1 + \theta _1^2 + \cdots + \theta _q^2}},1 \leq k \leq q} \\ {0,k > q} \end{array}} \right.$$

The second is the image method, since the sample autocorrelation and partial autocorrelation functions approximately obey a normal distribution, they can be judged by the 2-fold standard deviation range. If the autocorrelation coefficient or partial autocorrelation coefficient falls significantly outside the 2-fold range before the mst order, and rapidly decays to within the 2-fold range from the mnd order, and fluctuates around 0, the autocorrelation function or partial autocorrelation function is said to be truncated at the mrd order. And if the correlation coefficients of the samples falling outside the 2-fold range exceed 5% or decay slowly, the autocorrelation function or partial autocorrelation function is said to be trailing.

There are many methods for estimating the parameters of the model coefficients, mainly: spectral analysis, great likelihood estimation, moment estimation, and least squares. Currently widely used is the least squares estimation. For ARMA(p, q) model, set: (7)β^=(ϕ1,ϕ2,⋯,ϕp,θ1,θ2,⋯,θq)'$$\hat \beta = {\text{ }}({\phi _1},{\phi _2}, \cdots ,{\phi _p},{\theta _1},{\theta _2}, \cdots ,{\theta _q})'$$ (8)Ft(β^)=ϕ1xt−1+⋯+ϕpxt−p−θ1εt−1−⋯−θqεt−q$${F_t}(\hat \beta ) = {\phi _1}{x_{t - 1}} + \cdots + {\phi _p}{x_{t - p}} - {\theta _1}{\varepsilon _{t - 1}} - \cdots - {\theta _q}{\varepsilon _{t - q}}$$

Then the residual is: (9)εt=xt−Ft(β^)$${\varepsilon _t} = {x_t} - {F_t}(\hat \beta )$$

This gives the residual sum of squares: (10)Q(β^)=∑t=1nεt2=∑t=1n(xt−ϕtxt−1−⋯−ϕpxt−p−θqεt−1−⋯−θqxt−q)$$Q(\hat \beta ) = \sum\limits_{t = 1}^n {\varepsilon _t^2} = \sum\limits_{t = 1}^n {({x_t} - {\phi _t}{x_{t - 1}} - \cdots - {\phi _p}{x_{t - p}} - {\theta _q}{\varepsilon _{t - 1}} - \cdots - {\theta _q}{x_{t - q}})}$$

Then the least squares estimate of β^$$\hat \beta$$ is the set of parameter values that minimizes the sum of squares of the residuals.

The significance test of the model is mainly to test the validity of the model, i.e., to judge whether the model adequately and completely extracts the intrinsic laws of the time series. Therefore, the model test is mainly carried out by determining whether the residual series obtained from the model calculation is white noise. If the residuals are white noise, the model fit is significant and valid, and the model can be used to predict the effects of music learning. Otherwise, the model fit is not significant and the sequence information is not fully extracted, thus the model cannot be used to make effective predictions for the future. There are two main methods for residual white noise testing. One method is to judge by analyzing the characteristics of the residual sequence plot. The other method is to apply the hypothesis testing statistic for hypothesis testing, i.e., set the original hypothesis and alternative hypothesis respectively: (11)H0:ρ1=ρ2=⋯=ρm=0$${H_0}:{\rho _1} = {\rho _2} = \cdots = {\rho _m} = 0$$

H₁ There exists at least some ρ_k ≠ 0, ∀m ≥ 1, k ≤ m.

Because: (12)LB=n(n+2)∑k=1m(ρk2^n−k)Submit to x2(m),∀m>0$$LB = n(n + 2)\sum\limits_{k = 1}^m {(\frac{{\widehat {\rho _k^2}}}{{n - k}})} \:{\text{Submit to }}{x^2}(m)\:,\:\forall m > 0$$

So when the probability P value of the LB statistic is greater than 0.05, i.e., the original hypothesis is accepted at the 95% confidence level, the residual series is white noise. If when the value of P is less than 0.05, the original hypothesis is rejected and the residual series is a non-white noise series.