# Automatic Clustering for Improved Radio Environment Maps in Distributed Applications
## 3 Related Work
### Kriging Technique  
To estimate the RSRP of UE located at \(g\in\mathbb{R}^{2}\) and served by a base station (BS), we use the Kriging geostatistical approach. The purpose here is to build a map allowing the estimation of the RSRP given by  
\[z(g)=\underbrace{p_{BS}+10\mu\log_{10}(d(g))+s(g)}_{q(g)}+\varepsilon(g), \tag{1}\]  
where \(P_{BS}\) is the BS transmission power, \(d(g)\) is the geometrical distance between BS and UE, and \(\mu\) is the path loss exponent. The value \(s(g)\) represents the effect of shadowing in a wireless environment. It is modeled through a log-normal distribution. It is also associated with the stationary covariance matrix \(\mathbf{M}_{\mathbf{\kappa}}\), where \(\mathbf{\kappa}\) is the set of parameters. Finally, \(\varepsilon(g)\) represents the error of measurements and is modeled by means of a centered uncorrelated Gaussian process with variance \(\sigma^{2}\). In this study, we realize a lognormal random process. This is performed using a standard deviation of 6 dB to model the shadowing. With this setting, we consider a Non-LOS Urban Macro-cell environment [51], where the shadowing decorrelation distance is equal to 20 m [52]. Furthermore, we assume that \(s(g)\) and \(\varepsilon(g)\) are independent processes. We also assume that the RSRPs are collected by \(K\) sensor nodes (SNs) located at \(g_{k},k=1,2,\ldots,K\). Here, the wireless sensor network is modeled using a connectivity graph \(G(\mathbf{S},\mathbf{L})\), in which the SNs are denoted as \(\mathbf{S}=\{1,2,\ldots,K\}\) and the links are defined by \(\mathbf{L}\in\{L_{n},m\}_{K,K}(n,m\in\mathbf{V})\). The spatial location of SN\({}_{k}\) is denoted by \(g_{k}=(x_{k},y_{k})\) and SN\({}_{k}\) measures the RSRP value at \(g_{k}\). Thereafter, the collected RSRP values are transferred to the edge cloud. Next, the edge cloud builds an REM with the collected RSRPs. The reported RSRP measurements by the \(K\) SNs can be defined as  
\[\mathbf{r}=\mathbf{F}\mathbf{v}+\mathbf{s}+\mathbf{e}, \tag{2}\]  
where \(\mathbf{r}=(r(g_{1}),\ldots,r(g_{K}))^{T}\) represents a \(K\times 1\) vector, \(\mathbf{F}\in\mathbb{R}^{K\times 2}\) represents a deterministic matrix, in which the first column contains \"1s\", while the second is filled with \(10\log_{10}(d(g_{k}))\). \(\mathbf{v}=(p_{BS},\mu)^{T}\) represents a \(2\times 1\) vector, and \(\mathbf{s}=(s(g_{1}),\ldots,s(g_{K}))^{T}\) and \(\mathbf{e}=(\varepsilon(g_{1}),\ldots,\varepsilon(g_{K}))^{T}\) are \(K\times 1\) vectors. The RSRP vector, \(\mathbf{r}\in\mathbb{R}^{K\times 1}\), is considered to be a multivariate Gaussian random variable, characterized by the mean \(\mathbf{F}\mathbf{v}\) and covariance matrix \(\mathbf{M}_{(\mathbf{\kappa},\sigma)}=\mathbf{M}_{\mathbf{\kappa}}+\sigma^{2}\mathbf{ I}_{K}\). In this work, we use the Kriging technique to interpolate the RSRP at a location in the area of interest \(g_{0}\
otin\{g_{1},\ldots,g_{K}\}\). As \((q(g_{0}),\mathbf{r})^{T}\) is a Gaussian vector, the predicted RSRP value using the Kriging method at \(g_{0}\) is the conditional expectation of \(q(g_{0})\), given the measurement vector \(\mathbf{r}\). It can be expressed as [53]  
\[\hat{q}(g_{0})=\mathbf{f}(g_{0})\hat{\mathbf{v}}+\mathbf{m}_{\mathbf{\hat{\kappa} }}\mathbf{M}_{(\mathbf{\hat{\kappa}},\sigma)}^{-1}(\mathbf{r}-\mathbf{F}\hat{ \mathbf{v}}), \tag{3}\]  
where \(\mathbf{f}(g_{0})\) is the drift function vector at the location \(g_{0}\), \((\hat{\mathbf{v}},\hat{\mathbf{\hat{\kappa}}},\hat{\sigma})\) represents the estimated value of \((\mathbf{v},\mathbf{\alpha},\sigma)\) and \(\mathbf{m}_{\mathbf{\hat{\kappa}}}\) represents the vector of covariance between the target and the measurement values. In this work, we use the method of moments of [PERSON] [24; 54; 55; 56; 57] to estimate the Kriging parameters.