# Evaluating Slope Deformation of Earth Dams Due to Earthquake Shaking Using MARS and GMDH Techniques
## 3 Methodology  
### Group Method of Data Handling (GMDH)  
The GMDH method is a self-organized algorithm and can provide a good solution for problems that have unknown nature. In fact, this method is a branch of artificial neural network that models the behavior of a phenomenon based on the input and output dataset. Dataset is a pair of data that is composed of several input parameters and one corresponding output parameter (Xi,yi) (=1,2, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & **Height (m)** & **TD (s)** & **Ay (g)** & **Mw** & **Amax (g)** & **Tp (s)** & **Dave (m)** \\ \hline
**Min** & 2.500 & 0.050 & 0.000 & 4.900 & 0.010 & 0.250 & 0.001 \\
**Average** & 40.389 & 0.555 & 0.176 & 7.045 & 0.293 & 0.374 & 0.883 \\
**Max** & 235.000 & 2.740 & 0.550 & 8.300 & 0.900 & 0.700 & 6.546 \\
**Type** & Input & Input & Input & Input & Input & Input & Output \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Information on the parameters used in this study. Figure 1: Different distribution of the used parameters. \(\ldots\),M). Quadratic node is a computational function that is defined based on a transfer function in a feed-forward network. In the GMDH method, in order to define coefficients of this function, a regression analysis is used [82]. In a model that is built by the GMDH method, there are some neuron layers, and several data pairs in every layer are linked with each other by a quadratic polynomial. In this way, new neurons are made. Input and output data are linked and correspond with each other based on the same process [17; 83]. The structure of this process is shown in Figure 2. According to this figure, four parameters are first entered in the first layer, and some functions are defined in this layer. Then, based on this criterion, these functions are selected and transferred to the next layer. The output of the model is also a function that is built in the last layer. The general function of parameters is given in Equation (1). \[\hat{y}_{i}=f(X_{i1},\ X_{i2},\ X_{i3},\ \ldots X_{in}),\ (\mathrm{i}=2,3, \ldots\mathrm{M}) \tag{1}\]  
Obtaining the parameters in Equation (2) is one of the main steps in this modeling. \[y=a_{0}+\sum_{i=1}^{n}a_{i}x_{i}+\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}x_{i}x_{i}+ \sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}a_{ijk}x_{i}x_{k}+\ldots \tag{2}\]  
In fact, the above-mentioned parameters are obtained from the solution of a mathematical relation in which the most important parameter among them is the \"a\" coefficient. In each layer, several functions are defined to obtain the \"a\" parameter, and this process is repeated until Equation (3) is minimized. If needed, for more information about research in this area, see [39]. \[E=\frac{\sum_{i}^{M}=1\Big{(}y_{i}-y\Big{(}x_{i},x_{j}\Big{)}\Big{)}^{2}}{M} \tag{3}\]  
In each layer, with the help of neurons, the computational functions are defined. The selection pressure criterion is defined in several layers in order to select parameters. Based on the mentioned criterion, if the built function has the desired conditions, it is transferred to the next layer or otherwise deleted. Equation (4) shows the mentioned criterion in the form of a mathematical relation. The range of change of this parameter is [0, 1]. In order to select a function that has the minimum error, the value  
Figure 2: A proposed structure of the group method of data handling (GMDH) model. of \(\alpha\) is considered to be 1. On the other hand, if the value of \(\alpha\) is considered close to zero, more data are transferred to the next layer. \[e_{c}=\alpha\times RMSE_{min}+(1-\alpha)\times RMSE_{max} \tag{4}\]  
Figure 3 shows the modeling process in which all cases that are applied to run the GMDH model are considered. In order to evaluate the model results, two indices of the correlation coefficient and root mean square error (\(R^{2}\), RSME) were used. \[R^{2}=\left(1-\frac{\sum_{q=1}^{Num_{\text{model}}}\left(Y_{q}-\hat{Y}_{q} \right)^{2}}{\sum_{q=1}^{Num_{\text{model}}}\left(Y_{q}-\overline{Y}_{q} \right)^{2}}\right) \tag{5}\]