# Gaussian Process Modeling of Specular Multipath Components
## 3 Gaussian Process Regression
### GP Model  
Our goal is to model the angle-dependent SMC amplitude \(\psi(\phi_{k}(\mathbf{p}))\) and phase \(\zeta(\phi_{k}(\mathbf{p}))\) using a Gaussian process model. For the amplitude we have  
\[\psi(\phi_{k}(\mathbf{p}))\sim\mathcal{GP}\big{(}\mu_{\text{GP}}(\phi_{k}(\mathbf{p}) ),c_{\text{GP}}(\phi_{k}(\mathbf{p}),\phi_{k}(\mathbf{p^{\prime}}))\big{)}\,, \tag{11}\]  
where \(\mu_{\text{GP}}(\phi_{k}(\mathbf{p}))\): \(\mathbb{R}^{2}\to\mathbb{R}\) denotes the mean function and \(c_{\text{GP}}(\phi_{k}(\mathbf{p}),\phi_{k}(\mathbf{p^{\prime}}))\): \(\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}\) the covariance function (also called kernel). Equation (11) is used equivalently for the phase \(\zeta(\phi_{k}(\mathbf{p}))\). Covariance functions are a key component of GPR, since they provide the information of similarities between observations at different data points. Covariance functions are usually parameterized and these parameters, the mean function, and the noise are usually lumped together into a set called 'hyperparameters', which we will describe later in this section.  
In this work, the mean value is modeled by the constant, that is,  
\[\mu_{\text{GP}}(\phi_{k}(\mathbf{p}))=\beta_{k}^{\text{abs}} \tag{12}\]  
(and \(\beta_{k}^{\text{ph}}\) for the phase). A squared exponential kernel is used for the covariance,  
\[c_{\text{GP}}(\phi_{k}(\mathbf{p}),\phi_{k}(\mathbf{p^{\prime}}))=\sigma_{k}^{\text{ abs}\;2}\exp\left(-\frac{(\phi_{k}(\mathbf{p})-\phi_{k}(\mathbf{p^{\prime}}))^{2}}{2a_{k}^{ \text{abs}\;2}}\right)+\sigma_{\
u,k}^{\text{abs}\;2}\delta(\mathbf{p}-\mathbf{p^{ \prime}}), \tag{13}\]  
where \(\sigma_{k}^{\text{abs}}\) is the standard deviation of the correlation kernel, \(a_{k}^{\text{abs}}\) is the characteristic correlation angle, and \(\sigma_{\
u,k}^{\text{abs}}\) is the standard deviation accounting the DMC as shown in (9) (and \(\sigma_{k}^{\text{ph}}\), \(a_{k}^{\text{ph}}\), and \(\sigma_{\
u,k}^{\text{ph}}\) for the phase). Note the abuse of notation in the latter term, which is due to the mapping of \(\mathbf{p}\in\mathbb{R}^{2}\) to \(\phi_{k}(\mathbf{p})\in\mathbb{R}\). The Dirac function \(\delta(\mathbf{p}-\mathbf{p^{\prime}})\) indicates that the DMC are assumed to be spatially uncorrelated. Here, we have a pair of agent positions \((\mathbf{p},\mathbf{p^{\prime}})\) mapped to one covariance value, that is, \(\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}\).  
These models will be the basis for the GP modeling of the SMC amplitudes. Together with the data, the GP model reveals the angle-dependent functions \(\psi(\phi_{k}(\mathbf{p}))\) and \(\zeta(\phi_{k}(\mathbf{p}))\) as well as a variance that relates to the DMC \(S_{\
u}(\tau_{k};\mathbf{p})\). The full hyper-parameter vectors of the GP model which have to be learned from the amplitude and phase data of SMC \(k\) are given as  
\[\theta_{k}^{\text{abs}}=[a_{k}^{\text{abs}},\sigma_{k}^{\text{abs}},\sigma_{ \
u,k}^{\text{abs}},\rho_{k}^{\text{abs}}]^{\text{T}} \tag{14}\]  
and  
\[\theta_{k}^{\text{ph}}=[a_{k}^{\text{ph}},\sigma_{k}^{\text{ph}},\sigma_{\
u, k}^{\text{ph}},\rho_{k}^{\text{ph}}]^{\text{T}}\,, \tag{15}\]  
respectively.