## 3 Gaussian Processes for Mobile Networks  
In this work, the predictive mean and variance function are given by a GP regression model. The model presents a Gaussian distribution at each candidate location (\(\mathbf{x}\)). Observations (\(y=f(\mathbf{x})+\epsilon\)) are modeled with Gaussian white noise (\(\epsilon\sim\mathcal{N}(0,\,\sigma_{n}^{2})\)). The \(\sigma_{n}^{2}\) parameter is often called measurement noise and is one of the hyperparameters (\(\theta\)) of the GP. A Gaussian process prior specified by a mean function (\(m(\mathbf{x})\)) and a covariance function (\(k(\mathbf{x},\mathbf{x}^{\prime})\)) is used to model the unknown function (\(f(\mathbf{x})\)):  
\[f(\mathbf{x})\sim\mathcal{GP}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}^{\prime})). \tag{1}\]  
The predictive posterior distribution of the signal strength at a new location (\(\mathbf{x}^{*}\)), given the current dataset (\(X_{\mathrm{1:n}}=\{\mathbf{x}_{\mathrm{1:n}},y_{\mathrm{1:n}}\}\)), can be computed as  
\[\overline{f}^{*} =m(\mathbf{x}^{*})+k(\mathbf{x}^{*},\mathbf{x}_{i})K^{-1}(y-m( \mathbf{x}_{i})), \tag{2a}\] \[\mathbb{V}^{*} =k(\mathbf{x}^{*},\mathbf{x}^{*})-k(\mathbf{x}^{*},\mathbf{x}_{i })^{\top}K^{-1}k(\mathbf{x}^{*},\mathbf{x}_{i}). \tag{2b}\]  
where \(K\) denotes the covariance matrix with entries of \([K]_{ij}=k(\mathbf{x}_{i},\mathbf{x}_{j})+\sigma_{n}^{2}\delta_{ij}\), where \(\delta_{ij}=1\) for \(i=j\) and zero otherwise. \(k(\mathbf{x}^{*},\mathbf{x}_{i})\) results in the \(\mathrm{n}\times 1\) vector of cross-covariances between \(\mathbf{x}^{*}\) and the training inputs (\(\mathbf{x}_{i}\in X_{\mathrm{1:n}}\)). The Matern kernel for \(\
u=1/2\) defined as  
\[k_{M}^{1/2}(\mathbf{x},\mathbf{x}^{\prime})=\sigma^{2}\exp\biggl{(}-\frac{| \mathbf{x}-\mathbf{x}^{\prime}|}{\ell}\biggr{)}, \tag{3}\]  
is used as a covariance function. Here, \(\sigma\) represents the signal variance, \(\ell\) denotes the length scale, and \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\) are input vectors. The \(1/2\)-Matern kernel was chosen because it performs well compared to other common kernels, i.e., the squared exponential kernel used by [PERSON] et al. [7]. All evaluation calculations (see Section 4) were also performed for a squared exponential kernel, as well as \(3/2\)-Matern and \(5/2\)-Matern kernels. In order to highlight the influence of the mean functions, the results of the other kernels are not shown for the sake of clarity and brevity.  
The hyperparameters of the kernel are bounded to physically reasonable values for the mobile network setting. The kernel variance (\(\sigma\)) is bounded between 1 and 100, and  
Figure 3: Scatter plot of the received reference signal strength (RSRP) measurements, together with the antenna pattern of the base station, on a geospatial map. The base station is marked with a drop pin. Created using the MATLAB Antenna Toolbox. Background: © OpenStreetMap contributors, CC BY-SA.  
the length scale (\(\ell\)) is bounded between 0 and 250. Furthermore, the noise variance (\(\sigma_{n}\)) is kept fixed on an initial value computed using the measurement data. This avoids very high trained hyperparameter values, especially for the zero-mean GP, which may lead to numerical instability. High values of kernel hyperparameters can occur if the training data are not very informative.  
The GPs are implemented in Python 3.7 using the GPFlow package [16] in combination with TensorFlow [17].