## 5 Related Work  
Historically, the literature on time series forecasting has been dominated by linear forecasting methods such as ARIMA and its generalization for multivariate series VAR (or Vector Autoregression). The main reason for this fact is that those methods favor interpretability and are effective in many forecasting problems. Probably the most popular of these methods, ARIMA, has been used in a myriad of time series forecasting studies. For example, in [16] ARIMA has been used for performing forecast on international tourism time series data. The authors show that when finding a best-fitting ARIMA model, the forecasting error (RMSE) can be very low even for out-of-sample forecasts. The ARIMA model is also applied in [17] for forecasting short-term electricity demand and thus network load by considering weather-network load relationship. In their paper, the authors demonstrate the effectiveness of the method, by comparing the results of the transfer function model and the univariate ARIMA model with conventional linear regression models. Another interesting work applies a combination of ARIMA with Fuzzy Expected Intervals methodology for fisheries management [18]. Finally the study in [19] applied ARIMA models to remote sensing data in order to forecast periods of drought in the Guanzhong Plain, China. According to authors, ARIMA models developed during this work, can be successfully used for the drought forecasting in the interested area. Although successful in different scenarios, statistical methods such as the ones mentioned above depend too much on linear relationships for making predictions and this dependence may be a disadvantage. However, where statistical methods show limitations, deep learning approaches can be a valid alternative. This category of forecasting methods is relatively new but is quickly gaining popularity due to their ability to model complex nonlinear relationships and automatically learn arbitrary mappings from inputs to outputs. This wave of novel methods based on deep neural networks has sparked the debate inside the community of researchers about which category of methods performs best. The research in [20] was triggered by such debate. Although applying deep neural networks for time series forecasting is a relatively new trend, there are already thousands of works in the literature. Very recent work such as the one in [21] proposed an end-to-end deep learning-based  
Figure 21: Each boxplot graphics indicates the distribution of MAXAE error across cells for a low and high variability scenario respectively in green and blue color. The graphics shows how deep learning methods can outperform statistical methods in a high variance scenario. model to generate population flow from aggregated historical population variation data. The study in [22] shows how an ensemble of deep belief networks (DBN) can be successfully used for regression problems and time series forecasting. The study in [23] reports the results of the NN3 forecasting competition, highlighting the ability of neural networks to handle complex data, including short and seasonal time series. Finally, the work in [24] reviews the use of deep learning techniques for time-series analysis and forecasting. However, there is not only competition between statistical methods and deep learning ones but also a quest to integrate the two categories. For example, there have been attempts to combine neural networks with statistical methods such as ARIMA [25; 26; 27]. Finally, a third way to reason about time series forecasting is considering probabilistic models. Such models can define relationships between variables and thus they can be used to calculate probabilities on what will be the next observation. A probabilistic model that preserves known conditional dependence between variables are Bayesian networks, for which there exists a wide variety of works in the field of time series forecasting [28; 29; 30; 31; 32].