## 5 Explainability of the Proposed Model
Average building width: this feature refers to the arithmetic mean building width along the profile. It also implies the average street width. Figure 12 shows three selected path profiles with narrow buildings, wide buildings, and moderate buildings. These profiles were inputted into the network and their corresponding outputs were reordered. The subfigure in the bottom right of Figure 12 indicates a non-linear relationship between this feature and the predicted path loss
7. Distance to the nearest building from the Rx: in general, the path loss tends to be high when a high building is close to the Rx. This feature refers to the distance from the Rx to the nearest building along the profile. Figure 13 shows three selected path profiles with a building far from the Rx, near the Rx, and at a moderate distance. These profiles were inputted into the network and their corresponding outputs were reordered. The subfigure in the bottom right of Figure 13 indicates that this feature is relevant to the path loss prediction. 8. Height of the nearest building from the Rx: this feature is the height of the nearest building along the profile from the Rx. Figure 14 shows three selected path profiles with low, medium, and high values of this feature. These profiles were inputted into the network and their corresponding outputs were reordered. The subfigure in the bottom right of Figure 14 suggests a positive linear correlation between this feature and the predicted path loss. To evaluate the quality and appropriateness of the explainability of the proposed deep neural network, the aforementioned features were used to model the path loss based on the approach of linear regression. For single-frequency scenarios, the log-frequency term in Equation (2) can be merged into the left-hand side. Here, the coefficient \(\gamma\) in Equation (2) is set to two. The path loss for the kth Tx-Rx pair is given by  
\[PL_{k}^{XAI}=10\alpha log_{10}R_{k}+\beta+\sum_{i=1}^{8}u_{i}z_{i,k}+\sum_{i=1} ^{8}v_{i}z_{i,k}^{2} \tag{8}\]  
Figure 7: The relevance of the mean value of building height to the proposed path loss model. where \(z_{i,k}\), for \(1\leq i\leq 8\), stands for the eight aforementioned features. \(\alpha\), \(\beta\), \(u_{i}\), and \(v_{i}\) are model parameters to be optimized. Although some features seem to be linearly correlated with the path loss prediction, their quadratic forms are also considered in Equation (8), because they do no harm to the path loss model. To achieve the optimal model parameters based on the least-square regression, Equation (8) can be expressed in matrix form as:  
\[\mathbf{Y}=\mathbf{\Gamma}\boldsymbol{\omega} \tag{9}\]  
where:  
\[\mathbf{Y}=\left[ \begin{array}{cccccccc}y_{1}&y_{2}&\cdots&y_{M}\end{array}  \right]^{T} \tag{10}\]  
\[\mathbf{Z}=\left[ \begin{array}{cccccccc}10 log_{10}R_{1}&1&z_{1,1}&z_{2,1}& \cdots&z_{8,1}&z_{1,1}^{2}&z_{21}^{2}&\cdots&z_{8,1}^{2}\\ 10 log_{10}R_{2}&1&z_{1,2}&z_{2,2}&\cdots&z_{8,2}&z_{1,2}^{2}&z_{2,2}^{2}& \cdots&z_{8,2}^{2}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 10 log_{10}R_{M}&1&z_{1,M}&z_{2,M}&\cdots&z_{8,M}&z_{1,M}^{2}&z_{2,M}^{2}& \cdots&z_{8,M}^{2}\end{array} \right] \tag{11}\]  
\[\boldsymbol{\omega}=\left[ \begin{array}{cccccccc}\alpha&\beta&u_{1}&u_{2}& \cdots&u_{8}&v_{1}&v_{2}&\cdots&v_{8}\end{array} \right]^{T} \tag{12}\]  
As a result, the optimal parameter that minimizes the fitting error is given by [6]:  
\[\boldsymbol{\omega}_{\text{OPT}}=\left(\mathbf{Z}^{T}\mathbf{Z}\right)^{-1} \mathbf{Z}^{T}\mathbf{Y} \tag{13}\]  
Additionally, the predicted path loss can be expressed as:  
\[\hat{y}_{k}=\mathbf{z}_{k}\boldsymbol{\omega}_{\text{OPT}} \tag{14}\]  
where \(\mathbf{z}_{k}=\left[ \begin{array}{cccccccc}10 log_{10}R_{k}&1&z_{1,k}&z_{2, k}&\cdots&z_{8,k}&z_{1,k}^{2}&z_{2,k}^{2}&\cdots&z_{8,k}^{2}\end{array} \right]\). Figure 8: The relevance of the standard deviation of building height to the proposed path loss model.