# Gaussian Process Modeling of Specular Multipath Components
## 4 Results
### GPR of SMC Amplitudes  
Table 1 lists the learned hyperparameters \(\mathbf{\theta}_{k}^{\text{abs}}\) (or \(\mathbf{\theta}_{k}^{\text{ph}}\)) obtained from the measurements. It is shown that the characteristic correlation angle of the LOS GP amplitude has the highest angle correlation, that is, the estimated LOS amplitude shows a small variation over a wide range of angles. This might come from two factors, (i) the influence of DMC on the LOS amplitudes is rather small shown by the DMC standard deviation \(\sigma_{v,k}^{\text{abs}}\) in Table 1, and (ii) the estimated LOS amplitudes only depend on the antenna pattern \(A(\phi_{k}(\mathbf{p}))\), while the reflection coefficient \(\Gamma_{1}(\phi_{1}(\mathbf{p}))\) is defined to be one. The characteristic correlation angles of the SMC GP amplitudes are significantly smaller, that is, the estimated SMC amplitudes have a much larger variation over the range of angles. This is a strong indicator that the SMC amplitudes show an angle-dependent reflection coefficient \(\Gamma_{k}(\phi_{k}(\mathbf{p}))\) and the influence of DMC is larger, which is in agreement with increased DMC standard deviation \(\sigma_{v,k}^{\text{abs}}\) in Table 1. All these observations are supported by results shown in the figures below. In the following, all estimated and modeled SMC amplitudes are normalized as defined by Equation (9). Figure 3 shows the regression results for the LOS amplitude associated with Anchor 1 and some SMCs associated with Anchors 1 and 2. Particularly, it shows the estimated SMC amplitudes (colored markers) for the individual segments, the predicted mean given in (16) (red solid line), and the predicted standard deviation given in (17) (black dashed and blue dash-dotted lines) as a function of the angle of departure \(\phi_{2}(\mathbf{p})\). The deviation is shown as \(\pm 2\sigma\), representing the upper and lower limits that contain 95% of the data points. In Figure 3a, the LOS amplitudes are shown. It can be observed that the predicted mean value of the LOS GP amplitude only slowly changes with the angle \(\phi_{1}(\mathbf{p})\) according to the antenna pattern \(A(\phi_{1}(\mathbf{p}))\). The predicted standard deviation is rather small since the learned DMC standard deviation and the characteristic correlation angle are large. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & \multirow{2}{*}{**MPC**} & \multicolumn{2}{c}{**Constant Mean \(\mathbf{\beta}_{k}^{\text{abs}}\)**} & \multicolumn{2}{c}{**DMC Stdv. \(\sigma_{v,k}^{\text{abs}}\)**} & \multicolumn{1}{c}{**Char. Corr. Angle \(\mathbf{a}_{k}^{\text{abs}}\)**} & \multicolumn{1}{c}{**GP Stdv. \(\sigma_{k}^{\text{abs}}\)**} \\  & & **or \(\mathbf{\beta}_{k}^{\text{ph}}\) [Linear or Rad]** & **or \(\mathbf{c}_{\mathbf{v,k}}^{\text{ph}}\) [Linear or Rad]** & **or \(\mathbf{c}_{k}^{\text{ph}}\) [Deg.]** & **or \(\mathbf{c}_{k}^{\text{ph}}\) [Linear or Rad]** \\ \hline \multirow{4}{*}{**OMC**} & EPB—anchor 1 & 0.062 & 0.009 & 2.91 & 0.013 \\  & LOS—anchor 1 & 0.099 & 0.003 & 43.11 & 0.035 \\  & NGW—anchor 1 & 0.075 & 0.014 & 1.02 & 0.02 \\  & SW—anchor 2 & 0.09 & 0.013 & 2.17 & 0.03 \\ \hline \multirow{4}{*}{**OMC**} & EPB—anchor 1 & \(-\)0.956 & 0.387 & 9.83 & 1.03 \\  & EPB—anchor 2 & \(-\)12.74 & 0.779 & 85.99 & 15.57 \\ \cline{1-1}  & NGW—anchor 1 & \(-\)3.79 & 0.826 & 3.37 & 1.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the Gaussian Process (GP) model for the absolute value of the specular multipath component (SMC) amplitudes. Figure 3: GP regression of SMC amplitude for (**a**) LOS amplitude associated with anchor 1 at \(\mathbf{a}_{1}^{(1)}\), (**b**) the SMC amplitude of VA at \(\mathbf{a}_{2}^{(1)}\) associated with anchor 1 (east plaster board wall), (**c**) the SMC amplitude of VA at \(\mathbf{a}_{2}^{(2)}\) associated with anchor 2 (east plaster board wall), (**d**) the SMC amplitude of VA at \(\mathbf{a}_{3}^{(2)}\) associated with anchor 2 (south wall), (**e**) the SMC amplitude of VA at \(\mathbf{a}_{5}^{(1)}\) associated with anchor 1 (north glass window).