## 4 Analysis of Program Clusters Based on Audio Features
### Discriminant Analysis  
Discriminant analysis is often used to differentiate among groups based on a combination of independent variables [26]. The discriminant function is formed to create scores for each observation that maximally distinguishes between groups of observations. In this work, a two-group discriminant analysis was performed in SPSS using the program group as the dependent variable and the two principal components as the independent variables. One discriminant function was formed to predict and explain each program's membership. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature** & **Component 1** & **Component 2** \\ \hline Tempo & \(-0.185\) & \(0.692\) \\ Pulse clarity & \(-0.202\) & \(0.045\) \\ BE 400–800 & \(0.153\) & \(0.141\) \\ BE 800–1600 & \(0.015\) & \(0.450\) \\ BE \textgreater{} 1600 & \(-0.300\) & \(0.216\) \\ Skewness & \(0.265\) & \(-0.057\) \\ Kurtosis & \(0.235\) & \(-0.004\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The coefficient matrix of PCA. Figure 6: The score plot of Principal Component Analysis (PCA). Figure 8: Distribution of the two program clusters in the component map. Figure 7: The correlation between the original features and the two principal components. The underlying assumptions of discriminant analysis were tested and met at acceptable levels. The equality of covariance matrices of the independent variables across the groups was supported by the Box's M (\(p\) = 0.486). The normality of the independent variables was tested by the Kolmogorov-Smirnov method (\(p\) = 0.2). One discriminant function was derived, taking the following form:  
\[Z_{i}=a_{1}C_{1i}+a_{2}C_{2i} \tag{1}\]  
where  
\(Z_{i}\) Discriminant score for the program \(i\)  
\(a_{1}\), \(a_{2}\) Discriminant function coefficients (Here \(a_{1}=1.082\), \(a_{2}=0.98\))  
\(C_{1i}\), \(C_{2i}\) The two component scores for the program \(i\)  
According to the function, the discriminant score of each program can be calculated. The group mean of the discriminant score, defined as the group centroid, is also calculated, shown in Table 5. The membership of the program can be determined by comparing the discriminant score to the cutting score which represents the dividing point used to classify programs. The cutting score is measured based on the group centroid and the relative size of the two groups [20]. In this case, the cutting score was calculated as 0.9371. Thus, programs with a discriminant score less than 0.9371 were classified into C1, and programs with a discriminant score greater than 0.9371 were classified into C2. As can be seen, the predictive accuracy was 95% for the original grouped samples and 90% for the cross-validated grouped samples. The achieved accuracy was evaluated using the proportional chance criterion [26]. Usually, the predictive accuracy should exceed the proportional chance by 25% to prove its effectiveness. In this case, the proportional chance criterion is calculated as (14/20)? 2 + (6/20)? 2 = 58%, and the comparison threshold value is 72.5% (58% \(\times\) 1.25 = 72.5%). In both the original analysis sample and the cross-validated sample, the predictive accuracy level is above the threshold value, which validates the discriminant function. The Wilks' Lambda test examined the significance of the discriminant function, showing that the function is statistically significant in separating the clusters (Wilks' Lambda = 0.445, \(p\) = 0.001). In summary, using the two principal components extracted from the seven significant features, one discriminant function was derived, which could significantly separate the discriminating and undiscriminating program clusters with 95% classification accuracy.