## 3 Neural Network Modelling Method  
The proposed BPANN is an iterative gradient procedure designed to minimize the mean square error (MSE) between the actual output of the model and the desired output. Similar ANNs have already been used in many fields such as data modelling, classification, prediction, control, data and image compression and pattern recognition [3].  
The proposed BPANN model was based on a total of 96 group experiments, which included 80 group experiments conducted in this study and 16 group experiments conducted in our previous study [5]. Supervised learning was selected as the training method for the BPANN [20; 21; 22; 23]. To facilitate comparison between the BPANN model obtained in this study and our previously proposed regression models, the data obtained from the 80 group experiments were randomly divided into two subgroups consisting of 60 and 20 experimental datasets for conducting training and validation, respectively, and the 16 group experiments were employed as testing sets.  
The transfer function employed for the network is given as follows:  
\[f(x)=\frac{2}{1+e^{(-2x)}}-1 \tag{1}\]  
where \(x\) is the input value of the neuron and \(f(x)\) is the output of the neuron. This function is a hyperbolic tangent sigmoid, which is mathematically equivalent to a hyperbolic tangent function, but which provides for improved network functionality [20; 24]. All variable values were normalized within the interval (0, 1) to improve the effectiveness of the transfer function. While numerous methods are available for normalizing variable values, the present work employed the following equation [20; 24]:  
\[X_{i}=\frac{x_{i}-x_{\min}}{x_{\max}-x_{\min}} \tag{2}\]  
where \(X_{i}\) is the normalized element of any input value \(x_{i}\) and \(x_{\max}\) and \(x_{\min}\) are the maximum and minimum values of \(x_{i}\), respectively.  
The BPANN was trained using resilient backpropagation, which considerably improves the results of the training process for networks employing sigmoid transfer functions [22; 23]. The learning rate used was 0.01. The BPANN was constructed, trained and implemented using MATLAB 7.11.0 (R2010b) software, and statistical analyses were conducted using SPSS 20.0 software, all of which were run under a Microsoft Windows 7 operating system on a computer with an Intel(r) Core(tm) i3-470 CPU.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Research Detail** & **Description** \\ \hline Regression models & Linear and nonlinear regression models given by Equations (3) and (4), respectively, in this study. \\ \hline Comparison of the & \\ modelling precision & \\ with 8 new & \\ experimental & \\ conditions & \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._Overfitting is often the main problem associated with the training of ANNs [20; 21]. To avoid overfitting of the BPANN, the point at which overfitting occurred was detected by conducting a comparison of the training and validation set errors every 100 epochs to determine when an increase in the validation set error occurred in conjunction with a decrease in the training set error. In addition, a specific search program comprising a series of loops, one in each sublayer, was proposed to obtain the optimum structure of the BPANN [20; 21]. The number of neurons in each sub-layer was then successively increased automatically during training. Numerous methods can be employed to evaluate the performance of a model. The simplest method is based on the value of the correlation coefficient \(R\) for a plot of predicted value versus experimental output [1]. When all the loops have finished, the program considers the optimum structure to be that obtaining an absolute value of \(R\) that is closest to one for the validation set [20; 21], where \(|R|=1\) corresponds to perfect correlation. In addition, the determination coefficient \(R^{2}\) is another important value for evaluating the performance of a model, where an \(R^{2}\) value closest to one implies an optimum model.