## 4 Experimental Results  
In this section, we present the forecast results obtained on the dataset described in Section 3.1 by the strategy we propose. We also present a comparison with different methods, both standard and ML based. In order to assess the predictions produced by our proposal, we used the \(MRE\) measure, as defined in Equation (2). \(MRE\) represents the ratio of the forecasting absolute error to the observed value. Before presenting the comparison with other methods, we inspect the results obtained by the proposed strategy for each historical window value used (\(w\)) and each subproblem (\(h\)). Figure 3 shows a graphical representation of the results obtained, showing the associated MRE for different values of \(w\), when varying the length of \(h\). We can see that the best results were achieved when the forecasting is based on more historical data, i.e., for higher values of \(w\). In fact, the best results were obtained for \(w=168\). Analogously, the MRE increases as \(h\) becomes longer. The proposed strategy obtains similar results for \(w=\{168,144,120\}\) on all the considered values of the prediction horizon \(h\). It can be noticed that there is a significant increment in the error when the historical window size is lower. In particular, when \(w\) is set to 24 or 48, the predictions degenerates evidently. We can also notice that performances of the proposed strategy deteriorates, i.e., the achieved MRE is higher, as the values of \(h\) increase. This means that it is more difficult to predict further in the future. Table 4 shows the parameters selected by the GA for each \(h\) when a historical window of 168 was used. We can notice that the number of layers range between 27 and 98, and the number of neurons per layer between 478 and 942. It does not seem that this parameter is connected with the value of \(h\). Parameters \(\lambda\), \(\rho\) and \(\epsilon\) assume almost the same values on all the cases, while the \(Maxout\) is the activation function mostly chosen. The GA selected two possibilities as distribution functions, namely the _Gaussian_ and the _Huber_ function. The end metric selected, on the other hand, presents more variations. This could suggest that we could perhaps fix some of the parameters, e.g., \(\epsilon\), in order to reduce the search space. Figure 3: Results obtained for each value of \(h\) and \(w\). As previously stated, in order to globally assess the performance of our proposal, we compared the results achieved by our methodology (NDL) with the results obtained by other strategies commonly used for time-series forecast. In particular, we considered Random Forest (RF), Artificial Neural Networks (NN), Evolutionary Decision Trees (EV), the Auto-Regressive Integrated Moving Average (ARIMA), an algorithm based on Gradient Boosting (GBM), three Deep Learning models (FFNN, Feed-Forward Neural Network; CNN, Convolutional Neural Network; LSTM, Long Short-Term Memory), decision tree algorithm (DT) and an ensemble strategy that was proposed in [14], which combined regression trees-based, artificial neural networks and random forests (ENSEMBLE). For ARIMA, we used the tool in Ref. [52] for determining the order of auto-regressive (AR) terms (\(p\)), the degree of differencing (\(d\)) and the order of moving-average (MA) terms (\(q\)). The values obtained are \(p=4\), \(d=1\) and \(q=3\). The value for the auto-regressive parameter and the degree of differencing confirm that the time-series is not stationary, as indicated in Section 3.1. The deep learning models were designed using \(H2O\) framework of R [49]. The difference between NDL and DL, is that in the latter case, the network is trained with stochastic gradient descend using back-propagation algorithm. In order to set the parameters for DL, we used a grid search approach. As a consequence, we used a hyperbolic tangent function as activation function, the number of hidden layer was set to 3 and the number of neurons to 30.