# Assessing Ground Vibration Caused by Rock Blasting in Surface Mines Using Machine-Learning Approaches: A Comparison of CART, SVR and MARS

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}of 0.951, followed by SVR (R

^{2}= 0.87), CART (R

^{2}= 0.74) and empirical predictors. Based on the large-scale cases and input variables involved, the developed models should lead to better representative models of high generalization ability. The proposed MARS model can easily be implemented by field engineers for the prediction of blasting vibration with reasonable accuracy.

## 1. Introduction

Authors | Models | Input Parameters | No. of Datasets | Best Model | Performance Indices |
---|---|---|---|---|---|

Ke et al. [13] | SVR, GEP, ANN-SVR, Empirical predictor | HDM, BH, HD, B, S, Hc, PF, MCPD, D | 297 | ANN-SVR | R^{2} = 0.887RMSE = 1.232 |

Nguyen and Bui [14] | HGS–ANN, GOA–ANN FA–ANN, PSO–ANN | HD, MCPD, B, PF, D, SL, S NDS, DTS | 252 | HGS– ANN | R^{2} = 0.922RMSE = 1.761 |

Singh [15] | ANN | HDM, NH, HD, B, S, SL, Hdis, Rdis | 200 | ANN | R^{2} = 0.83 |

Nguyen et al. [16] | MARS, ANN, PSO–ANN, MARS-PSO–ANN, Empirical predictor | MCPD, D, HD, B, S, SL, PF | 193 | MARS-PSO–ANN | R^{2} = 0.902RMSE = 1.569 |

Singh et al. [17] | ANFIS, MVRA | MCPD, D | 192 | ANFIS | R^{2} = 0.98 |

Lawal et al. [18] | ANN, BK, GEP, MLR | S/B, BH/B, B/HDM, SL/B, SD/B, UCS, ρr, MCPD, D | 191 | ANN | R^{2} = 0.948RMSE = 0.0008 |

Singh and Verma [19] | ANFIS | B, S, D, IS, TC | 187 | ANFIS | R^{2} = 0.77 |

Monjezi et al. [20] | ANN | HD, T, MCPD, D | 182 | ANN | R^{2} = 0.949(ANN) |

Khandelwal and Singh [21] | ANN, MVRA | HD, S, D, E, P-wave, B, MCPD, BI, µ, VOD | 174 | ANN | R^{2} = 0.98 |

Khandelwal [22] | SVM, MVRA, Empirical predictor | MCPD, D | 174 | SVM | R^{2} = 0.96,MAE = 0.257 |

Khandelwal and Singh [23] | ANN | TC, D | 170 | ANN | R^{2} = 0.998 |

Monjezi et al. [24] | MLPNN, RBFNN, GRNN | D, B/S, MCPD, NHPD, UCS, DPR | 169 | MLPNN | R^{2} = 0.954,RMSE = 0.03 |

Yu et al. [25] | ELM, HHO–ELM, GOA–ELM, | D, HD, B/S, MCPD, PF | 166 | GOA–ELM | R^{2} = 0.9105RMSE = 2.855 |

Mohamed [26] | FS, ANN, MVRA | D, MCPD | 162 | FS | RMSE = 0.17 VAF = 87(%) |

Bayat et al. [1] | GEP | B, S, T, D, MCPD | 154 | GEP | R^{2} = 0.91RMSE = 5.78 |

Khandelwal and Kumar [27] | ANN, Empirical predictor | MCPD, D | 150 | ANN | R^{2} = 0.919, RMSE = 0.352 |

Singh et al. [28] | GA, MVRA, ANN, ANFIS, SVM | UCS, ρr, Hc, ɳ, ABS, FRC | 150 | GA | MAPE = 0.198 |

Zhou et al. [29] | RF, ANN, XGBoost, AdaBoost, Bagging, Jaya-X-GBoost Empirical predictor | HDM, HD, CPH, S, B, CL, BI, E, D, µ, P-wave, VOD, ρe | 150 | Jaya-XGBoost | R^{2} = 0.957RMSE = 4.088 |

Mohamed [30] | ANN | P-wave, HDM, VOD, B, S, BH, HI, D, ρe, ρr, MCPD, E, TC, ɳ, UCS, | 149 | ANN | R^{2} = 0.94,MSE = 0.00920 |

Rana et al. [31] | CART, ANN, MVRA, Empirical predictor | TC, TS, MCPD, NH, HDM, D, HD, CPH | 137 | CART | R^{2} = 0.95,RMSE = 1.56 |

