## 2 Materials and Methods
### The Non-Acoustic Parameter and Its Distribution Function  
In order to predict traffic noise for a given road stretch, when a direct measurement is not practicable, we need to define a non-acoustic parameter, which we denote generically as \(x\) and is related to model calculations of the traffic flow on that road stretch. Here, we have used the traffic model developed by the municipal agency for mobility, environment and territory (Agenzia Mobilita Ambiente e Territorio, AMAT), in charge of the transportation policy in the city of Milan. The model  
Figure 3: Shapiro–Wilk results for the normal distribution test of \(\delta_{kk}\) for Cluster 1 (**a**) and Cluster 2 (**b**) The red line indicates the threshold at the \(\alpha\) = 0.05 significance level. Figure 2: Mean cluster normalized equivalent level profiles (continuous lines), \(\overline{\delta}_{kk}\) (dB), for the two clusters, \(k\) = (1,2), as a function of hour \(i\) of the day (also indicated with the letter \(h\)). We show the corresponding dispersion of the hourly values within \(\sqrt{2}\) standard deviation, \(\sigma\) (colored bands). We will use also the notation \(\delta_{\mathrm{C1}}(h)\) and \(\delta_{\mathrm{C2}}(h)\) for the mean equivalent noise levels of Clusters 1 and 2 as a function of hour \(h\). Cluster 1 has 56 elements and Cluster 2 has 37 components. Details on the measurements and analysis related to this figure are reported in [29; 33; 39]. provides mean hourly values of traffic flow for each road stretch n in the entire urban zone of Milan. The hourly behavior of the traffic noise for a given road stretch n, characterized by a value \(x_{n}\), can be described in terms of the distribution functions of the variable \(x\), \(P(x)\), obtained from the roads belonging to Clusters 1 and 2. The corresponding distribution functions, denoted as \(P_{1}(x)\) and \(P_{2}(x)\), are shown in Figure 4, for the choice of \(x\) given by the logarithm of the total daily traffic flow rate \(x=\text{Log}(\text{T}_{\text{T}})\). Other choices of \(x\) lead to similar qualitative behavior [39; 40]. As one can see from Figure 3, there is a conspicuous overlap between the two distributions, suggesting that a sharp separation into two clusters is not possible in general. We conclude that a given value of \(x\) has components in both clusters, meaning that the temporal evolution of the noise for a given road stretch is partly due to Cluster 1 and partly due to Cluster 2. The idea of the method is to evaluate the probability \(\beta_{1}\) that \(x\) belongs to Cluster 1 and the probability \(\beta_{2}=1-\beta_{1}\) that it belongs to Cluster 2. The corresponding values of \(\beta\) are given by the following relations:  
\[\beta_{1}(x)=\frac{P_{1}(x)}{P_{1}(x)+P_{2}(x)} \tag{1}\]  
\[\beta_{2}(x)=\frac{P_{2}(x)}{P_{1}(x)+P_{2}(x)}.\]  
Using the values of \(\beta_{1,2}\) we can predict the hourly variations \(\delta_{x}(h)\) for a given value of \(x\) according to  
\[\delta_{x}(h)=\beta_{1}(x)\;\delta_{\text{C1}}(h)+\beta_{2}(x)\;\delta_{\text{ C2}}(h) \tag{2}\]  
with \(\delta_{\text{C1}}(h)\) and \(\delta_{\text{C2}}(h)\) representing the mean hourly values of the equivalent level (Figure 2) for both Clusters 1 and 2, respectively. The error made in using Equation (2) can be estimated (see also Section 3) by calculating the square deviation \(\varepsilon^{2}\) of the prediction \(\delta_{x}(h)\) from the measured values \(\delta_{meas}(h)\), that is  
\[\varepsilon^{2}=\frac{1}{24}\sum_{h=1}^{24}[\delta_{x}(h)-\delta_{meas}(h)]^{ 2}. \tag{3}\]  
Figure 4: Distribution functions \(P_{1}(x)\) and \(P_{2}(x)\) for \(x=\text{log}(\text{T}_{\text{T}})\) on Clusters 1 and 2, respectively. The continuous lines represent logarithmic fits to the actual histograms using the cumulative distribution functions and are used here to accurately determine \(\beta_{1,2}\) using Equation (1).