# A Statistical Approach to Violin Evaluation
## 2 Materials and Methods
### Mid-Frequency Features  
To define a set of features modeling the FRF at higher frequencies, we took inspiration from statistical approaches to the subject [9]. We chose to represent the energy distribution of a given bridge admittance \(H_{ij}(\omega)\) by computing its relative power (RP) as  
\[RP(\omega)=\frac{P(\omega)}{P_{tot}}=\frac{\int_{\omega_{min}}^{\omega}|H_{ij} (\omega)|^{2}\,d\omega}{\int_{\omega_{min}}^{\omega_{max}}|H_{ij}(\omega)|^{2 }\,d\omega}, \tag{4}\]  
where \(P(\omega)\) is the cumulative power and \(P_{tot}\) is the total power of the signal, i.e., computed within the range \([2\pi f_{min},2\pi f_{max}]\) with \(f_{min}=230\) Hz and \(f_{max}=4500\) Hz. In the frequency interval considered, \(RP(\omega)\) ranges between 0 and 1. The slope of this descriptor is characterized by \"bumps\" where the corresponding bridge admittance exhibits high-energy components (see Figure 1b as an example). In particular, two main slope changes typically occur in violin bridge admittances after 500-600 Hz and between 2 and 3 kHz. The first relevant increase in the RP value, encountered at low frequencies, quantifies the amount of energy of the signature modes. Indeed, the RP curve starts to flatten at the frequency of mode B1+. Starting from the antiresonance after mode B1+, the RP shows a different growth rate for each violin in the dataset. Here, the RP curve exhibits a characteristic S-shape strongly correlated to a high-energy region in the corresponding bridge admittance. To obtain a compact description of this behavior in the mid-frequency range, being it the interval between \(\omega_{mid}\) and \(\omega_{max}\), we modelled the RP curve by means of a sigmoid function with tunable curvature based on [25], namely  
\[\sigma(\omega,\omega^{*},k^{*})=\frac{\tilde{\omega}-k^{*}\tilde{\omega}}{k^{ *}-2k^{*}|\tilde{\omega}|+1}\quad\text{where}\quad\tilde{\omega}=\frac{ \omega-\omega^{*}}{\overline{\Delta\omega}}, \tag{5}\]\(\omega^{*}\) is the center frequency of the sigmoid, \(k^{*}\) the control parameter for its curvature and \(\overline{\Delta\omega}=min(|\omega^{*}-\omega_{mid}|,|\omega_{max}-\omega^{*}|)\) such that the domain of the sigmoid is symmetric with respect to \(\omega^{*}\). To better fit the RP values at the extremes of the mid-frequency range, the output of the function \(\sigma(\omega,\omega^{*},k^{*})\) is scaled and shifted leading to  
\[ \begin{split}\sigma^{\prime}(\omega,\omega_{mid},\omega^{*},k^{* })&=\sigma(\omega,\omega^{*},k^{*})\frac{RP(\omega^{*}+\overline{ \Delta\omega})-RP_{mid}}{\bar{\sigma}_{max}-\bar{\sigma}_{min}}+|RP_{mid}- \bar{\sigma}_{min}|,\\ RP_{mid}=RP(\omega_{mid}),\\ \bar{\sigma}_{max}=max(\sigma(\omega,\omega_{mid},\omega^{*},k^{* })),\\ \bar{\sigma}_{min}=min(\sigma(\omega,\omega_{mid},\omega^{*},k^{* })).\end{split}  \tag{6}\]  
The fitting was performed by searching for the values of \(\omega^{*},k^{*}\) such that the MSE between the measured RP and the predicted sigmoid has its minimum. The minimization was carried out iteratively using the simplex algorithm. Interestingly enough, the values found for \(k^{*}\) at the end of the optimization follow a Gaussian distribution centered around \(-0.7\). Two outliers were detected, with \(k^{*}\) equal to \(-0.25\) and \(-0.05\). The advantage of this representation is that the resulting optimal sigmoid model can be easily projected back to the frequency-magnitude space by computing its derivative with respect to \(\omega\). We thus decided to use the parameters of Equation (6) as features for the mid-frequency range. Along with these features, we added two additional descriptors to the set. On the one hand, we took the MSE between the measured RP and the sigmoid fit as a metric of the fitting accuracy. We will denote it as the \"area difference\" \(\Delta A\) for its geometric interpretation (see Figure 1b).