## 4 Experiments
### 24-h-Ahead Forecasting Analysis  
In this subsection, we describe the forecasting performance of the statistical and deep learning methods for a multi-step prediction task. In particular, we consider 24-step-ahead forecasting, with a step equal to one hour. We test the forecasting methods illustrated above with each time series of each cell in our dataset. As the ours is a multi-step forecasting process, we compute the forecasting error by summing up the errors committed at the intermediate time steps. In this way, we obtain three types of forecasting errors (MAPE, RMSE and MAXAE) for each cell. Given that our dataset is a grid of cells and we compute the three errors for each cell, there will be a distribution of values for each of the three errors containing as many elements as the number of cells in the dataset. We represent this distribution with a boxplot as in Figure 11, which shows the distribution of RMSE for each forecasting method. More precisely, each boxplot contains as many elements as the number of cells, where each element is the RMSE value obtained for the time series of that cell with the corresponding method. Something we can immediately notice from this plot is that statistical methods typically perform better, with ARIMA being the best performing approach. Sometimes, even the naive baseline model computing the seasonal average can outperform more sophisticated approaches such as CNN and LSTM, which typically require large datasets for training. The CNN-LSTM hybrid model is the best performing model among the machine learning methods, performing much better in comparison to statistical models, such as ETS and AR. The same conclusions can be drawn by looking at the MAPE performance measure, which is shown in Figure 12. However, something worth noticing in this graph is that for this kind of error, the performance of the CNN-LSTM model is quite similar to that of ARIMA. Figure 11: The distribution of RMSE across cells in a 24-h-ahead forecasting scenario. Figure 10: Two different types of cells with very different mobile activity patterns. The cell in (**a**) has a periodic, regular pattern, whereas the cell in (**b**) has an irregular pattern containing local and global trends with activity levels going up and down. When looking at MAXAE, represented in Figure 13, the gap between statistical and deep learning approaches is reduced, with the hybrid model CNN-LSTM performing better on average than ARIMA or any other statistical method. The MLP model also shows a better performance when compared to statistical methods such as ETS or AR. This behavior suggests that, in general, deep learning approaches are more robust to spikes and sudden changes in the time series. This point is important, because having an upper bound about the maximum error that we can make can give to the researcher the possibility to choose in which scenario to use a given forecasting method. Table 1 summarizes all the results contained in the boxplot graphics presented above. By computing the mean and the standard deviation of each error distribution, we give an even more synthetic overview of the forecasting performance of each method. In particular, ARIMA and CNN-LSTM hybrid model show a similar performance for both MAPE and RMSE erros while for the MAXAE error the CNN-LSTM shows to perform better than ARIMA. More in general, deep learning methods outperform statistical methods when it comes to containing the maximum error.