## 3 Model Construction
### Construction of Combined Prediction Models
#### 3.2.2. A Combined Residual Correction Prediction Model of Optimal Machine Learning  
Based on the IOWA Operator  
\(x_{t}\) represents the actual value observed at time \(t\), \(x_{it}\) denotes the forecasted value generated by the \(ith\) forecasting approach at time \(t\), \(i=1,2,\ldots,m\); \(t=1,2,\ldots,N\), and \(a_{it}\) is a metric that quantifies the accuracy of the \(ith\) forecasting technique at time \(t\).  
\[a_{it}=\begin{cases}1-|\left(x_{t}-x_{it}\right)/x_{t}|&,\left|\left(x_{t}-x_{ it}\right)/x_{t}\right|<1\\ 0&,\left|\left(x_{t}-x_{it}\right)/x_{t}\right|\geq 1\end{cases} \tag{5}\]  
The prediction accuracy, denoted as \(a_{it}\), is derived from the predicted value \(x_{it}\). As a result, the accuracy of \(m\) individual forecasting methods at time \(t\) and their corresponding forecasted values form \(m\) two-dimensional arrays: \(\langle a_{1t},x_{1t}\rangle\), \(\langle a_{2t},x_{2t}\rangle\), \(\ldots,\langle a_{mt},x_{mt}\rangle\).  
Let the true value of a time series at time \(t\) be \(Q_{act}\), let the predicted value of the GRNN model for a time series at time \(t\) be \(Q_{GRNN}\), and let the residual value at time \(t\) be \(Q_{e}\). Then, the residual \(Q_{e}\) is as follows.  
\[Q_{e}=Q_{act}-Q_{GRNN} \tag{6}\]  
Let the prediction values of the residuals \(Q_{e}\) at time \(t\) for the two SSA-SVR and SSA-ELM models be \(\Delta E_{1,t}\) and \(\Delta E_{2,t}\), respectively. The prediction accuracies of the two models at time \(t\) are calculated according to Equation (5) as \(\delta E_{1,t}\) and \(\delta E_{2,t}\), respectively. Considering the sequence of prediction accuracies \(\delta E_{i,t}\) as the induced values of the residual prediction \(\Delta E_{i,t}\), the accuracy of the SSA-SVR and SSA-ELM models' predictions at time \(t\), along with their respective predicted values, can be arranged into a two-dimensional array: \(<\delta E_{1,t}\), \(\Delta E_{1,t}>,<\delta E_{2,t}\), \(\Delta E_{2,t}>\). According to Definition 2 for the IOWA operator principle, a combined residual correction prediction model of optimal machine learning based on the IOWA operator can be established.  
\[\begin{array}{ll}\Delta E_{c,t}&=IOWA(<\delta E_{1,t},\Delta E_{1,t}>,< \delta E_{2,t},\Delta E_{2,t}>)\\ &=\sum\limits_{i=1}^{2}\omega_{i}\Delta E_{v-index(it)}\end{array}, \tag{7}\]  
where \(\Delta E_{c,t}\) is the combined prediction residual at time \(t\). The prediction accuracies \(\delta E_{1,t}\) and \(\delta E_{2,t}\) of the two forecasting methods at time \(t\) are ordered from largest to smallest, and \(v-index(it)\) is the subscript of the \(ith\)-largest prediction accuracy. Moreover, \(W=(\omega_{1},\omega_{2})^{T}\) is the weighted vector of the model at time \(t\), which satisfies \(\omega_{1}+\omega_{2}=1\) and \(\omega_{1},\omega_{2}\geq 0\).  
An optimal machine learning-based combined residual correction prediction model is developed, aiming to minimize the sum of squared errors and determine the optimal weight vector. This can be formulated as follows:  
\[\begin{array}{l}minS(E)=\sum\limits_{t=1}^{n}\left(Q_{e}-\sum\limits_{i=1}^ {2}\omega_{i}\Delta E_{v-index(it)}\right)^{2}\\ \sum\limits_{i=1}^{2}\omega_{i}=1,\ \omega_{i}\geq 0,\ i=1,2\end{array} \tag{8}\]  
The elements of the weight vector \(\omega_{i}(i=1,2)\) are determined through nonlinear programming to minimize the sum of the squares of the combined residual prediction errors.  
During the actual forecasting process, modeling the residual \(Q_{e}\) helps refine the final outputs of the GRNN prediction model, thereby improving the accuracy of the forecasts. Based on Equation (6), the following combined residual correction model can be established:  
\[Q_{GRNN,fin}=Q_{GRNN}+\Delta E_{c,t}=Q_{GRNN}+\sum\limits_{i=1}^{2}\omega_{i} \Delta E_{v-index(it)}, \tag{9}\]  
where \(Q_{GRNN,fin}\) is the final prediction at time \(t\) corrected by the combined residuals.