## 2 Methods
### Bb-Ann
Furthermore, in terms of the prediction error (i.e., the lower the error, the better the model), the lowest value was found for BBO-ANN (MAE = 0.022, RMSE = 0.026, RSR = 0.109) compared to DIRECT-ANN (MAE = 0.024, RMSE = 0.036, RSR = 0.151), PSO-ANN (MAE = 0.034, RMSE = 0.041, RSR = 0.174), MPMR (MAE = 0.034, RMSE = 0.040, RSR = 0.169), ELM (MAE = 0.037, RMSE = 0.045, RSR = 0.188), USBM (MAE = 0.100, RMSE = 0.117, RSR = 0.494), Indian standard (MAE = 0.087, RMSE = 0.105, RSR = 0.444), and Ambraseys-Hendron (MAE = 0.105, RMSE = 0.123, RSR = 0.517). To check the consistency of the developed models, the degree of agreement (\(d\)) was calculated using Equation (12) and the higher value was recorded for the BBO-ANN model. Therefore, it can be concluded that the BBO-ANN model (\(d\) = 0.997) had the best performance, followed by DIRECT-ANN (\(d\) = 0.994), PSO-ANN (\(d\) = 0.991), MPMR (\(d\) = 0.992), ELM (\(d\) = 0.991), USBM (\(d\) = 0.924), Indian standard (\(d\) = 0.943), and Ambraseys-Hendron (\(d\) = 0.914). Moreover, for a better representation in terms of model deviations, the receiver operating characteristic (ROC) curve was plotted (Figure 7). It is evident that all the models captured a good relationship when determining the PPV during training, and the lowest deviation was recorded for BBO-ANN followed by DIRECT-ANN, PSO-ANN, MPMR, and ELM. During the testing period, the BBO-ANN model outperformed the rivals in terms of all the fitness parameters. The results analyzed showed a consistent performance of BBO-ANN during both training and testing periods. Furthermore, during the training period, the convergence curve of two metaheuristic based optimized ANN models (Figure 8) showed that the BBO has a lower MSE (0.000899) compared to PSO (0.001777). Figure 8 shows the convergence plot of both hybridized ANN models. From analysis, it became evident that the BBO-ANN model reduces the fitness parameter MSE significantly compared to PSO-ANN for the same number of iterations. Therefore, based on the above analysis, it was found that BBO-ANN can be a new reliable technique for PPV analysis. In the present study, a sensitivity analysis was also performed using [PERSON]'s [57] method to assess the impact of input parameters on PPV. This method has been used in some studies [58; 59; 60], and is formulated as:

\[r_{ij}=\frac{\sum_{k=1}^{n}(y_{ik}\times y_{ok})}{\sqrt{\sum_{k=1}^{n}y_{k}{}^{2 }}\sum_{k=1}^{n}{y_{ok}{}^{2}}} \tag{13}\]

where \(n\) is the number of data values (this study used 80 data values), \(y_{ik}\) and \(y_{ok}\) are the input and output parameters. The value of \(r_{ij}\) for each input parameter varied between zero and one, and the highest \(r_{ij}\) values indicated the most effective output parameter (which was PPV in this study). Figure 9 shows the \(r_{ij}\) values for all input parameters. From Figure 9, it can be seen that the W with \(r_{ij}\) of 0.986 was the main parameter influencing PPV. Figure 7: ROC plot displaying the accuracy versus deviation for predicted PPV values on the training and testing dataset using the predictive models. ## 7 Conclusions

Ground vibration is considered to be the most adverse result induced by blasting. Accordingly, predictions of ground vibration are necessary, and this issue requires the application of appropriate prediction models. In this study, PPV was used as a descriptor to evaluate ground vibration. To predict the blast-induced PPV, this study proposed two novel hybrid AI models, namely the BBO-ANN and DIRECT-ANN models. In other words, one stochastic metaheuristic algorithm, namely BBO, and one deterministic optimization algorithm, namely DIRECT, were combined with the ANN model. To the best of our knowledge, this is the first work that predicted a blast-induced PPV using the DIRECT-ANN and BBO-ANN models. To demonstrate model reliability and effectiveness, the PSO-ANN, MPMR, ELM, and three empirical models, namely USBM, Indian Standard and Ambraseys-Hendron were also employed.