## 3 Results
#### Time History Plot Comparison and Forecast Errors Analysis  
Analysis of the error of the three proposed models gives information on attitudes of the three strategies to forecast \(\mathrm{L_{Aeq,16\,h}}\) in periods not used in the calibration phase. In Figure 11a, it is easy to see that the DD-TSA model underestimates the real value of acoustic emission in every 100 periods of the performance test. To correct this wrong behavior, it may be advisable to recalculate the trend by cutting part of the oldest periods of the calibration series. The weekly periodicity of the series is reproduced with good approximation, but some emission peaks are not adequately predicted by the model because of the completely random nature of these events. \begin{table}
\begin{tabular}{c c c} \hline
**Model** & **MPE** & **CVE** \\ \hline DD-TSA & 0.000 & 0.012 \\ DD+ANN (_t_-1) & \(-\)0.030 & 0.007 \\ DD+ANN (_t_-7) & 0.012 & 0.009 \\ \hline \end{tabular}
\end{table} 
Table 12: MPE and CVE values for DD-TSA, DD+TSA (_t_-1) and DD6+TSA (_t_-7). Figure 10: Correlogram plot for the residuals, evaluated in the comparison dataset of the calibration phase (1087 data): on the left, (**a**) the DD-TSA model, in the center, (**b**) the DD+ANN (_t_-1) model, and on the right, (**c**) the DD+ANN (_t_-7) model. The value of the autocorrelation coefficient is plotted as a function of the lag. Blue dotted lines represent the threshold of 0.1, under which the data are considered not autocorrelated. In particular, an unexpected low \(\text{L}_{\text{Aeq,16 h}}\) is observed in period 1119: this event is not predicted by any of the three models. This low \(\text{L}_{\text{Aeq,16 h}}\) is observed during a national Italian holiday called \"celebration of the Republic\". In the following time interval, in which there is a gradual recovery of normal traffic, the emission level also increases, but it is still lower than the average of the period. In this case, the DD+ANN model (\(t\)-1), being the fastest to follow the sudden and unexpected changes, manages to resemble the actual observed data. The effect of reduction due to the low value observed during June 2 in the \(\text{L}_{\text{Aeq,16 h}}\) foreseen by the DD + ANN model (\(t\)-7) occurs only after seven periods, with a negative result on the error of prediction of the model at that time period. In general, to solve the problem of forecasting during public holidays, it would be appropriate to include in the neural network a \"flag\" type input to distinguish working days from holiday periods. In Figure 11b, the forecasting errors provided by the three models in the 100 periods of the performance test are shown. In particular, the errors of the DD-TSA model were evaluated according to Formula (3), while the errors of the two hybrid models with ANNs in cascade have been calculated according to Formula (5). It can be noticed that the errors of the DD-TSA model are not random but still exhibit a weekly periodicity which was not adequately reproduced by the model. In addition, most of the errors are positive, further confirming that the model underestimates the measured data during this time interval. The plots of hybrid models show a greater dispersion around zero but are nevertheless distributed more evenly between positive and negative values. Table 13 shows the statistics of the model errors on 100 periods of the performance test. The mean and median of the error for the DD\(-\)TSA model are different from zero, due to the underestimation compared to the observed data. On the contrary, for hybrid models, both the mean and the median are very low and close to zero. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Mean** & **Std. Dev.** & **Median** & \multirow{2}{*}{**Skew**} & \multirow{2}{*}{**Kurt**} \\  & **[dBA]** & **[dBA]** & **[dBA]** & \\ \hline DD-TSA & 0.53 & 0.49 & 0.53 & \(-\)0.93 & 0.98 \\ DD+ANN (\(t\)-1) & 0.04 & 0.59 & 0.12 & \(-\)0.33 & 0.34 \\ DD+ANN (\(t\)-7) & 0.05 & 0.63 & 0.08 & \(-\)0.23 & 0.29 \\ \hline \hline \end{tabular}
\end{table}