## 3 Methodology
### Hybrid Model
Three separate lists, A, B, and C, are generated from list E. While lists A and B represent underestimated and overestimated values of E, list C holds neutral error values. The content of E is directly copied to A, B, and C to initialize the lists. Then, with the guidelines given in Equation (8), adjustments are made to each list. In list A, which represents underestimated values, all category A values are kept as they are, but everything else is set to zero. Similarly, list B, representing overestimated errors (i.e., category B), retains its original values but overwrites everything else with zero. For list C, all elements with over- and underestimated values are set to zero. Figure 3: Proposed hybrid structure. \[\text{Underestimation list: A}=\left\{ \begin{array}{l}\text{if }e_{i}\in \text{category A}\Rightarrow e_{i}=e_{i}\\ \text{else }e_{i}=0\end{array} \right\}\] \[\text{Overestimation list: B}=\left\{ \begin{array}{l}\text{if }e_{i}\in \text{category B}\Rightarrow e_{i}=e_{i}\\ \text{else }e_{i}=0\end{array} \right\} \tag{8}\] \[\text{Neutral list: C}=\left\{ \begin{array}{l}\text{if }e_{i}\
otin\text{category A}\text{ AND }e_{i}\
otin\text{category B}\\ \Rightarrow e_{i}=e_{i}\\ \text{else }e_{i}=0\end{array} \right\}\]  
Step 4. The Second Model  
The second model performs a complementary function. It uses leftover error values from the basic model to extract additional explanatory information. As explained in the previous step, the error output is categorized into three separate lists. The lists A and B represent under- and overestimated values, and they are the inputs for the second model. List C contains error values that remain below the threshold (i.e., DLE) and are excluded from the run. The second model includes an artificial neural network (ANN) to identify nonlinear patterns and any high-order linear relationships that the basic model missed. In the original approach proposed by [PERSON] [26], the primary and secondary models are assigned specific tasks. While the primary model deals with the linear component of the data, the leftover error is assumed to be the nonlinear component. This type of classification of the domain may not be applicable in every situation, and the residual error of the primary model cannot be presupposed as a nonlinear component. It is likely that the leftover error values are a combination of linear and nonlinear error values. The ANN model is run separately on A and B lists to generate two sets of forecasts, representing the under- and overestimation portions of the model. As stated in Equations (9) and (10), the ANN uses \(n\) period-lagged values from \(t\) to \(t-n\) to generate the forecast for \(t+1\). \[N_{A(t+1)}=e_{A(t+1)}=f_{A}\Big{(}e_{At},e_{A(t-1)},e_{A(t-2)},\dots,e_{A(t-n)} \Big{)}+\varepsilon_{A(t+1)} \tag{9}\]  
\[N_{B(t+1)}=e_{B(t+1)}=f_{B}\Big{(}e_{Bt},e_{B(t-1)},e_{B(t-2)},\dots,e_{B(t-n)} \Big{)}+\varepsilon_{B(t+1)} \tag{10}\]  
where \(f_{A}\) and \(f_{B}\) represent functions used by ANN. The terms \(\varepsilon_{A(t+1)}\) and \(\varepsilon_{B(t+1)}\) are random errors. Step 5. Hybridization  
Hybridization involves integrating basic and secondary models in a schema to produce a single (i.e., hybrid) forecast. In the proposed schema, the hybrid forecast for \(t+1\) is the sum of outputs generated by \(M_{t}\) and \(N_{t}\). \[\hat{Y}_{A(t+1)}=M_{(t+1)}+N_{A(t+1)} \tag{11}\]  
\[\hat{Y}_{B(t+1)}=M_{(t+1)}+N_{B(t+1)} \tag{12}\]  
Forecasts given in (11) and (12) can be considered a modified version of the basic model forecast. The \(\hat{Y}_{A(t+1)}\) in (11) is the corrected version when the basic model underestimates the true value. Underestimation produces positive errors in the basic model. When positive values are fed into the secondary model, the secondary model, in turn, is likely to produce positive-valued forecasts. This could be considered compensation for the underestimation of the basic model. A reverse mechanism works in (12). When the basic model overestimates, errors are likely to be negative.