Verma and Singh [32] | SVM, ANN, MVRA | HD, B, S, T, MCPD, TC, D | 137 | SVM | MAPE = 0.001 |

Verma and Singh [33] | GA, ANN, MVRA, Empirical predictor | HD, B, S, T, MCPD, TC | 127 | GA | R^{2} = 0.99,MAPE = 0.088 |

Ghasemi et al. [34] | FS, MRA, Empirical predictor | B, S, T, NHPD | 120 | FS | R^{2} = 0.945,RMSE = 2.73 |

Ghasemi et al. [35] | ANFIS-PSO, SVR | B, S, T, NH, MCPD, D | 120 | ANFIS-PSO | R^{2} = 0.957,RMSE = 1.83 |

Bui et al. [36] | ANN, SVM, Tree-based ensembles, CSO–ANN Empirical predictor | MCPD, CPH, D, B, S, PF | 118 | CSO–ANN | R^{2} = 0.99RMSE = 0.246 |

Dehghani and Ataee-pour [37] | ANN, Empirical predictor, Dimensional analysis | S, B, DPR, NH, PF, D, CPD, MCPD, PLI | 116 | ANN | R^{2} = 0.945, RMSE = 0.0245 |

Zhongya [38] | BPNN, MVRA, ELM- FA MIV | D, MCPD, B/S, NHPD, UCS, DPR | 108 | ELM-FA MIV | R^{2} = 0.96, RMSE =0.21 |

Armaghani et al. [39] | MPMR, LSSVM, GPR PSO–ELM, AGPSO–ELM | B/S, MCPD, D, T, PF, HD | 102 | AGPSO–ELM | R^{2} = 0.90RMSE = 0.08 |

Faradonbeh et al. [40] | GEP, NLMR | T, B/S, PF, D, HD, MCPD | 102 | GEP | R^{2} = 0.874 |

Mokfi et al. [41] | GMDH, GEP, NLMR | MCPD, PF, T, B/S, D, HD | 102 | GMDH | R^{2} = 0.874,RMSE = 0.963 |

Ismail et al. [42] | GEP, ANFIS, SCA-ANN Empirical predictor | D, MCPD, ρr, SRH | 100 | SCA-ANN | R^{2} = 0.999RMSE = 0.0094 |

Hajihassani et al. [43] | ICA-ANN, ANN, MLR | B/S, T, MCPD, P-wave, E, D | 95 | ICA-ANN | R^{2} = 0.97 |

Chen et al. [44] | FA–SVR, PSO–SVR, GA–SVR, FA–ANN, PSO–ANN, GA–ANN, MFA–SVR | B/S, T, MCPD, D, E, P-wave | 95 | MFA–SVR | R^{2} = 0.984RMSE = 0.614 |

Peng et al. [45] | ANN, ANN-PSO, ANN-GA, ANN | MCPD, D, PF, SD, RQD, B, S | 93 | ANN-PSO | R = 0.945 RMSE = 0.680 |

Hasanipanah et al. [46] | CART, MLR, Empirical predictor | MCPD, D | 86 | CART | R^{2} = 0.95,RMSE = 0.17 |

Hudaverdi and Akyildiz [47] | ANN, MLR Empirical predictor | MCPD, D, B, S | 86 | ANN | RMSE = 5.28 |

Zhu et al. [48] | ANN, ANFIS, RANFIS CRANFIS, CRANFIS-PSO, Empirical predictor | B, S, T, PF, MCPD, D | 84 | CRANFIS-PSO | R^{2} = 0.997RMSE = 0.076 |

Shahnazar et al. [49] | PSO-ANFIS, ANFIS | D, MCPD | 81 | ANFIS-PSO | R^{2} = 0.984,RMSE = 0.4835 |

Hasanipanah et al. [50] | SVM, Empirical predictor | MCPD, D | 80 | SVM | R^{2} = 0.96,RMSE = 0.34 |

Abbaszadeh Shahri et al. [51] | GFFN-FA, GFFN-ICA, GFFN | B, S, TC, D, MCPD | 78 | GFFN-FMA | R^{2} = 0.97RMSE = 0.187 |

Saadat et al. [52] | ANN, Empirical predictor | MCPD, D, SL, HD | 69 | ANN | R^{2} = 0.95,RMSE = 8.79 |

Álvarez-Vigil et al. [53] | ANN, MLR | RMR, BCPRA, D, HDM, S, HD, B, MCPD, VOD, TC, NH | 60 | ANN | R^{2} = 0.96,RMSE = 0.65 |

