# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 3 The Proposed Method Artificial Neural Network with Convex Combination (ANN-CC)
However, this method does not guarantee the mathematical coherence of predicted bounds. That is, the predicted lower bounds of intervals should be smaller than the predicted upper ones. Otherwise, the unnatural interval crossing problem will occur, which leads to an invalid interval prediction [27]. Furthermore, the forecasting performance can be impaired if there is not a clear dependency between the respective bounds of output and input [28; 29]. Instead of restricting the weight of interval-valued data \((\mathbf{w}=(\mathbf{w}_{1},\mathbf{w}_{2}))\) to be 0.5 (mid-point), in this study, we consider the convex combination method to get the reference point of the interval-valued data. Thus, in this estimation, the weights of neuron \(j\)\((\omega_{j})\) are dependent on the given weight \(\mathbf{w}\). Since there is no close form solution for this weight parameter of the interval data, we thus employ a grid search selection of \(\mathbf{w}\) that minimizes the sum of squared errors, denoted as \(loss(\mathbf{w})\). Then, we can rewrite our loss function in Equation (19) as  
\[loss(\mathbf{w})=\frac{1}{T}\underset{l=1}{\overset{T}{\sum}}\;(\gamma^{cc} -\hat{\gamma}^{cc}(\mathbf{w}))^{2}. \tag{20}\]  
This loss function is computed using two steps of estimation. Firstly, it is important to solve a nonlinear optimization to obtain \((\omega_{j})\) which depends on the candidate \(\mathbf{w_{i}}\). In the second step, the loss function is then minimized with respect to the candidate \(\mathbf{w_{i}}\). Then, we introduce another candidate \(\mathbf{w}\) to repeat the first step. After \(loss(w)\) is computed for all candidates \(\mathbf{w_{i}}\), the minimum value of \(loss(w)\) is preferred. Thus, the optimal \(\mathbf{w}=(\mathbf{w}_{1},\mathbf{w}_{2})\) is obtained by  
\[\widehat{\mathbf{w}}=\underset{\mathbf{w}}{\text{argmin}}=loss(\mathbf{w}). \tag{21}\]  
We note that ANN estimates over a grid search between 0 and 1. Finally, following the CC method in Section 2.3, we can predict the lower and upper bounds as follows. \[\hat{\gamma}^{l}=f\Bigg{(}\underset{j=1}{\overset{l}{\sum}}\;H_{j}^{cc,l}( \widehat{\mathbf{w}}_{1})\widehat{\omega_{j}}+\widehat{b}_{j}^{\circ}\Bigg{)} -\hat{\gamma}^{r}(\widehat{\mathbf{w}_{2}}), \tag{22}\]  
Figure 1: The network architecture of the interval-valued data artificial neural network with convex combination (ANN-CC) model (one hidden layer is assumed. \[\hat{Y}^{u}=f\left(\sum_{j=1}^{l}H_{j}^{cc,I}(\widehat{\mathbf{w}_{1}})\widehat{ \omega}_{j}^{o}+\widehat{b}_{j}^{o}\right)+\hat{Y}^{r}(1-\widehat{\mathbf{w}_{2 }}), \tag{23}\]  
where \(H_{j}^{cc,I}(\widehat{\mathbf{w}_{1}})=g\bigg{(}x^{cc}(\widehat{\mathbf{w}_{1}} )\overset{\prime-I}{\omega}_{j}+\widehat{b}_{j}^{-I}\bigg{)}\) is the \(j\)th hidden neuron's input \(X^{cc}(\widehat{\mathbf{w}_{1}})\) and the range prediction is computed by  
\[\hat{Y}^{r}=f\left(\sum_{j=1}^{l}H_{j}^{u,I}\widehat{\omega}_{j}^{o}+\widehat{ b}_{j}^{o}\right)-f\left(\sum_{j=1}^{l}H_{j}^{l,I}\widehat{\omega}_{j}^{o}+ \widehat{b}_{j}^{o}\right), \tag{24}\]  
where \(H_{j}^{l,I}=g\bigg{(}X^{l,r}\widehat{\omega}_{j}^{I}+\widehat{b}_{j}^{-I} \bigg{)}\) and \(H_{j}^{u,I}=g\bigg{(}X^{u,r}\widehat{\omega}_{j}^{I}+\widehat{b}_{j}^{-I} \bigg{)}\). By making this prediction, the predicted lower bounds of intervals should not cross over the corresponding upper bound. We also note that \(\widehat{\omega}_{j}^{I}\) and \(\widehat{\omega}_{j}^{o}\) have contained the information of the weight parameter of the interval data \(X\) and \(Y\). In our work, the ANN-CC methodology includes the grid search for the parameter \(\widehat{\mathbf{w}}=\left\{\widehat{\mathbf{w}}\,,\widehat{\mathbf{w}}_{2}\right\}\). Note that grid search was executed before estimating the weight parameters \(\widehat{\omega}_{j}^{o}\) and \(\widehat{\omega}_{j}^{o}\) in the ANN structure. A pseudo-code for the performed grid search and ANN-CC estimation is presented in Algorithm 1.