## 4 Analysis of Program Clusters Based on Audio Features
### Principal Component Analysis  
The Principal Component Analysis (PCA) is a classic dimension-reduction technique, which can create an entirely new set of variables with a reduced number but also retain the nature and character of the original variables [26]. PCA is often used in the preprocessing of the data. In this work, the predictive model of the program's discrimination ability is built by using the linear discriminant analysis (LDA). Before using the seven significant features as the independent variables in the discriminant analysis, it was found that some of these features were highly correlated (e.g., skewness and kurtosis: r = 0.945, \(p<0.001\); BE \(>\) 1600 and skewness: r = \(-0.864\), \(p<0.001\)). The strong correlation indicates that multicollinearity exists among the seven features, which contradicts the assumption of discriminant analysis [26]. To solve the problem, a PCA was performed to reduce the dimensions of the feature space so that the assumption of discriminant analysis could be met. The appropriateness of PCA was examined by the Bartlett test of sphericity and the Kaiser-Meyer-Olkin measure of sampling adequacy. The Bartlett test of sphericity tests the overall significance of the correlation matrix, which finds that the correlations between these features are significant at the 0.001 level (\(\chi^{2}\) = 101.04, \(p<0.001\)). The Kaiser-Meyer-Olkin measure of sampling adequacy (MSA) quantifies the degree of intercorrelations among the features. The MSA value is 0.613, which is above the recommended value of 0.5 [26]. Both test results indicated that the PCA would be useful. In order to determine the number of components extracted, both the latent root criterion and the scree test criterion were used, and the results were the same. The latent root criterion only keeps components having eigenvalues greater than 1. The scree test criterion determines the cutoff point from the shape of the scree plot. Figure 6 shows the scree plot which plots the eigenvalue against the number of components in their extraction order. Obviously, the first two components have eigenvalues greater than 1, and from the third component, the curve begins to straighten out. Therefore, the first two principal components are considered significant and shall be kept. The Varimax rotation method was applied to interpret the two components better. The result of PCA shows that the first two principal components explain 76.04% of the variance across features in which the first component accounts for 53.67%, and the second component explains the left 22.37%. Table 4 presents the coefficient matrix to compute the two principal component scores. Figure 7 shows the correlation between the seven features and the two principal components. A longer vector represents that the feature is more correlated with the two components. By comparing the projections of the feature vector on Component 1 and Component 2 axes, the component to which the feature is more correlated can be determined. In addition, when two features are close in the correlation map, they are highly correlated (e.g., skewness and kurtosis). It can be seen that five features (i.e., BE \(>\) 1600, pulse clarity, kurtosis, skewness, and BE 400-800) are more associated with Component 1. In particular, BE \(>\) 1600 and pulse clarity show negative correlations with Component 1 while others are positively related to the component. BE 800-1600 is correlated to both components. It is also notable that tempo shows a strong correlation with Component 2 and is barely related to Component 1. As described above, BE \(>\) 1600, BE 400-800, and BE  
800-1600 represent the proportion of energy in that specific band. Kurtosis and skewness describe the shape of the spectrum. Pulse clarity, although as a rhythmic feature, indicates the strength of the beats in the music. Hence, it can be safely drawn that the first principal component (Component 1) represents features describing the energy distribution of the program from different aspects.