## 3 Methodology
### Hybrid Model  
The proposed hybrid arrangement combines ARIMA and ANN models in series. As given in Equation (5), the original time series is considered the sum of two components: one is linear and the other is nonlinear. The linear component is represented by a multi-period lagged ARIMA model (M\({}_{t}\)). This component captures the linear relationship in the data. The second component (N\({}_{t}\)) is built on the leftover error of the first model. Whatever cannot be explained by ARIMA is fed into the ANN. \[\mathbf{Y_{t}}=\mathbf{M_{t}}+\mathbf{N_{t}} \tag{5}\]  
Based on the assumption that, the data \(Y_{t}\) can be split into two components, representing linear (\(M_{t}\)) and nonlinear (\(N_{t}\)) portions, a hybrid forecasting approach is developed to exploit the unique characteristics of each component. Processes involved in hybrid modeling can be divided into six sub-tasks performed in sequential order: (1) The routine starts with the _basic model_; (2) the error output of the basic model is processed in the _classification_; (3) based on the classification, _the input lists are prepared for the second model_; (4) _the secondary model_ runs on each list separately; (5) the output of the second model is consolidated in _hybridization_; (6) in the final stage, projections are generated with the hybrid model. Figure 3 shows tasks and relationships graphically. Step 1. The Basic Model  
The modeling process starts with the division of the original time series into training and test series. Next, the basic model is developed using only the training data, which includes the first portion of the data set (i.e., the chronologically older). In terms of size, the training data are typically several times larger than the test portion. In selecting the split point, we followed the guidelines used in previous studies on the same data sets. It should be noted that the number of observations in the training data is further reduced because of the lag effect in the data. The formula for the basic model is given in Equation (6). \[Y_{t}=M_{t}+e_{t} \tag{6}\]where \(Y_{t}\) and \(M_{t}\) represent the actual and forecast values at point t. The offset between the two is the error \((e_{t})\), and it is used as a basis to construct a new time series for the second component. \[e_{t}=Y_{t}-M_{t} \tag{7}\]  
Step 2. Classification  
In the second stage, the error output of the first component (Equation (7)) is classified into three distinct groups. The classification is done according to an a priori selected threshold value. Over-shooters include non-negative errors that are greater than the threshold. Under-shooters are negative errors that are smaller than the negative value of the threshold limit. Anything that falls between these two limits is categorized as neutral. This type of categorization was first proposed by [PERSON] et al. (2018), where the threshold limits are defined as the desired level of error (DLE). The value of this parameter could be set by the user based on the analysis of the error coming from the basic model. In particular, the distribution of error coming from the basic model can provide insight regarding when or how the basic model over- or under-estimates. If there is no insight into the performance of the basic model, it could be set to a fixed value. For example, the DLE is set to 5% in experimental runs by [PERSON] et al. (2018). The error output of the basic model is categorized into three groups: A, B, and C, with the following definitions:  
* \(\{e_{i}|e_{i}>DLE\}\), Category A
* \(\{e_{i}|e_{i}<-DLE\}\), Category B
* \(\{|e_{i}|\leq DLE\}\), Category C  
Step 3. Input Lists for the Second Model  
Based on the categorization of the second step, three separate data lists are constructed from the original error output in Equation (7). To explain the process, let us assume that the output of the basic model is stored in a list, E, with n members, \(E=\{e_{1},e_{2},e_{3},\ldots,e_{n}\}\).