# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 3 The Proposed Method Artificial Neural Network with Convex Combination (ANN-CC)
 Note: “in” and “out” denote in-sample and out-of-sample forecasts, respectively. Bold number indicates the lowest MAE, MSE, and RMSE in all cases. The number in the bracket indicates the \(p\)-value of MCS test results by using bootstrap simulation 1 data of 16 monitoring stations in Hong Kong. In this study, we also considered the data from Central/Western Station and downloaded the hourly data ranging from 1 January 2020, to 31 December 2020. Then, we aggregated the hourly data to the minimum and maximum form according to each day's record. There were seven air quality indicators in the database. However, we considered some of them and selected the respirable suspended particulates (RSP) as the interval-valued response variable and selected dinitrogen tetroxide (NO\({}_{2}\)) and sulfur dioxide (SO\({}_{2}\)) as the interval-valued explanatory variables. Again, each dataset was split into 80% for training and 20% for testing. Thus, both in-sample and out-of-sample forecasts were conducted in this comparison. The performance evaluation of the interval-valued data forecasting models ANN-CC under CAPM was accomplished through MAE, MSE, and RMSE. As shown in Table 8, similar to the first example, our proposed ANN-CC model was the best in terms of MAE, MSE, RMSE, and MCS's _p_-value. To illustrate the performance of our ANN-CC models, we show the out-of-sample forecasting result of our model on the Thai stocks and RSP in Figure 5. For clarity, only 10% of the out-of-sample forecasting result are shown. In this figure, the red vertical line segment represents a predicted interval-valued data, while the gray vertical line segment represents a predicted interval-valued data; the extremes correspond to the minimum and maximum interval values. The comparison between actual values and predicted values indicates the quality of modeling and the prediction task. The results show that the predicted values were very close to the actual values, indicating the goodness of fit of our model. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & **Number** & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**MSE**} & \multicolumn{2}{c}{**RMSE**} \\ \hline
**PTT** & **Activation** & **Hidden Neuron** & **out** & **in** & **out** & **in** & **out** & **in** \\ \hline ANN-CC & sigmoid & 2 & **2.6374** & **2.7092** & **7.3232** & **7.4085** & 2.7078 & 2.7225 \\  & & & **(1.0000)** & **(1.0000)** & **(1.0000)** & **(1.0000)** & **(1.0000)** & **(1.0000)** \\ ANN-Center & sigmoid & 2 & 2.9833 & 2.8363 & 8.1092 & 8.3233 & 2.8483 & 2.8854 \\  & & & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) \\ RANN-LU & sigmoid & 3 & 2.7762 & 2.7423 & 7.6403 & 7.5409 & 2.7656 & 2.74613 \\  & & & (0.0000) & (0.0000) & (0.0000) & (0.0510) & (0.0000) & (0.0000) \\ IF & & & 2.8532 & 2.7821 & 8.5110 & 8.3368 & 2.9161 & 2.8866 \\  & & & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) \\ PM & & & 2.7011 & 2.6893 & 7.5215 & 7.4212 & 2.7427 & 2.7240 \\  & & & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) \\ Center & & & 3.0029 & 2.9861 & 8.7334 & 8.7433 & 2.9543 & 2.9571 \\  & & & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) \\ Center-range & & 2.8763 & 2.7823 & 8.5532 & 8.3499 & 2.9239 & 2.8883 \\  & & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) \\ \hline \end{tabular} Note: “in” and “out” denote in-sample and out-of-sample forecasts, respectively. Bold number indicates the lowest MAE, MSE, and RMSE in all cases. The number in the bracket indicates the _p_-value of MCS test results by using bootstrap simulation at 1000 times.

\end{table} 
Table 8: Summary of the forecasting performance and MCS test of the second dataset. Figure 5: The out of sample interval forecasts by ANN-CC (red) vs. the actual data (gray) for example data. ### Discussion

Although the traditional models like ANN-Center and RANN-LU provide acceptable prediction results in all stocks and RSP, they still face some limitations.