## 4 Hybrid Models  
Attempting to combine the best modeling features of both classical statistical algorithms and machine learning models, a large part of the time series forecasting scientific literature is concerned with the development of combinations of forecasting models. We will initially refer to the motivations behind the development of hybrid models and, indicatively, emphasize some applications from various fields of research while focusing on the characteristics of the given problem and how these are modeled by the trained prediction models. Hybrid time series forecasting models are developed in the scientific literature based on three main factors:  
First, because of the practical difficulty in determining whether the time series under consideration has been produced by linear or non-linear underlying processes, as well as the difficulty in choosing one forecasting method over the others for a particular task and forecasting environment. The usual practice to deal with this particular problem is essentially to develop, train (in the case of machine learning algorithms), and test more than one predictive model, while factors such as sampling uncertainty and the dispersion of the sampling process make it difficult to generalize the chosen model. Consequently, combining several algorithms to create a complex prediction model can facilitate the selection process. The second reason for developing hybrid models is the fact that time series produced by real processes, in the majority of cases, do not have a purely linear or non-linear profile but contain a combination of linear and non-linear patterns. In these cases, single statistical or neural models are not sufficient for time series modeling and forecasting since the simple version of the ARIMA model cannot deal with non-linear relationships, while the neural network model alone is not able to handle linear and non-linear patterns equally well. Therefore, by combining ARIMA models with machine learning models, the complex autocorrelation structures in the data can be modeled more accurately. The third factor to consider is the fact that in the scientific literature on time series forecasting, it is almost universally accepted that there is no one forecasting method that is better than all others for every forecasting situation. This is largely due to the fact that a real problem is often complex in nature, and any single model may not be able to capture the different patterns equally well. Therefore, the combination of different models can increase the probability of detecting different patterns in the data and improve the prediction performance. In addition, the combinatorial model is more robust to a possible structural change in the data [4]. Regarding the combination of different forecasting algorithms in building an efficient hybrid model, various workflows are proposed, according to the forecasting problem at hand as well as its scientific approach. In the classic work by [PERSON] et al. [4], a hybrid model combining ARIMA and neural networks is proposed, aiming to exploit the capabilities of each model in terms of linear and non-linear modeling, respectively. The proposed methodology consists of two steps: In the first step, the ARIMA model is used to analyze the linear part of the problem, while in the second step, a neural network is developed to model the residuals of the ARIMA model. Since the classical ARIMA model cannot capture the non-linear structure of the data, the residuals of the linear model will contain information about the non-linearity. The outputs of the neural network can be used as predictors for the error terms of the ARIMA model. The results of this research showed that the hybrid model improves the prediction accuracy of both individual models. The proposed hybrid approach is presentedgraphically in Figure 6. A similar approach is used by [PERSON] et al. [57], [PERSON] et al. [58], and [PERSON] et al. [5] in different research fields and with a different selection of machine learning algorithms, combined with the ARIMA model.