## 4 Applied Methods
### Group Method of Data Handling
To do this, considering the previous studies (e.g., [21]), values of 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 were selected to be used as number of neurons in the parametric study and results of GMDH models based on R\({}^{2}\) are shown in Table 4. Among the obtained results, as it can be seen in Table 4, GMDH model number 8 with 16 neurons shows the best prediction performance and due to that neuron number of 16 was selected in the rest of modelling process. It should be mentioned that this parametric study was conducted using selection pressure of 50% and layer number of 5. In the last step of GMDH modelling, number of layers should be determined through the conduct of another parametric study. For this purpose, amounts of 2, 3, 4, 5, 6, 7, 8, 9, and 10 were set as possible layers number. This was chosen according to previous investigations and their suggestions (e.g., [21]). According to the obtained results of this parametric study (Table 5), R\({}^{2}\) of 0.962, 0.968, 0.970, 0.978, 0.976, 0.974, 0.980, 0.977, and 0.978 were achieved for layer numbers of 2, 3, 4, 5, 6, 7, 8, 9, and 10, respectively. The results have clearly confirmed that the layer number 8 (GMDH model number 7) with R\({}^{2}\) of 0.980 receives the best performance prediction. Therefore, in design of GMDH model in this study, values of 50%, 16 and 8 were selected for selection pressure, neuron number and layer number, respectively. More discussions about the performance of selected GMDH model will be given in the next section. and variance account for (VAF) (see Equations (1)-(4)) were chosen and calculated for all developed models i.e., SVR, ANN, CART, and GMDH. Obviously, higher VAF values and lower error values give a model with higher performance prediction. \[\text{RMSE}\ =\ \sqrt{\frac{1}{\text{N}}\sum_{i\ =\ 1}^{\text{N}}(y-y^{\prime})^{2}} \tag{1}\]  
\[\text{E}_{a}\ =\ \left|y-y^{\prime}\right| \tag{2}\]  
\[\text{MAPE}\ =\ \frac{1}{N}\sum_{i\ =\ 1}^{n}\left|\frac{y-y^{\prime}}{y}\right| \tag{3}\]  
\[\text{VAF}\ =\ \left[1-\frac{\text{Var}(y-y^{\prime})}{\text{var}(y^{\prime})} \right]\times 100 \tag{4}\]  
where, y and y\({}^{\prime}\) are the measured and predicted values, respectively. In addition, N in total number of datasets [120]. In order to validate/test the proposed models, the selected datasets which were not used in the modelling process (23 databases from January 2014 to November 2015) were applied and then predicted. Subsequently, the obtained results from testing step were compared with the measured iron ore price values as shown in Figures 10-14 for ARIMA, CART, ANN, SVR, and GMDH models, respectively. As shown in these figures, the measured and predicted values by GMDH are closer to each other compared to other proposed models. This confirmed that the GMDH model which is a new model in the area of this study is able to perform well for prediction of iron ore price. It is a powerful and practical predictive model for such purposes. Figure 10: Comparison between the measured and the predicted by ARIMA model for test stage. Figure 11: Comparison between the measured and the predicted by CART model for test stage. Figure 12: Comparison between the measured and the predicted by ANN model for test stage. Figure 13: Comparison between the measured and the predicted by SVR model for test stage. Aside from that, the obtained results of performance prediction for all proposed techniques are tabulated in Table 4.0 As it can be seen in this table, higher performance prediction is related to results of GMDH model in both train and test phases. Although the other methods e.g., SVR and ANN are also suitable for prediction purposes and they can provide an acceptable level of accuracy, the highest prediction capacity was achieved developing GMDH predictive approach. Hence, this technique can be introduced as a new predictive model in estimating iron ore price.