## 3 Results
### Investigation of the Training Process  
Considering these models' respective training phases, we noticed that GP perfectly fitted (Figure 5a) its training data (RMSE = 0.00, \(R^{2}\) = 1.00), thereby yielding an overfitted model. On the other hand, we observed that SVR learning process overtly discarded (Figure 5b) the differences between these facial pre-touch distances (RMSE = 7.69, \(R^{2}\) = 0.10). The SVR's performance can be explained in light of its optimization criterion that attempts to minimize the orthogonal distances between data and its fitted hyper/plane. Considering the distribution of facial-area pre-touch distances around the face area (i.e., an approximately hemispheric surface), it is foreseeable for SVR's optimization criterion to find one such candidate hemisphere (i.e., analogous to a plane in 3D that is bent along the azimuth angle). On the other hand, Lasso (Figure 5c) whose only criterion was to penalize the growth of its learned weight matrix appeared to achieve a balance between bias and variance (RMSE = 4.90, \(R^{2}\) = 0.64). This suggestion is clarified by comparing its RMSE and \(R^{2}\) in the case of full data and its average performance on test cases in Sections 3.1 and 3.2 (M\({}_{RMSE}\) = 5.79 and M\({}_{R^{2}}\) = 0.44).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Models** & \(p\)\(<\) & **W(26)** & **r** & \(p\)\(<\) & **W(26)** & **r** \\ \hline Lasso & 0.001 & 27.39 & 0.87 & 0.001 & 47.84 & 1.51 \\ SVR & 0.001 & 4.51 & 0.14 & 0.001 & 37.27 & 1.18 \\ GP & 0.001 & 22.90 & 0.72 & 0.001 & 52.62 & 1.66 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Coefficient of determination (\(R^{2}\)). Comparison of the use of average with Lasso, SVR, and GP with and without polynomial degrees. Subscripts 0 and 4 correspond to the case of polynomial degree 0 (i.e., no polynomial degree) and 4.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Models** & \(p\)\(<\) & **W(26)** & **r** & \(\mathbf{M_{0}}\) & \(\mathbf{SD_{0}}\) & \(\mathbf{M_{4}}\) & \(\mathbf{SD_{4}}\) \\ \hline Lasso & 0.001 & 36.74 & 1.16 & \(-\)0.06 & 0.08 & 0.44 & 0.30 \\ SVR & 0.001 & 29.08 & 0.92 & \(-\)0.03 & 0.09 & 0.06 & 0.11 \\ GP & 0.001 & 10.29 & 0.33 & \(-\)2.47 & 0.93 & \(-\)2.05 & 0.84 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Effect of polynomial degree on improving the coefficient of determination (\(R^{2}\)) in Lasso, SVR, and GP. M\({}_{i}\) and SD\({}_{i}\), \(i=0,4\) refer to the mean and the standard deviation of RMSEs associated with these models while using polynomial degrees 0 (i.e., no polynomial degree) and 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & \(p_{0}\)\(<\) & **W(26)** & **r\({}_{0}\)** & \(p_{4}\)\(<\) & **W(26)** & **r\({}_{4}\)** \\ \hline Lasso vs. GP & 0.001 & 12.22 & 0.86 & 0.001 & 12.21 & 0.86 \\ Lasso vs. SVR & 0.001 & 4.95 & 0.35 & 0.001 & 6.84 & 0.48 \\ Lasso vs. Fixed & 0.001 & 14.11 & 1.00 & 0.001 & 14.11 & 1.00 \\ SVR vs. GP & 0.001 & 12.22 & 0.86 & 0.001 & 12.20 & 0.86 \\ SVR vs. Fixed & 0.001 & 14.11 & 1.00 & 0.001 & 14.11 & 1.00 \\ GP vs. Fixed & 0.001 & 14.11 & 1.00 & 0.001 & 14.11 & 1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Pairwise model’s \(R^{2}\) comparison. Subscripts 0 and 4 correspond to the case of polynomial degree 0 (i.e., no polynomial degree) and 4.  
Although we observed a high variability in under/overfitting the pre-touch distances by these models, it was interesting to observe that the [PERSON]'s test (Figure 6) identified a non-significant difference between these models based on the difference between their estimations and the actual pre-touch distances (\(p=0.95\), H(2, 16,991) = 0.10, r = 0.02). However, this failure is clarified once one observes that the difference between the estimations and actual pre-touch distances by GP and Lasso yielded zero-mean differences (M\({}_{GP}\) = 0.00, SD\({}_{GP}\) = 0.00, M\({}_{Lasso}\) = 0.00, SD\({}_{Lasso}\) = 4.92) with a negligibly above zero in the case of SVR (M\({}_{SVR}\) = 0.11, SD\({}_{SVR}\) = 7.72).