# Automatic Clustering for Improved Radio Environment Maps in Distributed Applications
## 6 Numerical Results
To evaluate the performance of prediction, the RMSE was used as the performance metric. It is given by:  
\[\text{RMSE}=\Big{(}\frac{1}{\text{card}(\mathcal{T})}\sum_{g_{0}\in\mathcal{ T}}\big{(}\hat{q}(g_{0})-q(g_{0})\big{)}^{2}\Big{)}^{\frac{1}{2}}. \tag{10}\]  
Figure 4: Real coverage map. where \(\mathscr{T}\) is the testing set. \(\hat{q}(g_{0})\) and \(q(g_{0})\) are the estimated and the real RSRP at \(g_{0}\), respectively. We also used the R\({}^{2}\) metric expressed as  
\[\text{R}^{2}=1-\sum_{g_{0}\in\mathscr{T}}\frac{\big{(}q(g_{0})-\hat{q}(g_{0}) \big{)}^{2}}{\big{(}q(g_{0})-\hat{q}\big{)}^{2}}, \tag{11}\]  
where \(\bar{q}\) is the mean of RSRP values. In the following, we evaluate the benefit of using clustering in RSRP map construction. To this end, we carried out a comparative study between the following two cases: (i) without clustering, and (ii) with two clusters. For the first case, we estimated the predictor parameters using a learning set that considered the entire area of interest, while for the second case, for each cluster, we estimated the predictor parameters using a learning set that considered the area covered by the cluster. The results of the prediction when considering the first case are presented in Figure 6. Here, the performance of the RSRP prediction when considering the entire area as one cluster is shown. As seen from this result, one can clearly see that there was a difference with the real coverage presented in Figure 4. In fact, an RMSE of 6.70 dB was observed in this case (i.e., without clustering). For the second case, we estimated the predictor parameters of each cluster using its associated learning set. Then, the prediction for each testing set, associated with each cluster, was performed, based on the related estimated parameters. By combining the results for individual testing sets, we obtained the performance of the prediction for the entire testing set. The results of the prediction for each cluster are presented in Figure 7. In fact, we show, in Figure 7a,b, the constructed REMs for the first and second clusters, respectively. By substituting the two sub-REMs in Figure 7c, we obtain the REM for the whole area of interest. In Table 5, we present the comparison between the two cases, i.e., with and without clustering, in terms of RMSE. As seen from these results, the clustering offered a gain of about 3.3 dB and, consequently, led to a more accurately constructed map. Here, the standard deviation (std) of the RMSE was about 0.009 dB. To further examine the approach, we used R\({}^{2}\). Similarly, we compared the two cases with and without clustering. The R\({}^{2}\) of the generated model using the proposed approach achieved significant prediction results. Table 6 shows results of about 98%, which indicated high accuracy in terms of REM construction. Figure 5: Clustering of the environment. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Without Clustering** & **With Clustering** \\ \hline R\({}^{2}\) & 0.921 & 0.979 \\ \hline  & 1 st cluster & 2 nd cluster \\ \hline  & R\({}^{2}\) per cluster & 0.980 & 0.979 \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of the proposed approach with the case without clustering using R\({}^{2}\). Figure 6: Predicted map without clustering. Figure 7: Prediction of the: (**a**) first cluster, (**b**) second cluster (**c**) both clusters.