Lawal et al. [3] | ANN, GEP, MFO-ANN, MLR, Empirical predictor | HD, CPD, NH, TC, D, RMR | 56 | MFO-ANN | R^{2} = 0.957MSE = 0.0008 |

Amini et al. [54] | ANN | D, ρe Ve, B, S, TC | 51 | ANN | R^{2} = 0.96 |

[55] | CART, MR, Empirical predictor | MCPD, D | 51 | CART | R^{2} = 0.92,RMSE = 0.97 |

Iphar et al. [56] | ANFIS, MLR | MCPD, D | 44 | ANFIS | R^{2} = 0.98,RMSE = 0.80 |

Armaghani et al. [57] | BP-ANN, PSO–ANN | HDM, HD, MCPD, S, B, SL, PF, ρr, SD, NR | 44 | PSO–ANN | R^{2} = 0.93 |

Lapčević et al. [58] | ANN | CPH, DT, MCPD, TC, D | 42 | ANN | R^{2} = 0.95 |

Mohamadnejad et al. [59] | SVM, GRNN, Empirical predictor | MCPD, D | 37 | SVM | R^{2} = 0.89,RMSE = 1.62 |

Monjezi et al. [60] | GEP, MLR, NLMR | D, MCPD | 35 | GEP | R^{2} = 0.918, RMSE = 2.321 |

Li et al. [61] | SVM, Empirical predictor | MCPD, D | 32 | SVM | R^{2} = 0.945 |

Ravilic et al. [62] | MCPD, D, TC | ANN, Empirical predictor | 32 | ANN | R^{2} = 0.9 RMSE = 0.018 |

Monjezi et al. [12] | ANN, Empirical predictor | TC, MCPD, D | 20 | ANN | R^{2} = 0.924,RMSE = 0.071 |

Ragam and Nimaje [63] | GRNN, Empirical predictor | D, MCPD | 14 | GRNN | R^{2} = 0.999, RMSE = 0.0001 |

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Methods

#### 2.2.1. Empirical Methods

#### 2.2.2. Multiple Linear Regression (MLR)

#### 2.2.3. Classification and Regression Tree (CART)

_{1}, R

_{2}..., R

_{m}and the output as a constant Cm in each region, the adaptive basis function framework of the recursive partitioning can be represented as in Equation (3) [66].

_{m}is the mth region and C

_{m}is the mean response in a given region (scalar for regression, class probabilities for multi-class classification).

#### 2.2.4. Support Vector Regression (SVR)

#### 2.2.5. Multivariate Adaptive Regression Splines (MARS)

## 3. Results

#### 3.1. MLR

− 0.01635 × CPH + 0.0003 × TC − 0.000019 × MCPD − 0.011698 × D

^{2}of 0.384 and 0.4 for training and testing, respectively. This shows that MLR poorly explains the relationship between PPV and the predictor variables and confirms the non-linear interaction between variables.

#### 3.2. Empirical Methods

^{2}on the testing dataset, followed by the USBM and CMRI predictor, respectively.

#### 3.3. CART Model for the Prediction of PPV

^{2}of 0.744 was obtained on the testing dataset (Figure 9 and Table 7). Therefore, the value of 0.01 was considered the optimum ccp_alpha parameter. The performance indices for all iterations are presented in Table 7. Although the pruning stage decreases the performance of the training set, the model with ccp_alpha of 0.01 can yield efficient prediction on a new dataset and be considered the optimum CART model. The structure of the corresponding regression tree is presented in Figure 10.

^{2}of 0.74 on unseen data (test dataset) outperformed the best empirical predictor (Ambraseys–Hendron equation, R

^{2}= 0.67) and multiple linear regression (R

^{2}= 0.4). It can be employed to estimate PPV with a prediction accuracy of over 74%.

#### 3.4. SVR Model for the Prediction of PPV

^{2}and RMSE of 0.9007 and 1.0047 for the training dataset and 0.876 and 0.9981 for testing datasets. The relationship between measured and predicted PPV is presented in Figure 14. The results indicate better accuracy of the SVR model as compared to MLR, empirical, and CART models.

