# A Statistical Approach to Violin Evaluation
## 3 Results
### Violin Clustering  
Given the features defined in Sections 2.3 and 2.4, the resulting feature space considered has thus 19 dimensions, leading to a feature matrix in \(\mathbb{R}^{M\times 19}\), where \(M=25\). Z-score normalization [27] has been applied over each feature vector inside the feature matrix to equalize values within a common scale. Once normalized, an Euclidean distance can be computed over the feature vectors characterizing each pair of admittances, leading to a distance matrix \(\mathbf{D}\in\mathbb{R}^{M\times M}\). Notice that all the features have the same relative importance to define our distance. In perceptual terms this could not hold, as some features can have a far larger influence than others. However, further studies are needed in this sense to define a \"perceptual distance\" between instruments.  
Figure 3a shows the distance matrix \(D\), with the elements sorted following the hierarchy of a dendrogram [28]. The dendrogram is computed iteratively using farthest neighbor clustering to define the hierarchical relations [29]. In this way, similar violins (i.e., pairs of instruments with a small distance) are clustered together.  
We decided to cut the dendrogram at a value of the farthest neighbor distance equal to 50 (vertical cyan line) as the link between H3, M22 and the remaining Stradivari violins occurs at around 48. As a result, four clusters with varying sizes can be observed: (i) M17-M13-M18-M5-M16-M3-M8-M6-M7, (ii) M12-M14-M4-M1-M19-M11-M20-M10-M9, (iii) M21-M15, and (iv) H1-M2-H2-H3.  
It is noteworthy that the distance between M22 and H1 is lower than the one assessed between M22 and its model H3. The similarity between M22 and H3 was already confirmed up to 1400 Hz in [16] (referred to as \"V1\" and \"Strad\" ), where the first ten peaks were chosen as features. To investigate which features the similarity between M22 and H1 mostly relies on, we visually compare the bridge admittance of the three violins.  
Figure 3b shows the three FRFs overlapped. Looking at the low-frequency range, the modes of M22 (dash-dotted blue line) resemble both those of its original H3 (dotted red line) and H1 (dashed yellow line). In terms of Euclidean distances, the distance computed considering only the low-frequency features yields comparable values: 7.72 between M22 and H1, and 8.92 between M22 and H3. However, a huge difference can be observed at mid frequencies, where H3 is characterized by a high-energy region centered at a lower frequency (1632 Hz against 2402 Hz for M22 and 2634 Hz for H1) and more limited in amplitude with respect to M22 and H1 (\(-\)24.2 dB against \(-\)16 dB for M22 and \(-\)14.9 dB for H1). Due to this difference, the proposed similarity metric varies greatly between the two pairs of violins: 9.51 between M22 and H1, 17.88 between M22 and H3.  
The same considerations can be inferred more clearly by inspecting the corresponding feature-based reconstructions in Figure 3c. In particular, the reconstruction of the high-energy region better highlights its position and width and thus makes the visual comparison between different instruments much easier.