# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 3 The Proposed Method Artificial Neural Network with Convex Combination (ANN-CC)
We constructed a deep neural network using stock returns from the Stock Exchange of Thailand (SET), the major stock market in Thailand. We have selected three major stocks with high market capitalization at the beginning of the sample period. We collected SET50 (proxy of the market) and three company stocks, PTT, SCC, and CPALL, over the period from 4 January 2012 to 30 December 2019. The interval-valued data were constructed from a daily range of selected price indexes, i.e., the lowest and highest trading index values for the day were calculated to define the movement on the market for that day. Then, the construction of our interval-valued prediction for these three stocks would be made based on the models developed in Section 3. We note that all the interval price series were transformed to be interval returns following Equations (34) and (35). The descriptive statistics, namely the mean, standard deviation, minimum value, and maximum value of the variables for the full sample, are summarized in Table 3. We observe that the returns exhibited negative skewness for lower bound returns and positive for upper bound returns. In addition, the values of skewness on both sides were asymmetric, indicating that the gain and the loss in the Thai stock market were quite different. For illustration, a visual relationship between the stock market and each stock is shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **SET\_\_u** & **SET\_l** & **PTT\_\_u** & **PTT\_\_l** & **SCC\_\_u** & **SCC\_\_l** & **CPALL\_u** & **CPALL\_\_I** \\ \hline Mean & 0.012 & \(-\)0.011 & 0.019 & \(-\)0.018 & 0.018 & \(-\)0.017 & 0.021 & \(-\)0.019 \\ Median & 0.010 & \(-\)0.009 & 0.016 & \(-\)0.015 & 0.016 & \(-\)0.015 & 0.017 & \(-\)0.016 \\ Maximum & 0.097 & 0.025 & 0.136 & 0.033 & 0.113 & 0.027 & 0.185 & 0.032 \\ Minimum & \(-\)0.019 & \(-\)0.106 & \(-\)0.027 & \(-\)0.128 & \(-\)0.016 & \(-\)0.105 & \(-\)0.058 & \(-\)0.172 \\ Std. Dev. & 0.010 & 0.010 & 0.016 & 0.015 & 0.014 & 0.013 & 0.017 & 0.016 \\ Skewness & 1.894 & \(-\)1.812 & 1.576 & \(-\)1.564 & 1.653 & \(-\)1.475 & 2.038 & \(-\)2.280 \\ Kurtosis & 12.247 & 11.383 & 7.981 & 8.775 & 8.337 & 7.522 & 12.824 & 15.743 \\ Jarque—Bera & 7659.845 & 6398.846 & 2665.961 & 3308.472 & 3022.738 & 2235.950 & 8677.507 & 14,052.600 \\ MBF Jarque-Bera & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ Observations & 1841 & 1841 & 1841 & 1841 & 1841 & 1841 & 1841 & 1841 \\ Unit root test & \(-\)2.230 & \(-\)3.454 & \(-\)2.125 & \(-\)2.125 & \(-\)2.661 & \(-\)2.385 & \(-\)2.307 & \(-\)3.255 \\ MBF-unit root & 0.083 & 0.003 & 0.105 & 0.105 & 0.029 & 0.058 & 0.070 & 0.005 \\ \hline \end{tabular} Note: \(\_u\) and \(\_l\) are upper and lower bounds, respectively.

\end{table} 
Table 3: Data description. Moreover, a unit root test was also conducted to examine whether a time series variable was nonstationary and possessed a unit root. In this study, we used the minimum Bayes factor (MBF) as a tool for making a statistical test. The MBF has a significant advantage over the \(p\)-value because the likelihood of the observed data can be expressed under each hypothesis [33]. If 1 \(<\) MBF \(<\) 1/3, 1/3 \(<\) MBF \(<\) 1/10, 1/10 \(<\) MBF \(<\) 1/30, 1/30 \(<\) MBF \(<\) 1/100, 1/100 \(<\) MBF \(<\) 1/300 and MBF \(<\) 1/300, there is a chance that the MBF favors, respectively, the weak evidence, moderate evidence, substantial evidence, strong evidence, very strong evidence, and decisive evidence for the null hypothesis. According to the results, all data series were decisive stationary, as shown by the low MBF values [33]. ### Comparison Results

This section shows the results of the models representing artificial neural networks for interval-valued data of Thailand's stock market. In this empirical example, we consider the number of hidden neurons between one and three hidden neurons with four activation functions. Again, one input layer, one output layer, and one hidden layer were assumed.