## 2 Material and Methods
### Models Design and Evaluation
#### 2.2.1 GMDH Model  
The GMDH is a self-organizing and complex approach through a complicated process by building a feed-forward network (FFN) function [15; 16; 25]. Although the GMDH has the same processing of ANN, it possesses some advantages compared by ANN, such as high speed and easier mathematical functions [25]. The current study developed GMDH model for estimating S and CS of SCC using eight input variables through a partial quadratic polynomials system.  
In general, the relationship between inputs and output variables can be represented as follows [16]:  
\[y=b_{0}+\sum_{1}^{n}b_{i}x_{i}+\sum_{1}^{n}\sum_{1}^{n}b_{ij}x_{i}x_{j}+\sum_{ 1}^{n}\sum_{1}^{n}\sum_{1}^{n}b_{ijk}x_{i}x_{j}x_{k}+\cdots \tag{1}\]  
where, \(y\) and \(x\) represent the output and input variables, respectively, \(n\) is the number of observations for each variable, and \(b\) notes model coefficients.  
Simply, a partial quadratic polynomials system for two inputs variables can be expressed as follows [15; 16]:  
\[y\Big{(}x_{i},x_{j}\Big{)}=b_{0}+b_{1}x_{i}+b_{2}x_{j}+b_{3}x_{i}x_{j}+b_{4}x _{i}^{2}+b_{5}x_{j}^{2} \tag{2}\]  
The least-square technique is used to calculate the model coefficients. \(\left(\begin{array}{c}n\\ 2\end{array}\right)=\frac{n(n-1)}{2}\) neurons can be generated in the first hidden layer of the FFN from the measurements \(\left\{\left(y_{i},x_{ip},x_{iq}\right)(i=1,2,\ldots,M\right]\) for different \(p,q\in\{1,2,\ldots,n\}\)[15; 16; 25; 16]. As a step for finding the unknown coefficients of the model, the inputs and output variables can be reconstructed in the following form.  
\[\left[\begin{array}{cccc}x_{1p}&x_{1q}&\ldots&y_{1}\\ x_{2p}&x_{2q}&\ldots&y_{2}\\ \vdots&\vdots&\vdots&\vdots\\ x_{Mp}&x_{Mq}&\ldots&y_{M}\end{array}\right]\]  
The quadratic sub-expression in Equation (2) use for each row of \(M\) to express the following matrix equation:  
\[Y=Ab \tag{3}\]where, \(b\) is the vector of model coefficients of quadratic polynomial and \(Y\) is the output vector. While matrix \(A\) can be expressed as follows, in the state of Equation (2), as an example:  
\[A=\left[\begin{array}{cccccc}1&x_{1p}&x_{1q}&x_{1p}x_{1q}&x_{1p}^{2}&x_{1q}^{2 }\\ 1&x_{2p}&x_{2q}&x_{2p}x_{2q}&x_{2p}^{2}&x_{2q}^{2}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 1&x_{Mp}&x_{Mq}&x_{Mp}x_{Mq}&x_{Mp}^{2}&x_{Mq}^{2}\end{array}\right] \tag{4}\]  
Therefore, based on the least-square method, the unknown coefficients can be obtained as follows:  
\[b=\left(A^{T}A\right)^{-1}A^{T}Y \tag{5}\]  
In this method, the connectivity topology of the network design based on three parameters, that are maximum neuron numbers (Nn), the maximum number of layers (Ln), and selection-pressure criteria (\(\alpha\)) [25]. The Nn can be calculated using \(\left(\begin{array}{c}n\\ 2\end{array}\right)=\frac{n(n-1)}{2}\) formula. While, the Ln and \(\alpha\) can be estimated based on the best determination of root mean square error (RMSE) and Pearson correlation coefficient (r) for the neurons; meanwhile, the network scheme was also designed based on the RMSE values, based on the best or worst RMSE, the neurons can be removed or accepted using RMSE and r values. More details for the estimating of these parameters can be found in et al. [25]. In this study, the best models for S and CS of SCC were generated and concluded in Table 4.