# Optimized Sensors Network and Dynamical Maps for Monitoring Traffic Noise in a Large Urban Zone
## 4 Results
### Correlations between Sensor Locations: How to Optimize the Network  
Thus far, we have discussed how the Dynamap network can be constructed, presenting detailed discussions of its design, and how its predictions can be assessed by reporting a number of tests for its validation. The general question arises, a posteriori, of how the design of the network can be optimized further. The way to do it is closely related to the concept of correlations among the noise station measurements. The idea is rather simple: one wishes to reduce to a minimum the amount of redundant information registered by the different stations. In other words, their locations should be identified such that they can collect as much distinct information as possible about the traffic noise within the urban zone. To this end, we study the sensor variations, \(\delta^{(\tau)}(t,s_{j}(g_{i}))\) (Equation (14)), at different time intervals \(\tau=(5,10,15,30,60)\) min, over a period of five consecutive working days, thus creating 24 contemporary times series for each value of \(\tau\). To determine the degree of correlation between, say, sensors \(s_{j}\) and \(s_{j^{\prime}}\), we calculate the Pearson's correlation coefficient (see, e.g., [60]), given by  
\[\rho(j,j^{\prime})=\frac{\text{cov}(j,j^{\prime})}{\sigma_{j^{\prime}}\sigma_ {j^{\prime}}}, \tag{17}\]  
where \(\text{cov}(j,j^{\prime})\) is the covariance between times series \(j\) and \(j^{\prime}\), defined in the standard fashion as the mean value of the product of the deviations from their mean values, and  
Figure 9: Comparison of equivalent noise levels \(L_{\text{eq}}^{(1\text{b})}(t,s)\) vs hour \(t\) of the day for: (**a**) Traffic noise measurements and ‘non-corrected’ Dynamap predictions for Site 10 (upper panel), and Site 12 (lower panel). The bands correspond to the statistical errors. (**b**) Traffic noise measurements and ‘corrected’ Dynamap predictions for the same sites. The bands correspond to the total errors [86]. (With permission from Elsevier). are the corresponding standard deviations of the time series. In this way, we can study both intragroup correlations (same sensor group indexes \(g_{i}\)) or intergroup ones (different \(g_{i}\)'s). To study the noise correlations among the 24 sensors using Equation (17), we employ two normalization procedures:  
* (P1) From each time series, we remove the hourly median value. In this way, we obtain what we called a de-trended time series. * (P2) From each time series, we remove the mean level, \(\left\langle L_{\text{eq}}^{\left(\text{1h}\right)}(t,s)\right\rangle\), calculated between 6:00-10:00 p.m. This normalization procedure retains both high and low frequency fluctuations. An illustrative example of each procedure is shown in Figure 10, for the noise station \(s136\) at time interval \(\tau=5\) min. To be noted is that, for P1, the resulting profile is quite flat \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Site** & \(g_{i}\) & \(\left\langle\boldsymbol{e_{\text{Leq}}}\right\rangle_{\text{N}}\) & \(\left\langle\boldsymbol{e_{\text{Leq}}}\right\rangle_{\text{C}}\) & \(\left\langle\boldsymbol{e_{\text{Leq}}}\right\rangle_{\text{M}}\) \\ \hline  
\hline
19 & 6 & 3.4 & 1.3 & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mean local deviations (in dB) of Dynamap predictions from measurements at different sites. The codes of the measurement sites (first column) are indicated by single numbers (see Table 4). The group index \(g_{i}\) is given for each site (second column). The remaining three columns report the mean deviations: \(\left\langle\boldsymbol{e_{\text{Leq}}}\right\rangle_{\text{N}}\) obtained without systematic error corrections, \(\left\langle\boldsymbol{e_{\text{Leq}}}\right\rangle_{\text{C}}\) obtained after systematic error corrections have been applied, and \(\left\langle\boldsymbol{e_{\text{Leq}}}\right\rangle_{\text{M}}\) corresponding to the median average of the corrected predictions [91].