# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 3 The Proposed Method Artificial Neural Network with Convex Combination (ANN-CC)
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\begin{tabular}{c c c c c c c c} \hline \hline CPALL & w1 & w2 & out & in & out & in & out & in \\ \hline tanh & 0.500000 & 0.5000000 & 0.014358 & 0.014825 & 0.000393 & 0.000414 & 0.019813 & 0.020359 \\ sigmoid & 0.500000 & 0.500000 & 0.020873 & 0.020682 & 0.000718 & 0.000674 & 0.026790 & 0.025967 \\ linear & 0.500000 & 0.500000 & 0.021531 & 0.020515 & 0.000791 & 0.000655 & 0.028131 & 0.025998 \\ exp & 0.500000 & 0.500000 & **0.014286** & **0.014245** & **0.0000413** & **0.000385** & **0.020323** & **0.019629** \\ \hline \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 each case.

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Table 5: _Cont._To make a clear explanation, the best prediction models for the three stocks are summarized in Table 7. We found that the exponential activation function was mostly selected (except for ANN-Center in the PTT case). This indicates a nonlinear pattern of these three stock companies. It is evident that the performance measures of the best specification of ANN-CC models were lower than those of the ANN-Center and RANN-LU for all stock indices, meaning that the ANN-CC model was superior to the conventional models in both in- and out-of-sample forecast performances. Additionally, in this section, we also compared the performance of our ANN-CC against the methods proposed in the literature, namely, center and center-range methods of [PERSON] and [PERSON] (2017), PM method of [PERSON] et al. (2019), and IT method of [PERSON] et al. (2012). The result clearly confirmed the higher prediction performance of our proposed ANN-Center. A robustness check was also conducted to confirm the performance of our proposed model. As in practice, the simple loss functions, such as MAE, MSE, and RMSE, may not yield a piece of sufficient information to identify a single forecasting model as \"best\". Therefore, in this study, another accuracy measure, the model confidence set (MCS), was used to evaluate forecasting performance. [PERSON] et al. (2019) introduced the MCS test to validate the forecasting performance of forecasting models. In this study, our MCS tests were based on two loss functions, MAE, MSE, and RMSE. We note that when the \(p\)-value was higher, it meant that they were more likely to reject the null hypothesis of equal predictive ability. In other words, the greater the \(p\)-value, the better the model. For more details of the MCS test, we referred to [PERSON] et al. (2019). The statistical performance results for all competing models are provided in the bracket shown in Table 7. According to the MCS test results, the ANN-CC model clearly outperformed other competing models for both in- and out-of-sample forecasts. The \(p\)-values of the ANN-CC model were equal to one, but the other three models were less than the 0.10 threshold. It means that ANN-Center, RNN-LU, and linear regression forecasting models were removed in the MCS inspection process, and thus the ANN-CC was the only survivor model. ### Hong Kong Air Quality Monitoring Dataset

In the second dataset, we considered the Hong Kong air quality monitoring dataset as another example. This dataset is suggested in [PERSON] et al. (2021) and can be retrieved from [[http://www.epd.gov.hk](http://www.epd.gov.hk)]([http://www.epd.gov.hk](http://www.epd.gov.hk)) (accessed on 20 February 2021). They provide hourly air quality

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