# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 1 Introduction
[23] demonstrated the accuracy of ANN in predicting Shanghai Stock Exchange (SHSE) movement by comparing the neural networks and linear models under the capital asset pricing model (CAPM) and Fama and French's three-factor contexts, and they found that neural networks outperformed the linear models. As we mentioned above, this paper comes within the framework of interval-valued data forecasting by ANN. To the best of our knowledge, work-related to interval-valued data forecasting by neural networks is somewhat limited. Some attempts along this line include [PERSON] et al. [24], [PERSON] et al. [25], [PERSON] [PERSON] [26], and [PERSON] et al. [27]. [PERSON] et al. [25] proposed the ANN-Center method to predict the interval value by using the midpoint of the interval as the input. Later, [PERSON] and [PERSON] [26] introduced the ANN-LU method by using ANN-Center and ANN-Range (predicting the difference between upper and lower bounds) for predicting the lower and upper bounds of intervals separately. [PERSON] et al. [27] suggested that ANN-LU may face the unnatural interval crossing problem when the predicted lower bounds of intervals are larger than the predicted upper ones and vice versa and thereby leading to an invalid interval prediction. Hence, they introduced the regularized ANN-LU (RANN-LU) model for interval-valued time series forecasting. Although ANN-center, ANN-LU, and RANN-LU are found to be superior to the classical linear regression models for interval-valued data prediction of [PERSON] and Diday [7], these models rely either on midpoint (ANN-Center) or lower and upper bounds of the interval for forecasting, which may not be a good input for predicting the future value of an interval. For example, in the case of RANN-LU, if we predict the future value of the upper bound and lower bound separately, the prediction may not be reliable, since the whole information of the interval is not taken into account [27; 28]. Although ANN-Center and ANN-LU consider both upper and lower bounds in the prediction process, the prediction still relies on the midpoint of the interval indicating the symmetric weight between lower and upper bounds. To overcome these problems, in this study, we introduce the convex combination (CC) approach to ANN for predicting interval-valued data, say, lower and upper bounds. Our model is a generalization of the ANN-Center, allowing the weight to be more flexible and not to be fixed at 0.5 (asymmetric weight). The novelty of the proposed ANN-CC can be summarized in the following two aspects: First, our method can construct prediction intervals based on the CC approach and ANN models. More specifically, this study proposes the novel CC method for interval ANN modeling. In this approach, the intervals of input and output variables are parametrized through the convex combination of lower and upper bounds. The proposed ANN-CC is a promising alternative to the existing approaches. With optimal reference points that better represent the intervals, we are able to find an efficient solution that improves the prediction accuracy of the lower and upper bounds. Second, to the best of our knowledge, there is no study extending the CC approach for interval ANN modeling. Our proposedmethod fills in such a literature gap and can capture both linear and nonlinear patterns within interval-valued data. The rest of this paper is organized as follows. Section 2 gives a brief review of interval-valued prediction methods. Section 3 presents the proposed ANN with the convex combination method. In Section 4, we provide a simulation study to assess the performance of our proposed method; Section 5 describes the data used in this study. The analytical results are presented in Section 5. Section 6 provides the conclusion of this study.