# Gaussian Process Modeling of Specular Multipath Components
## 4 Results
### Measurement Pre-Processing  
The agent trajectory is divided into 7 segments shown with different colors. The spacing between trajectory points was 2 cm for segments 1 and 2, and 4 cm elsewhere. During the experiment, the exact positions of the trajectory points were unknown. Hence, in a pre-processing step, we used the SLAM algorithm [8,10] to estimate the trajectory positions \(\mathcal{L}=\{\mathbf{p}\}\) and the locations of the VAs and physical anchors \(\{\mathbf{a}_{k}\}\). The estimated trajectory and VAs are plotted in Figure 1. For Anchor 1, there are the VAs \(\mathbf{a}_{2}^{(1)}\), \(\mathbf{a}_{4}^{(1)}\) and \(\mathbf{a}_{5}^{(1)}\)--indicated here by additional superscript index \({}^{(1)}\)--, corresponding to single reflections on the EPB, WW and NGW respectively. For Anchor 2, there are the VAs \(\mathbf{a}_{2}^{(2)}\), \(\mathbf{a}_{3}^{(2)}\) and \(\mathbf{a}_{4}^{(2)}\), indicated by index \({}^{(2)}\), corresponding to single reflections on the EPB, SW and WW, respectively.  
Given the estimated positions of transmitters and receivers, the expected delays, that is, \(\tau_{k}(\mathbf{p})=\frac{1}{c}\|\mathbf{a}_{k}-\mathbf{p}\|\) are deduced. Then, the SMC amplitudes \(\delta_{k}(\mathbf{p})\) are estimated from the received signals, using these delays in (5). The use of (5) for estimating the SMC amplitudes relies on the assumption that individual SMCs do not overlap, reducing the interfering multipath to the DMC \(v_{k}(\mathbf{p})\). We remove overlapping SMCs from our data set if two delays have \(|\tau_{k}(\mathbf{p})-\tau_{l}(\mathbf{p})|\leq\frac{T_{p}}{2}\), for \(k\
eq l\), yielding the subsets \(\mathcal{P}_{k}\subseteq\mathcal{L}\) of trajectory points. Finally, the distance dependence will be removed from the SMC amplitudes to get the normalized amplitude data \(\psi(\phi_{k}(\mathbf{p}))\) and phase data \(\zeta(\phi_{k}(\mathbf{p}))\) according to Equations (9) and (10). Note that the trajectory positions and VA positions are mapped to the direction angles \(\phi_{k}(\mathbf{p})=\angle(\mathbf{p}-\mathbf{a}_{k})\) for the purpose of GPR.  
Several datasets are hence deduced, which are denoted as \(\mathcal{D}_{k}=\left\{\phi_{k}(\mathbf{p}),y_{k}(\mathbf{p})\right\}_{\mathbf{p}\in \mathcal{P}_{k}}\), containing the direction angles \(\phi_{k}(\mathbf{p})\) and the data points \(y_{k}(\mathbf{p})\). The data points are either the absolute values \(\psi(\phi_{k}(\mathbf{p}))\) or the phases \(\zeta(\phi_{k}(\mathbf{p}))\). The index \(k\) relates here to the different SMCs and the two anchors. Specifically, for Anchor 1, we analyze data sets corresponding to the LOS, the EPB, and the NGW, for Anchor 2, we analyze the EPB and the SW.  
On the basis of these data sets, the GP model is learned using the Matlab function _fitrgp_ (Matlab version R2018a; Statistics and Machine Learning Toolbox). Prediction is done using the function _predict_. The regression result will be discussed to see if it is sensible to use GP regression to  
Figure 2: Photos of the measurement hardware showing (**a**) the IImsens ultra-wideband (UWB) M-Sequence channel sounder and (**b**) a coin UWB dipole antenna.  
model the SMC amplitudes. We also evaluate the GPR performance numerically with the SMSE and the MSLL in Section 3.4.