## 2 Materials and Methods
### Artificial Neural Network Model  
ANNs are a type of model made by the union of simple mathematical functions acting in a parallel structure, able to simulate the behavior of the human brain. These functions are called \"artificial neurons\"; they have many inputs, with assigned weights, and a single output. The activation of the single neuron is a function of the weighted mean of inputs. This structure is able to learn and generalize, producing outputs even with unknown inputs. The simplest way to train an ANN is to give as input a series of data (training set) and to compare the response of the network with the true value. Once the difference between the two responses (error) is evaluated, the weights can be adjusted to minimize the error or to bring the error below a given threshold.  
The ANN implemented in this paper is a feedforward backpropagation network trained with the \"trainlm\" function of Matlab, which optimizes the error estimation through the Levenberg-Marquardt optimization. The adaptation learning function is the \"learndm\" one, which is the gradient descent with momentum weight and bias learning functions. The model has seven inputs and a single output, i.e., \(\hat{e}_{t}\), which represents the error of the network at the period \(t\). The hidden layer is single and is made of 140 neurons, i.e., 20 times the inputs. The transfer function for the hidden layer neurons is of the sigmoid type, while the one for the output neurons is linear.  
The objective function to minimize is the mean squared normalized error performance function, and the learning algorithm stops at the value of 0.01. The adopted strategy to improve the generalization performances of the network and to reduce overfitting risks is the \"_Cross-validation_\". The maximum number of accepted worsening in the validation phase, before the algorithm stops, is 10.  
Once the weights have been calculated, the ANN can give an output for the estimation of the random component \(\hat{e}_{t}\). For the purposes of the present work, two different neural networks have been designed and implemented, with different input vectors and, consequently, different prediction horizons: ANN (\(t\)-1) and ANN (\(t\)-7).  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{b_{0}}\) [dBA] & \(\mathbf{b_{1}}\) [dBA/Day] \\ \hline Trend component & 73.638 & \(-0.0019969\) \\ \hline \end{tabular}
\end{table}
Table 6: Model parameters estimated on the calibration dataset: \(b_{0}\) is the intercept, where \(b_{1}\) is the slope of the trend line.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(\mathbf{\overline{S}_{1}}\) & \(\mathbf{\overline{S}_{2}}\) & \(\mathbf{\overline{S}_{3}}\) & \(\mathbf{\overline{S}_{4}}\) & \(\mathbf{\overline{S}_{5}}\) & \(\mathbf{\overline{S}_{6}}\) & \(\mathbf{\overline{S}_{7}}\) \\  & **Friday** & **Saturday** & **Sunday** & **Monday** & **Tuesday** & **Wednesday** & **Thursday** \\ \hline Seasonal component & 1.00414 & 0.99805 & 0.98650 & 1.00143 & 1.00287 & 1.00323 & 1.00371 \\ \hline \end{tabular}
\end{table}
Table 7: Model parameters estimated on the calibration dataset: \(\overline{S}_{i}\) are the seasonal coefficients referring to the the seven days.  
The used dataset is divided into three parts: one for the training of the network (60%), one for the validation (20%) and one for the test (20%). Two cycles of training were executed, to obtain an optimum minimization of the objective function.