## 4 Modelling Process  
It is likely that several of the features account for the same variance in the listener ratings of hardness, and that others do not account for any variance. To develop a robust and generalisable model of hardness, features should only be included if they are necessary to account for variance within the data. Suitable features were chosen using an additive modelling method: initially creating a linear regression model with the one feature that best predicts the ratings of hardness, and iteratively adding the next-best performing feature. To avoid the use of multiple features describing similar variance within the data, potentially leading to an ungeneralisable model, the variance inflation factor (VIF) was calculated for each subsequent feature [33]. [PERSON] [34] states that a VIF of greater than 10 is too large. Therefore, if the VIF of any feature is greater than 10 during the addition of any feature, this newly added feature was removed from inclusion in future iterations.  
Figure 4: Weighting function for midband frequencies.  
Figure 5 shows how the \(R^{2}\) and root mean square error (RMSE) varied with each iteration. The performance generally increases with successive iterations, with several iterations offering a significant improvement, marked by grey vertical dashed lines. Several iterations where the model's performance increases are not marked since the added feature was removed for violating the VIF criterion. The model's performance appeared to stop improving after iteration 15, indicating that the features added beyond this point did not account for any additional variance in the data. The seven features added at the highlighted iterations are shown in Table 1, along with the model's performance at that iteration.  
From Table 1, the first two features that show an improvement of the hardness model are the logarithmic and linear version of the same feature: the weighted-average maximum bandwidth. The VIF metric used for removing features only tests whether a feature explains the same variance as a linear combination of other features; it cannot account for the nonlinear relationship between these two features. Therefore, the non-logarithmic-transformed version of this feature was removed from the final feature set. The final model of hardness is a multilinear regression of the log weighted average maximum bandwidth, the log mean attack centroid, the log weighted average midband level, the log percussive-to-harmonic ratio, the mean onset strength, and the log attack time.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Iteration Number** & **Feature** & \(\mathbf{R^{2}}\) & **RMSE** \\ \hline
1 & log weighted-average maximum bandwidth & 0.64 & 10.20 \\
3 & weighted-average maximum bandwidth & 0.66 & 10.00 \\
5 & log mean attack centroid & 0.66 & 9.91 \\
9 & log weighed average midband level & 0.69 & 9.53 \\
13 & log percussive-to-harmonic ratio & 0.72 & 9.08 \\
14 & mean onset strength & 0.74 & 8.79 \\
15 & log attack time & 0.76 & 8.38 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance of the features that show improved performance during the iterative modelling process.  
Figure 5: Performance of the multilinear regression model for each stage of the iterative modelling. The dashed grey lines represent iterations where the model shows marked improvement.