## 2 Materials and Methods
### Analysis
(**b**) 3-D heatmap of pre-touch surface around the face using the modified grid that includes 5664 facial-area pre-touch data points. The azimuth and elevations angles of these samples are within [\(-80.00\),..., \(+80.00\)] and [\(-32.00\),..., \(+53.00\)], respectively. The \(z\)-axis corresponds to the distance of the touches’ hand from the touches’ face (i.e., distance at which the touches were asked by the touches to stop approaching hand to get any closer). (**c**) Distribution of touches’ pre-touch data points around the touches’ face area. Points are plotted in different shades of grey for better visualization purpose. These data points correspond to the locations where the touches were asked by the touches to stop approaching their face. Coordinates of these data points correspond to toucher’s hand distance (\(x\)-axis, in cm), azimuth angle (\(y\)-axis, in degree), and elevation angle (\(z\)-axis, in degree) with respect to the touches’ face. (**d**) Schematic diagram of the face’s orientation with respect to the azimuth and elevation angles as shown in subplot (**b**). We also examined whether the use of polynomial degree on input features (i.e., azimuth and elevation angles) benefited these models' estimate of participants' facial-area pre-touch boundary. The use of polynomial degree was a natural consideration since any such pre-touch distance boundary was analogous to approximation of a three-dimensional volume around the face (e.g., a perfect spherical space in the case of fixed pre-touch distance). We examined polynomial degrees 0 (i.e., no polynomial degree or control condition) through 10. All models attained their best estimates using polynomial degree 4. Therefore, we used 0 (i.e., control setting in the case of the models) and 4 (i.e., feature vectors of length \(2^{4}=16\), given the azimuth and elevation angles as original input features) polynomial degrees in our analyses. For each of these two cases, we adopted a cross-validation strategy in which we randomly split the pre-touch surface that included 5664 distance data (Figure 2b) into 80.0% train (4531 pre-touch data) and 20.0% test (i.e., 1133) sets. We repeated this procedure for 1000 simulation runs during which we recorded the root-mean-squared-error (RMSE) and coefficient of determination (\(R^{2}\)) of the models as measures of their goodness of estimates. The RMSE allowed us to determine how well each model was estimating the pre-touch distance values on test set. On the other hand, the \(R^{2}\) enabled us to evaluate whether the use of averaged pre-touch distance in the train set might yield a better estimates of test set's pre-touch distances than the models' estimates based on their training. Precisely, in the case of regression, coefficient of determination (i.e., \(R^{2}=1-\frac{SS_{residual}}{SS_{total}}\) where \(SS_{residual}\) and \(SS_{total}\) refer to residual and explained sum of squares, respectively) measures how well the regression predictions approximate the real data points [28; 29]. As a result, \(R^{2}<0.0\) would mean that the use of the averaged pre-touch distances as their estimated pre-touch boundary was better than the model estimates, \(R^{2}=0.0\) would indicate no difference between the two, and \(R^{2}>0.0\) would justify the use of a model for estimating this boundary. For the analysis, we performed 1000 rounds of random sampling in which we selected (without replacement) 100 of these RMSE and \(R^{2}\) values for our statistical tests of significance. At each round, we first applied pairwise Wilcoxon rank sum between each pair of polynomial degrees (i.e., 0 versus 4) for each model. Next, we applied Wilcoxon rank sum test between models and the fixed pre-touch distance (i.e., 20.0 cm). In the case of models, we considered their pairs of equal degrees (e.g., GP vs. SVR using degree 0). In the case of fixed pre-touch distance, \(R^{2}\) indicated whether the use of averaged pre-touch distance might yield a better estimate than such a fixed spherical boundary.