# Gaussian Process Modeling of Specular Multipath Components
## 3 Gaussian Process Regression
### Evaluate the Quality of Prediction  
Given a prediction method, we can evaluate the quality of prediction in several ways. Perhaps, the simplest is the squared error loss. However, this quantity is sensitive to the overall scale of the target values, so it makes sense to normalize by the variance of the targets of the test cases to obtain the standard mean squared error (SMSE) [31]. Additionally, for the GPR on the absolute values of SMC amplitudes, since we produce a predictive distribution at each test input, we can evaluate the negative log probability of the target under the model. This loss should be standardized and averaged to obtain the mean standard log loss (MSLL) [31].  
In order to evaluate the quality, the data set will be divided into two sets, for training and testing. Besides the training data \(\mathcal{D}_{k}\), the set of test data is given by \(\mathcal{D}_{k}^{*}=\{\mathbf{\phi}_{k}^{*},\mathbf{y}_{k}^{*}\}\) with \(N^{*}\) measurements \(y_{k,i}^{*}\) at positions \(\mathbf{p}_{i}^{*}\). The SMSE of the predicted mean \(\mathbb{E}[\psi(\phi_{k}(\mathbf{p}_{i}^{*}))|\mathcal{D}_{k},\mathbf{\theta}_{k}^{ \text{abs}}]\) in (16) of the angle-dependent amplitude \(\psi(\phi_{k}(\mathbf{p}_{i}^{*}))\) (or phase \(\zeta(\phi_{k}(\mathbf{p}_{i}^{*}))\)) based on the learned parameters \(\mathbf{\theta}_{k}^{\text{abs}}\) (or \(\mathbf{\theta}_{k}^{\text{ph}}\)) in (16) is given by [31]  
\[\text{SMSE}^{\text{abs}}=\frac{\sum_{i=1}^{N^{*}}\left(y_{k,i}^{*}-\mathbb{E}[ \psi(\phi_{k}(\mathbf{p}_{i}^{*}))|\mathcal{D}_{k},\mathbf{\theta}_{k}^{\text{abs}}] \right)^{2}}{\sum_{i=1}^{N^{*}}\left(y_{k,i}^{*}-\overline{y}_{k,i}^{*}\right)^{ 2}}, \tag{21}\]where \(\vec{y}_{k,i}^{*}=1/N^{*}\sum_{i=1}^{N^{*}}y_{k,i}^{*}\) is the mean value of the measurements (similarly for the phase we have \(\text{SMSE}^{\text{ph}}\)). In (21), the numerator is the sum of the squared error between the measured test data \(y_{k}^{*}\) and the predicted mean. The denominator is the sum of squared error between the measured test data \(y_{k}^{*}\) and the mean value of the measurements \(\vec{y}_{k,i}^{*}\). Thus, an SMSE significantly smaller than one indicates a high prediction quality.  
Another measure of quality is the MSLL, which also considers the predicted variance, that is, \(\mathbb{V}[\psi(\phi_{k}(\mathbf{p}_{i}^{*}))|\mathcal{D}_{k},\mathbf{\theta}_{k}^{ \text{abs}}]\) in (17). It is calculated by averaging over the standard log loss of the amplitude \(\psi(\phi_{k}(\mathbf{p}_{i}^{*}))\) (or phase \(\zeta(\phi_{k}(\mathbf{p}_{i}^{*}))\)) based on the learned parameters \(\theta_{k}^{\text{abs}}\) (or \(\theta_{k}^{\text{ph}}\)) for each pair in \(\mathcal{D}_{k}^{*}\), that is,  
\[\text{MSLL} =\frac{1}{N^{*}}\sum_{i=1}^{N^{*}}-\log p(y_{k,i}^{*}|\phi_{k}( \mathbf{p}_{i}^{*}),\theta_{k}^{\text{abs}})\] \[=\frac{1}{N^{*}}\sum_{i=1}^{N^{*}}\frac{1}{2}\log(2\pi\sigma_{n,k,i}^{2})+\frac{\left(y_{k}^{*}-\mathbb{E}[\psi(\phi_{k}(\mathbf{p}_{i}^{*}))| \mathcal{D}_{k}^{*},\mathbf{\theta}_{k}^{\text{abs}}]\right)^{2}}{2\sigma_{n,k,i}^ {2}}\,, \tag{22}\]  
where \(\sigma_{n,k,i}^{2}=\mathbb{V}[\phi_{k}(\mathbf{p}_{i}^{*})|\mathcal{D}_{k},\mathbf{ \theta}_{k}^{\text{abs}}]+\sigma_{\epsilon}^{2}\). Large negative values of the MSLL indicate a high prediction quality.