# A Statistical Approach to Violin Evaluation
## 3 Results
### Correlation between Features and PCA  
To understand if these \"hand-crafted\" features are linearly independent and statistically significant, we computed the Pearson correlation coefficient between the column vectors of the feature matrix extracted from the dataset. Subsequently, we performed principal component analysis on the same matrix. Figure 4a shows the matrix of the Pearson correlation where the absolute value of the correlation coefficients is reported for each pair of features in the set. The values range between 0 and 1, and moderate or strong correlation occurs for values greater than 0.3 [30]. The signature mode frequencies (\(f_{1},\ldots,f_{4}\)) generate a cluster inside the matrix, with coefficients ranging between 0.5 and 0.65, therefore they are strongly correlated. Another cluster can be observed for (\(p_{1},\ldots,p_{4}\)). In this case, the signature mode amplitudes show a moderate correlation with values around 0.3, except for (\(p_{1},p_{2}\)) where the largest coefficient occurs in the matrix (0.75). Conversely, an interpretation regarding the correlation coefficients of the signature mode damping (\(q_{1},\ldots,q_{4}\)) is harder to infer from the matrix. It is interesting to note that indeed the frequency and the amplitude of the peaks seem to be _orthogonal_ descriptors of the FRF. Figure 3: (**a**) Euclidean distance matrix, with elements sorted following the dendrogram hierarchy. In this way, similar violins cluster together (colored in red). Four clusters are obtained by cutting the dendrogram at a farthest neighbor distance equal to 50 (vertical cyan line). (**b**) The bridge admittance of M22 (dash-dotted blue line) compared to those of H3 (dotted red line) and H1 (dashed yellow line), and (**c**) corresponding reconstructions from the feature space. Magnitudes are reported in [dB re 1 m s\({}^{-1}\) N\({}^{-1}\)]. The similarity between these violins leads to the cluster at the bottom-right corner of the distance matrix. The comparison of the three reconstructions highlights the different similarities: the signature modes of M22 are similar to those of both H3 and H1 while its mid-range resembles more that of H1. Furthermore, low correlation can be seen in general between the subsets of low-frequency and mid-frequency features. By looking at the submatrix on the bottom-left of Figure 4a (outlined by dashed red lines), the only correlated features are \(\omega_{mid},RP_{mid}\) and \(\omega^{*}\). Indeed, \(\omega_{mid}\) and \(\omega^{*}\) are linearly related to \((f_{1},\ldots,f_{4})\) while \(RP_{mid}\) is strongly correlated with \((p_{3},p_{4})\). If we limit the analysis to the subset of mid-frequency features, the sigmoid curvature \(k^{*}\) and the area difference \(\Delta A\) show a correlation coefficient above 0.3 with most of the other features. On one hand, the curvature of the RP curve (\(k^{*}\)) is correlated to its behavior at the extremes of the range: strong correlation with \(RP_{mid}\) (0.64) and moderate correlation  
Figure 4: (**a**) Matrix of the Pearson correlation between features. The absolute value of the coefficients, ranging between 0 and 1, is reported. Explicit values are shown only for combinations of features with moderate and strong correlation (\(\geq\)0.3). (**b**) The normalized cumulative sum of the eigenvalues obtained with PCA. A dashed red line highlights how many components are needed to recover 96.7% of the original information. (**c**) Feature importance in the first twelve principal components. Explicit values are shown only when greater than 0.05. Interestingly enough, frequencies and amplitudes in signature modes are almost independent and the mid-frequency features mainly constitute the second principal component. with \(k_{high}\) (0.43), \(\omega_{mid}\) (0.36), and \(\Delta A\) (0.36). On the other hand, strong correlation can be observed between \(\Delta A\) and features characterizing the central part of the sigmoid: \(\omega^{*}\) (0.52), \(RP^{*}\) (0.4), \(k^{*}\) (0.36), and \(RP_{mid}\) (0.32).