## 6 Results and Discussion
To clearly show the performance of the six models in the training and testing phases, the error histograms were described in Figure 10. The horizontal axis and the vertical axis represent the error between the predicted value and its total percentage (%), respectively. That is to say, increasing the tall error bars close to the origin, increases the prediction accuracy of the model. As can be seen in this figure, the percentage of the lowest error is highest in the KELM-GWO model and the lowest in the DELM model. Figure 11 shows the error curves of the six models in the testing phase. There are two types of curves in each diagram: the black curve represents observed values, and the other colored curves represent predicted values for each model. Taken as a whole, the predictive performance of each model in the testing phase is close. Nevertheless, magnified local errors can indicate differences in the predictive performance of all the models. As can be seen in these diagrams, there are smaller predicted errors from rock samples of No. 65 to No. 81 in the KELM-GWO model than the other five models, which means that the KELM-GWO model is more suitable for predicting UCS than other models. The graphical Taylor diagrams can be a concise tool for evaluating the performance of the models more comprehensively, as shown in Figure 12. A complete Taylor diagram consists of three parts: standard deviation (blue circles), RMSE (green circles), and correlation coefficient (black dotted lines). The RMSE and correlation coefficient of the observed value was 0 and 1; the standard deviation, RMSE, and correlation coefficient of all models can be calculated based on their predicted values. Accordingly, the highest similarity to the observed data corresponded to the best prediction model. As can be seen in the diagrams, the KELM-GWO model was the best model with the closest position to the observed values in both the training and testing phases. After this model, the ranking of the prediction performance is BPNN, KELM, ELM, empirical, and DELM. To compare and assess the prediction performance with the proposed model, the current and previous works are listed in Table 6. As can be seen in this table, the KELM model was almost never used to predict UCS and the KELM-GWO model proposed in this study is better than most of the single AI models. Although the performance results of several studies are better than this paper, the volume of data limits the difficulty of applying these models to the UCS prediction of other rocks. In addition, different prediction results can be obtained by using different input parameter combinations based on the same model. Figure 10: Illustration of error histogram in the training and testing phases. Therefore, it is necessary to obtain the importance of the different input parameters for the UCS prediction in this study. The mean impact value (MIV) has been reported to analyze the importance of input variables in prediction problems [98; 99; 100]. The core of the MIV technique is enlarging and reducing the input variable with the same ratio. Then, two new kinds of datasets were used to train and test based on the proposed models. Accordingly, two new outputs were obtained and the errors between them were named as impact values (IVs) of the input variable. Finally, the mean of those IVs (MIV) can be calculated and the largest MIV of the variable represents the highest importance for predicting UCS. The results of importance analysis for input variables are illustrated in Figure 13. As can be seen in this figure, the most important input variable is \(n\) with an MIV of 15.47; the remaining order of importance for the others is PLS of 10.74, SHR of 9.50, and \(\rm V_{p}\) of 8.47. Figure 11: The curves of predicting FD in the testing phase by all models.