# A Convex Combination Approach for Artificial Neural Network of Interval Data
## 3 The Proposed Method Artificial Neural Network with Convex Combination (ANN-CC)  
Artificial neural network (ANN) models can approximate various forms of nonlinearity in the data. There are various types of ANN, but the most popular one is the Multilayer Perceptron (MLP). ANN models have been successfully applied in a variety of fields such as accounting, economics, finance, and marketing, as well as forecasting [20, 26]. In this study, we use a three-layer ANN model in which both the inputs and outputs contain the lower and upper bounds of intervals. As shown in Figure 1, suppose the interval-valued data \((X,Y)\) consists of one predictor and one response. The first layer is the input layers, the middle layer is called the hidden layer, and the last layer is the output layers. All the data are expressed in the form of a lower-upper bound. Thus, we have \(\left\{X_{1}^{u},X_{1}^{l}\right\}\) and \(\left\{Y^{u},Y^{l}\right\}\). For convenience, we use the notation \(X^{cc}\)\(=\)\(\mathrm{w}_{1}X_{1}^{l}+(1-\mathrm{w}_{1})X_{1}^{u}\) for input variable and \(Y^{cc}\)\(=\)\(\mathrm{w}_{2}Y^{l}+(1-\mathrm{w}_{2})Y^{u}\) for output variable. Note that ANN-center is the particular case of ANN-CC if we set \(\mathrm{w}_{1}=\mathrm{w}_{2}=0.5\). Each layer contains \(j\)th neurons which are connected to one another and are also connected with all neurons in the immediate next layer. The neuron input path has a signal on its \(X^{cc}\), and the strength of the path is characterized by a weight of neuron \(j\) (\(w_{j}\)). The neuron is modeled as summing the path weight times the input signal over all paths and adding the node bias \((b)\). Note that the neural network consists of input function and output function; thus, we can express these functions as:  
\[H_{j}^{cc,l}=g\Big{(}X^{cc}\omega_{j}^{l}+b_{j}^{l}\Big{)}, \tag{17}\]  
where \(H_{j}^{cc,l}\) is the \(j\)th hidden neuron's input, \(\omega_{j}^{l}\) is the weight vector between the hidden layer and the input layer. \(g(\cdot)\) is the activation function for the hidden layer. \(b_{j}^{l}\) is the bias term of input layer. Then, \(H_{j}^{cc,l}\) is transformed into output \(Y^{cc}\) with the activation function of the output layer. Thus, the model can be written as:  
\[\hat{Y}^{cc}=f\Bigg{(}\sum_{j=1}^{J}H_{j}^{cc,l}\omega_{j}^{o}+b_{j}^{o}\Bigg{)}, \tag{18}\]  
where \(\omega^{o}=\left\{\omega_{1}^{o},\dots,\omega_{j}^{o}\right\}\) is the weight vector between the hidden layer and the output layer. \(b_{j}^{o}\) is the bias term of output layer. \(f(\cdot)\) is the activation function of the output layer. A challenge in ANN design is the selection of activation function. It is also known as transfer function, and can be basically divided into four types: tanh or hyperbolic tangent activation function (tanh), sigmoid or logistic activation function (sigmoid), linear activation function (linear), and exponential activation function (exp). Learning occurs through the adjustment of the path weights and node biases. The most common method used for this adjustment is backpropagation. In this method, the optimal weights \(\omega_{j}^{l}\) and \(\omega_{j}^{o}\) are estimated by minimizing the squared difference between the model output and the estimated. We formulate the loss function as follows,  
\[loss=\frac{1}{T}\sum_{t=1}^{T}\left(Y^{cc}-\hat{Y}^{cc}\right), \tag{19}\]  
where \(\hat{Y}^{cc}\) and \(Y^{cc}\) are the estimated output and observed output, respectively. In addition to the weights of neuron \(j\) (\(\omega_{j}\)), our estimation also considers the weight parameter of the interval data (\(\mathrm{w}\)) to determine the reference of the output and input variables. As the interval data is manipulated as a new type of numbers represented by an ordered pair of its minimum and maximum values, the numerical manipulations of interval data should follow \"interval calculus\" [25]. In practice, we can separately predict the lower and upper bounds of the interval using the min-max method.