# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 3 The Proposed Method Artificial Neural Network with Convex Combination (ANN-CC)
The ANN-Center relies on the midpoint of the data, which may not reflect real behavior of the data during the day. Likewise, the RANN-LU predicts future observation based on either lower or upper bounds, separately, clearly leaving out the information within the interval-value data. Thus, our proposed ANN-CC model is used to solve such a challenging task and consider the whole information during the interval. According to the above experimental results, we can draw the following conclusions:

1. Regarding the prediction performance, our ANN-CC was superior to other traditional models in all datasets. 2. We note that the symmetric weight within the interval data should not be \(\mathrm{w}=0.5\). We found that the prediction result was sensitive to the weight \(\mathrm{w}\); thus, the weight should not be a fixed parameter. 3. Our model outperformed the ANN-LU and RANN-LU models in situations in which the interval series had linear and nonlinear behavior. 4. We also studied the sensitivity of each activation function and found that the quality of the prediction model was not very sensitive in many cases. However, careful assessment needs to be made when choosing the activation function. 5. Even though the exponential activation function seemed to be the best fit one in the ANN architecture, it was noticed that other activation functions performed well in some cases. Although the exponential activation function performed very well in the selected three stocks, it may not be reliable in other stocks or under other ANN structures. 6. However, we can draw an important conclusion that our ANN-CC is a promising model for interval-valued data forecasting. The ANN-CC method has the advantages of not assuming constraints for the weight nor fixing reference points. The ANN-CC model is adaptive and adjusts itself for the best fit. The fitted model allows the behavior analysis of response lower and upper bounds based on the variation of the reference points of input and output intervals. ## 6 Conclusions

This paper proposed an artificial neural network with a convex combination (ANN-CC) method for interval-valued data prediction. Simulation and experimental results on real data showed that the proposed ANN-CC model is a useful tool in interval-valued prediction tasks, especially for complicated nonlinear datasets. Moreover, the proposed ANN-CC model fills the research gap by considering interval-valued data using the convex combination method. Our proposed model was examined by comparing its performance with conventional ANN with the center method (ANN-Center) and regularized ANN-LU (RANN-LU), linear regression with the center method, center-range, PM, and IT methods. We considered three stock returns in the Thai stock market and Hong Kong air quality monitoring dataset in our empirical comparison. According to the in-sample and out-of-sample forecasts, we found that the performances of various ANN-CC specifications were not much different. However, we observed that the tanh activation function performed well in the in-sample and out-sample forecast, while the linear activation function performed relatively well in the in-sample forecast. In addition, we could also confirm the higher performance of our ANN-CC compared to that of the ANN-Center and RANN-LU. Experimental results on two real datasets also confirmed that the proposed ANN-CC model outperformed the conventional models. In this study, a neural network with one hidden layer was assumed. However, real-world data is quite complex, and one hidden layer may not be enough to learn the data. Thus, a deep neural network (more than two hidden layers) should be more promising in the approximation performance. Another meaningful future work is to employ other deep learning methods, such as recurrent neural networks and long short-term memory. These models handle incoming data in time order and learn about the previous time to predict future value.