# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 3 The Proposed Method Artificial Neural Network with Convex Combination (ANN-CC)
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**ANN-CC**} & \multicolumn{3}{c}{**ANN-Center**} & \multicolumn{3}{c}{**RANN-LU**} \\ \hline _Scenario 1_ & MAE & MSE & RMSE & MAE & MSE & RMSE & MAE & MSE & RMSE \\ \hline tanh & 2.4011 & 10.8590 & 3.2951 & **2.3154** & **9.4115** & **3.0681** & 3.1244 & 14.2251 & 3.7729 \\  & (0.1548) & (1.3251) & (1.2434) & (0.1215) & (1.1154) & (1.0859) & (1.1245) & (2.3414) & (1.5521) \\ sigmoid & 2.5870 & 10.9894 & 3.3154 & **2.4023** & **9.5558** & **3.0917** & 3.1148 & 12.5441 & 3.5423 \\  & (0.1211) & (1.3511) & (1.2433) & (0.1011) & (1.1584) & (1.0756) & (1.3584) & (3.6974) & (1.2532) \\ linear & 2.7244 & 12.0524 & 3.4723 & **2.4488** & **10.3554** & **3.2184** & 3.5415 & 14.3554 & 3.7890 \\  & (0.1513) & (1.1125) & (1.3234) & (0.1254) & (1.0015) & (0.028) & (1.2121) & (1.5487) & (1.1112) \\ exp & 2.5980 & 11.3486 & 3.3693 & **2.4223** & **10.0215** & **3.1661** & 3.1057 & 12.5015 & 3.5361 \\  & (0.1148) & (1.981) & (1.2113) & (0.1011) & (1.1057) & (1.0723) & (1.2554) & (3.4548) & (0.9723) \\ \hline _Scenario 2_ & MAE & MSE & RMSE & MAE & MSE & RMSE & MAE & MSE & RMSE \\ \hline tanh & **2.1350** & **7.0789** & **2.6610** & 3.1332 & 9.0173 & 3.0021 & 3.5445 & 15.1541 & 3.8935 \\  & (0.1254) & (1.3258) & (1.1123) & (2.1254) & (2.4848) & (1.4434) & (1.3145) & (2.6879) & (1.2329) \\ sigmoid & **2.3328** & **8.6030** & **2.9337** & 4.2214 & 10.6030 & 3.2565 & 2.5847 & 9.6984 & 3.1154 \\  & (0.1158) & (1.2217) & (1.0283) & (3.3141) & (1.6278) & (1.3233) & (1.2597) & (3.3354) & (1.2810) \\ linear & **2.7825** & **8.9356** & **2.9894** & 4.3112 & 10.9356 & 3.3078 & 4.1125 & 14.0778 & 3.7533 \\  & (0.1698) & (1.1369) & (1.0022) & (2.6545) & (2.3679) & (1.2232) & (1.3541) & (1.1548) & (1.0012) \\  & **2.4625** & **8.5072** & **2.9173** & 3.5845 & 10.4072 & 3.2269 & 3.4797 & 10.3155 & 3.2129 \\  & (0.1354) & (1.2589) & (0.9233) & (2.3651) & (2.2589) & (1.1129) & (1.5479) & (1.1554) & (0.9928) \\ \hline _Scenario 3_ & MAE & MSE & RMSE & MAE & MSE & RMSE & MAE & MSE & RMSE \\ \hline tanh & **2.1049** & **6.9441** & **2.6352** & 3.4488 & 9.9410 & 3.1539 & 2.4141 & 8.5454 & 2.9245 \\  & (0.1112) & (1.5159) & (0.6333) & (2.1254) & (4.3549) & (1.6727) & (1.0413) & (2.5444) & (1.5529) \\ sigmoid & **2.1249** & **7.0088** & **2.6468** & 3.9784 & 15.0113 & 3.8730 & 2.5454 & 10.5544 & 3.2456 \\  & (0.1874) & (1.6511) & (0.7843) & (2.3743) & (5.4035) & (1.2270) & (1.1115) & (2.1125) & (2.1323) \\ linear & **2.8524** & **12.9612** & **3.6013** & 3.8411 & 14.9023 & 3.8607 & 2.9445 & 13.1154 & 3.6210 \\  & (0.1369) & (1.3594) & (1.0091) & (2.6588) & (5.8941) & (1.3410) & (0.8797) & (2.1112) & (1.2221) \\ exp & **2.4489** & **10.4617** & **3.2344** & 4.8778 & 16.4107 & 4.0511 & 3.0124 & 11.3547 & 3.2683 \\  & (0.1364) & (1.5114) & (0.8833) & (4.2643) & (5.4113) & (2.0013) & (1.0694) & (1.9967) & (1.1009) \\ \hline \hline \end{tabular} Note: The parenthesis denotes the standard deviation. The bold number presents the lowest values of mean absolute error (MAE), mean squared errors (MSE), and root mean squared errors (RMSE).

\end{table} 
Table 1: Experimental results for the linear case. _Scenario 3_: Deviate from the center to the upper of the interval: w\({}_{1}=0.8\), w\({}_{2}=0.8\)

\[\left[(0.8)Y^{l}+(1-0.8)Y^{u}\right]=1+3e^{\left[(0.8)X^{l}+(1-0.8)X^{u}\right]^{ 2}}+\varepsilon. \tag{33}\]

For each scenario, we performed 100 replications. This data was more complicated than that in the linear structure case, as it showed a nonlinear relationship between the dependent and independent variables, as shown in Figure 3. The results are summarized in Table 2.

Table 2 presents the simulation results based on the nonlinear case. Similar results were obtained, as the proposed ANN-CC showed higher performance in _Scenarios 2_ and 3. We observed that ANN-Center still performed poorly in _Scenarios 2_ and 3. The reason is simple.