## 2 Materials and Methods
### Analysis  
Our earlier results [27] based on the present dataset indicated that such factors as gender and approaching angles did not induce any significant difference on the evaluators' comfortable distances and that although the acclimation effect showed a trend, its difference was less than 2.0 cm. Therefore, we combined the data from all four distinct male/female pairs in the present analyses. First, we constructed (Figure 2a) the grid of the participants' pre-touch distance (in cm) based on their associated azimuth and elevation angles (both angles within [\(-\)89.00,...,+89.00] in degree). Next, we modified this grid to eliminate the entire row that included a cell with no pre-touch distance (i.e., grey cells in Figure 2a). The resulting modified grid included 5664 facial-area pre-touch samples whose azimuth and elevations angles were [\(-\)80.00,..., +80.00] (mean (M) = 0.54, median (Mdn) = 0.10, standard deviation (STD) = 38.99, 95.0% confidence interval (CI\({}_{95.0\%}\)) = [\(-\)0.11 1.20]) and [\(-\)32.00,..., +53.00] (M = 9.83, Mdn = 9.17, STD = 24.17, Cl\({}_{95.0\%}\) = [9.33 10.31]). We then used this modified grid to construct the pre-touch surface (Figure 2b). Considering the mean and median of the azimuth angles associated with these data points that are closely centered about zero, the participants' data around azimuth axis was uniformly distributed between the two sides of the face. In the case of elevation angles, we observed that the touchers' data were mostly approached the touchee's face above the lips area. Figure 2c visualizes the distribution of touchers' pre-touch distances around the touchees' face area (i.e., the location at which the touchers were asked by the touches to stop approaching). The coordinates of these data points correspond to the toucher's hand distance from the touchee's face (\(x\)-axis, in cm),azimuth angle (\(y\)-axis, in degree), and elevation angle (\(z\)-axis, in degree) with respect to the touches' face. Figure 2d shows the head direction. We considered two pre-touch boundary estimation scenarios. In the first scenario that was based on our earlier results [27], we used a fixed pre-touch distance of 20.0 cm (i.e., a sphere of diameter 40.0 cm) around the face. In the second scenario, we used the data associated with participants' pre-touch surface (Figure 2b) and used regression models for estimating this boundary. We considered three regression models: (1) Gaussian processes regression (GR) to evaluate whether assumption of normal distribution in participants' pre-touch distances warrants a reliable estimate of this pre-touch boundary, (2) support vector regression (SVR) to determine whether estimating the pre-touch boundary by minimizing the orthogonal distance between participants' pre-touch data and its corresponding pre-touch boundary can yield a better result, and (3) ordinary regression with Lasso regularization (Lasso) to validate the utility of a non-parametric regressor with a simple criterion i.e., L1-regularization (i.e., sum of the model's absolute weight values) for estimating this pre-touch boundary. For all these models, we used the azimuth and elevation angles as input features and their corresponding pre-touch distances as output values that these models needed to estimate. The preprocessing of the models' input features included the scaling of the azimuth and elevation angles within [0,..., 1] using \(\frac{\alpha-min(A)}{max(A)-min(A)}\) where \(A\) is the set of either azimuth or elevation angles and \(\alpha\) identifies a specific angle that to be scaled. Figure 2: Pre-touch distance surface. (**a**) Full participant’s pre-touch distance (cm) grid along the azimuth and elevation angles. Azimuth and elevation angles are within \(-89.00\) and \(+89.00\) degrees. The color-bar on the side indicates the pre-touch distance value (in cm) in each cell of the azimuth-elevation grid whose cells are constructed by 1-degree increment along these angles’ directions. Grey cells in this subplot show the areas in which no pre-touch distance was available.