# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 1 Introduction  
Time-series point (single-value data) forecasting normally fails to reflect the range of fluctuation or uncertainty for economic, financial, and environmental data. Moreover, the existing model of interval forecasting is still incomplete, complex, and has relatively low accuracy. Thus, interval-value data forecasting has become an important issue to be investigated [1, 2]. This study comes within interval-valued time series forecasting framework by introducing the convex combination (CC) method designed to choose the reference points that better represent the interval-valued data. The CC method automatically explores the set of reference points from input and output variables to build the neural networks (NN) models. This is an enhancement and a generalization over existing methods. Interval-valued data forecasting serves the needs of investors and data scientists who are sometimes interested in single value-data and the variability of value intervals in the data. Interval-valued data provides rich information that can help investors and data scientists make an accurate decision [3]. With the advance in data science, enormous information and big data can be collected nowadays. However, conventional forecasting methods cannot be effectively implemented to deal with this big data to yield accurate results. Furthermore, these methods are generally proposed to forecast future observations using only the point-valued data, which gives rise to a higher computation cost when dealing with big data. Moreover, it is sometimes difficult to express the real behaviorof the variable using only point-valued data. Thus, interval-valued data, such as the range of temperature, stock returns, and willingness to pay (minimum and maximum), is generally used to predict uncertainty in many situations. Interval analysis suggested by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) assumed that observations and estimations in the world are usually uncertain, incomprehensive, and do not precisely represent the real behavior of the data. This is quite true, as our use of point-valued data will entitle us to lose substantial information about prices or values during the day or the week. With interval-valued data, however, we can capture more realistic movement or information of the variables and can also handle big data forecasting at the same time. Thus, the interval approach should be considered to explain real data behavior under the context of big data. Interval-valued data is a special type for symbolic data analysis (SDA), composed of lower and upper bounds of an interval. The objective of SDA is to provide a way to construct aggregated data described by multivalued variables, and thereby provide an efficient way to summarize the large data sets by some reference value of symbolic data. Thus, the estimation tools for interval-valued data analysis are very intensively required in recent studies. The use of interval-data to represent uncertainty is common in various situations, such as in coalitional games where payoffs of coalitions are uncertain and could be modeled as intervals of real numbers, see, e.g., [PERSON] et al. [PERSON] et al. (2018) and [PERSON] et al. (2019). From a methodological point of view, interval-valued data is generally transformed into point-valued data or reference point data by some techniques; then, this reference point data is furthered used as a variable in the models. One of the famous approaches for dealing with interval-valued data is the mid-point method which was introduced by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018). They analyzed these data using ordinary least squares regression on the midpoints of the intervals, namely the lower and upper bounds of the independent and dependent variables. [PERSON] and [PERSON] and [PERSON] (2018) improved this approach by presenting a new method based on two linear regression models (center-range method). The first regression model is fitted at the midpoints of the intervals and the second one on the ranges of the intervals.