# A Statistical Approach to Violin Evaluation
## 2 Materials and Methods
### Feature-Based Reconstruction  
Given the models for the different components of the bridge admittance in Equations (1) and (6), its magnitude can be reconstructed starting from the corresponding values of low and mid-frequency features as  
\[\left|H_{ij}(\omega)\right|\sim\left|\sum_{r=1}^{4}\frac{\Phi_{ri}\Phi_{rj}}{ \omega_{r}^{2}+2j\omega\xi_{r}\omega_{r}-\omega^{2}}\right|+C\frac{\delta \sigma^{\prime}(\omega,\omega_{mid}\omega^{*},k^{*})}{\delta\omega},\quad \omega\in[\omega_{min},\omega_{max}], \tag{7}\]  
where \(C\) is a constant found through iterative optimization such that the MSE between the second term of Equation (7) and \(\left|H_{ij}(\omega)\right|\) is minimized around \(\omega=\omega^{*}\).  
Figure 2 shows the application of the proposed methodology. Here, one of the historical violins (i.e., H1) is taken as an example. Figure 2a depicts its FRF magnitude. It can be seen that the four signature modes are easily identifiable up to around 700 Hz, while modes are more overlapped for higher frequencies. Using these modes, the reconstruction obtained through modal superposition (Figure 2b) approximates the FRF up to \(\omega_{mid}\).  
The second component in Equation (7) accounts for the reconstruction of the mid-frequency portion of the FRF (Figure 2c). The parameters controlling \(\sigma^{\prime}(\omega,\omega_{mid},\omega^{*},k^{*})\) are the result of the fitting performed on the RP curve of H1.  
The combination of the two models in Figure 2d resembles a \"cleaner\" version of the original measured signal. Indeed, the reconstructed signal keeps the same energy balance between low and mid components. By computing the residual as the relative difference between measure and reconstruction, as depicted in Figure 2e, we can assess where the model fails at recovering the original signal measured.  
Figure 2: Example of measured bridge admittance and relative feature-based reconstruction. From top to bottom, the original signal (**a**), its analytical approximation in the low-frequency range through modal superposition (**b**), its reconstruction based on the sigmoid model in the mid-frequency range (**c**), and their linear combination (**d**) are shown. The relative difference between measured and reconstructed signals is assessed in percentage as a function of frequency (**e**). Magnitudes are reported in [dB re 1 m s\({}^{-1}\) N\({}^{-1}\)].  
In the specific case of H1, the reconstructed signature modes resemble the original ones and the residual oscillates around 5% close to antiresonance frequencies. A relevant increase of the residual up to 40% is then encountered around 1 kHz. The frequency span characterized by large values of the residuals covers the portion of FRF dominated by transitional modes. For a subset of FRFs in the dataset, transitional modes exhibit low mode overlapping and modal superposition may model the FRF better than the proposed energy-based model. However, a common trend was not found throughout the dataset and further research is needed to improve the model within this frequency range.  
In the case of H1, the proposed reconstruction is also able to model the region of the FRF between 1500 and 3000 Hz. Here, the residual stabilizes between 5% and 10% except for spikes corresponding to the presence of peaks or deeps in this portion of the spectrum, which cannot be recognized however as individual modes.  
Finally, the reconstruction fails at explaining the trend of the admittance for higher frequencies, i.e., between 4000 and 4500 Hz. In this range, the FRF exhibits a more linear trend, and the residual reaches values around 20%. The feature \(k_{high}\) has been indeed introduced to overcome this lack of information, defined as the slope of the linear fit limited to the high frequency range mentioned.