# 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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Require : \(\mathbf{w}=\{\mathbf{w}_{1},\mathbf{w}_{2}\}\), where \(\mathbf{w}_{1}\) and \(\mathbf{w}_{2}\) are the set of candidate weight of input \(\left\{X_{1}^{u},X_{1}^{l}\right\}\) and output \(\left\{Y^{u},Y^{l}\right\}\), respectively, within [0, 1]. #Serach the optimal \(\widehat{\mathbf{w}}\) For each \(w_{i1},w_{2}\) in \(\mathbf{w}_{1},\mathbf{w}_{2}=[0.001,0.002,\dots,1]\) Calculate\(\gamma_{i}^{cc}=\mathbf{w}_{2}Y^{l}+(1-\mathbf{w}_{2})Y^{u}\)and\(X_{i}^{cc}=\mathbf{w}_{i1}X_{1}^{l}+(1-\mathbf{w}_{i1})X_{1}^{u}\) #Define the loss function of ANN structure \(\omega_{i}^{o}=\left\{\omega_{i}^{o},\ldots,\omega_{i}^{o}\right\}\), \(\omega_{i}^{I}=\left\{\omega_{i1}^{lo},\ldots,\omega_{i}^{lo}\right\}=\)Parameters( ) \(H_{ij}^{cc,I}=g\left(x_{i}^{cc}\omega_{ij}^{l}+b_{ij}^{lo}\right)\) \(\widehat{Y}_{i}^{cc}=f\Big{(}\sum_{j=1}^{lo}H_{ij}^{cc,I}\omega_{ij}^{o}+b_{ij }^{lo}\Big{)}\) \(loss_{i}(\mathbf{w}_{i})=\left\|Y_{i}^{cc}-\widehat{Y}_{i}^{cc}\right\|^{2}\) # Follow the gradients until convergence \(\mathbf{\omega}_{i}=(\omega_{i}^{o},\omega_{i}^{l})\) repeat\(\mathbf{\omega}_{i}=\mathbf{\omega}_{i}-\rho\
abla\mathbf{\omega}_{i}^{Loss}\) until convergence endfor #Choose the \(\mathbf{



## 4 Simulation Study

To examine the performance of our proposed method, we conducted a simulation study. We considered two data generation processes which are different in structure: linear and nonlinear. ### Linear Structure

The simple interval generation process is conducted, and one independent variable is assumed. We considered three typical data generation processes with different weight parameters. For this purpose, we considered the following three Scenarios of weight in the intervals:

_Scenario 1_: Center of interval data: \(\mathrm{w}_{1}=0.5\), \(\mathrm{w}_{2}=0.5\)

\[\left[(0.5)Y^{l}+(1-0.5)Y^{u}\right]=1+5\Big{[}(0.5)X^{l}+(1-0.5)X^{u}\Big{]}+\varepsilon. \tag{25}\]

_Scenario 2_: Deviate from the center to the lower of the interval data: \(\mathrm{w}_{1}=0.2\), \(\mathrm{w}_{2}=0.2\)

\[\hat{Y}^{cc}=f\Bigg{(}\sum_{j=1}^{J}H_{j}^{cc,l}\omega_{j}^{o}+b_{j}^{o}\Bigg{)}, \tag{18}\]0

_Scenario 3_: Deviate from the center to the upper of the interval data: \(\mathrm{w}_{1}=0.8\), \(\mathrm{w}_{2}=0.8\)

\[\hat{Y}^{cc}=f\Bigg{(}\sum_{j=1}^{J}H_{j}^{cc,l}\omega_{j}^{o}+b_{j}^{o}\Bigg{)}, \tag{18}\]1

For each scenario, we performed 100 replications. In each simulation, we proceeded as follows. (1) Generate the error from the normal distribution with mean zero and variance one. (2) Generate the upper bound of the independent variable \(X^{u}\), from the uniform (1,3). Then, we computed the lower bound of the independent variable \(X^{l}=X^{u}-r_{x}\), where \(r_{x}\sim U(0,2)\) denotes the range between the upper and lower bounds. (3) Compute the expected independent variable \(X^{cc}\) of the intervals that have been computed by

\(X^{cc}=(\mathrm{w}_{1}X^{u})+(1-\mathrm{w}_{1})X^{l}\). Then, we could generate the expected dependent variables \(Y^{cc}=X^{cc}\beta+\varepsilon\). (4) Finally, we derived the upper and lower bounds of intervals by \(Y^{l}=Y^{cc}-r_{y}\), where \(r_{y}\sim U(0,2)\) and \(Y^{u}=\left(Y^{cc}-(1-\mathrm{w}_{2})Y^{l}\right)/\mathrm{w}_{2}\). We note \(r_{x}\) and \(r_{y}\) are a random number for simulating the interval \(X\) and \(Y\), respectively. This guaranteed that the bounds are not crossing each other. In this simulation study, we performed 100 replications with a sample size n = 1000 for all three scenarios. Each simulation dataset a randomly split, with 80% for training and 20% for testing. Our ANN with the convex combination method (ANN-CC) was then compared with two conventional models, namely the ANN-Center and the RANN-LU method of [PERSON] et al. [27]. In this simulation study, four transfer functions, namely tanh or hyperbolic tangent activation function (tanh), sigmoid or logistic activation function (sigmoid), linear activation function (linear), and exponential activation function (exp) were considered.