# 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 & 3.6341 & 14.8140 & 3.8491 & **3.1258** & **9.8741** & **3.1438** & 3.9874 & 14.9874 & 3.8715 \\  & (0.3015) & (2.3840) & (1.0023) & (0.2474) & (2.0126) & (0.9343) & (1.4126) & (3.2158) & (0.9323) \\ sigmoid & 3.4558 & 10.9894 & 3.3155 & **3.2154** & **9.9741** & **3.1582** & 4.9845 & 15.3898 & 3.9245 \\  & (0.3114) & (5.0114) & (0.9833) & (0.2099) & (2.3654) & (0.8834) & (1.9136) & (5.1145) & (0.9823) \\ linear & 5.3155 & 17.1148 & 4.1377 & **5.2145** & **16.4213** & **4.0523** & 5.4136 & 17.8854 & 4.2295 \\  & (1.4259) & (5.6584) & (1.0824) & (1.3978) & (3.3665) & (1.1112) & (2.0113) & (5.7897) & (1.2334) \\  & 4.6211 & 16.3123 & 4.0391 & **3.1158** & **14.1314** & **3.7599** & 4.8654 & 17.0198 & 4.1256 \\  & (0.8797) & (2.8557) & (1.0067) & (0.9788) & (2.6547) & (0.9823) & (1.3541) & (3.9844) & (1.2220) \\ \hline _Scenario 2_ & MAE & MSE & RMSE & MAE & MSE & RMSE & MAE & MSE & RMSE \\ \hline tanh & **3.2250** & **9.1588** & **3.0372** & 3.9788 & 10.5448 & 3.2475 & 3.7888 & 9.8368 & \\  & (0.8453) & (3.6941) & (0.8832) & (2.5481) & (4.6641) & (1.1234) & (1.1255) & (3.6879) & \\ sigmoid & **4.0581** & **13.5154** & **3.6765** & 5.3698 & 20.3688 & 4.5139 & 4.3781 & 16.1556 & 4.0197 \\  & (1.3511) & (3.6444) & (0.8734) & (2.4145) & (4.3658) & (2.4449) & (1.8746) & (5.1034) & \\ linear & **4.8744** & **17.9981** & **4.2428** & 6.1142 & 27.6548 & 5.2597 & 5.6684 & 22.8314 & 4.7783 \\  & (2.5584) & (4.3398) & (0.9734) & (3.9451) & (10.4891) & (2.1240) & (2.9876) & (6.3598) & 4.7783 \\  & **4.1158** & **15.4458** & **3.9311** & 5.4101 & 20.0123 & 4.4744 & 4.2666 & 14.9155 & \\  & (0.7446) & (3.3658) & (0.9872) & (2.3155) & (4.4115) & (2.1098) & (1.3659) & (1.4231) & 3.8628 \\ \hline _Scenario 3_ & MAE & MSE & RMSE & MAE & MSE & RMSE & MAE & MSE & RMSE \\ \hline tanh & **3.4589** & **9.3158** & **3.0544** & 5.8115 & 21.1930 & 4.6033 & 3.5012 & 9.4125 & 3.0681 \\  & (0.7894) & (4.1158) & (0.9239) & (3.0125) & (10.3661) & (1.2323) & (1.2458) & (4.2320) & (1.9383) \\ sigmoid & **4.1585** & **14.3651** & **3.7901** & 5.9884 & 20.0113 & 4.4730 & 4.3685 & 14.8554 & 3.8544 \\  & (1.3841) & (3.9887) & (1.1112) & (2.3688) & (10.4035) & (1.2223) & (1.5645) & (4.3598) & (1.2234) \\ linear & **4.8664** & **16.6557** & **4.0814** & 5.9424 & 25.1253 & 5.0131 & 4.9785 & 16.9974 & 4.1229 \\  & (2.1355) & (6.0123) & (1.0389) & (2.6871) & (10.3211) & (1.4980) & (1.8994) & (5.3145) & (1.4409) \\ exp & **4.2556** & **14.4456** & **3.8009** & 6.6698 & 31.4107 & 5.6044 & 4.8664 & 15.5024 & 3.9377 \\  & (1.2154) & (4.1106) & (0.9227) & (5.1155) & (12.9987) & (1.3409) & (2.0115) & (2.1258) & (1.2284) \\ \hline \hline \end{tabular} Note: One hidden layer node is assumed for ANN in this simulation study. The parenthesis denotes the standard deviation. The bold number presents the lowest values of MAE, MSE, and RMSE.

\end{table} 
Table 2: Experimental results for the nonlinear case. defensive stock (low risk) of excess stock return. In this study, we preserve the interval format of the stock \(i\) as

\[\left[r_{it}^{l},r_{it}^{u}\right]=\left[\frac{P_{it}^{l}-P_{t-1}^{A}}{P_{it-1}^{A }},\frac{P_{it}^{u}-P_{it-1}^{A}}{P_{it-1}^{A}}\right], \tag{35}\]

where \(P_{it}^{u}\), \(P_{it}^{l}\) and \(P_{it}^{A}\) are maximum, minimum, and average prices of the individual stock \(i\) at time \(t\), respectively. Note that the risk-free rate \(r_{ft}^{l}\) and \(r_{ft}^{u}\), in this empirical study, is assumed to be zero. Again, we preserve the interval format of the market return as

\[\left[r_{mt}^{l},r_{mt}^{u}\right]=\left[\frac{P_{mt}^{l}-P_{mt-1}^{A}}{P_{mt- 1}^{A}},\frac{P_{m,t}^{u}-P_{mt-1}^{A}}{P_{mt-1}^{A}}\right], \tag{36}\]

where \(P_{mt}^{u}\), \(P_{mt}^{l}\) and \(P_{mt}^{A}\) are maximum, minimum, and average prices of the interval of the market at time \(t\), respectively.