#### 3.5. MARS Model for the Prediction of PPV

^{2}= 0.951) on training and testing datasets (Figure 16). The optimum hyperparameter values yielded a total of 55 candidate BFs, as shown in Figure 17. It is worth noting that the generalized cross-validation (GCV) method is applied to remove insignificant BFs during the backward stage. Figure 17 indicates 32 prominent numbers of terms with the highest general co-efficient of determination (R

^{2}). The remaining terms (BFs) does not influence the model performance as the R

^{2}remains relatively unalterable with further BFs (Figure 17). Therefore, the insignificant BFs were removed (pruned) and the final MARS model involves 32 imperative BFs. A similar methodology was employed by Abdulelah Al-Sudani et al. [72] and Chen et al. [73] in previous research to identify the optimum MARS model. The elected 32 BFs and their corresponding coefficients are presented in Table 10 alongside the general regression equation. The application of this equation consists of summing the regression equations of each spline (BF). The obtained value represents the target response PPV based on the proposed MARS model. From Table 9, it can be seen that the performance of both training and testing data are similar, suggesting a good generalization ability of the proposed MARS model.

## 4. Discussion

## 5. Conclusions

- ▪
- Based on 1001 datasets, the effective parameters on PPV were assessed using sensitivity analysis. PPV depends upon various blast-design parameters such as hole diameter, hole depth, number of holes, burden, spacing, stemming length, charge per hole, total charge, maximum charge per delay, and monitoring distance.
- ▪
- Machine-learning techniques outperformed traditional prediction techniques including empirical and statistical methods and better explain the non-linear interaction between input variables and the response PPV.
- ▪
- A comprehensive quantitative interaction between input variables and the response PPV is obtained from CART and MARS models, and can be easily employed to predict PPV with reasonable accuracy.
- ▪
- Despite using many datasets and input variables, the study shows that the MARS model can be easily employed to estimate PPV with high prediction accuracy (R
^{2}= 0.951; RMSE = 0.227) compared to CART and SVR.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ABS | Absorption | MCPD | Maximum charge per delay |

AGPSO | Autonomous groups particles swarm optimization | MFA | Modified firefly algorithm |

A–H | Ambraseys–Hendron predictors | MFA | Modified firefly algorithm |

ANFIS | Adaptive neuro-fuzzy inference system | MFO | Moth-flame optimization algorithm |

ANN | Artificial neural network | MIV | Mean impact value |

B | Burden | ML | Machine learning |

BCPRA | Blast-control point relative angle | MLPNN | Multilayer perceptrons neural network |

BF | Basis function | MLR | Multiple linear regression; |

BI | Blasting index | MPMR | Minimax probability machine regression |

BIGV | Blast-induced ground vibration | MR | Multiple regression; |

BK | Buckingham π (pi) theorem | MVRA | Multivariate regression analysis |

BP | Backpropagation | NDS | Number of blasting groups |

BPNN | Backpropagation neural network | NH | Number of holes |

CART | Classification and regression tree | NHPD | Number of holes per delay |

CL | Average charge length | NLMR | Non-linear multiple regression |

CMRI | Central Mining Research Institute predictor | NR | Number of rows |

CPH | Average explosive charge per hole | PF | Powder factor |

CRANFIS | Chaos recurrent adaptive neuro-fuzzy inference system | PLI | Point load index |

CSO | Cuckoo search optimization | PPV | Peak particle velocity |

D | Distance | PSO | Particle swarm optimization |

DPR | Delay per row | P-wave | P-wave velocity |

DTS | Time delay for each group, | R^{2} | Co-efficient of determination |

E | Young’s modulus | RANFIS | Recurrent adaptive neuro-fuzzy inference system |

ELM | Extreme learning machine | RBFNN | Radial basis function neural network |

FA | Firefly algorithm | Rdis | Radial distances |

FRC | Fracture roughness co-efficient | RF | Random forest |

FS | Fuzzy system | RMR | Rock mass rating |

GA | Genetic algorithm | RMSE | Root-mean-square error |

GCV | Generalized cross-validation | RQD | Rock quality designation |

GEP | Gene-expression programming | S | Spacing |

GFFN | Generalized feed-forward neural network | SCA | Sine cosine algorithm |

G–D | Ghosh–Daemen empirical predictor | SD | Sub-drilling; |

GMDH | Group method of data handling | SL | Steaming length |

GOA | Grasshopper optimization algorithm | SRH | Schmidt rebound hardness value |

GPR | Gaussian process regression | SVM | Support vector machine |

GRNN | General regression neural network | SVR | Support vector regression |

H | Bench height | TC | Total charge |

Hc | Hardness co-efficient | TS | Tunnel cross-section |

HD | Hole depth | UCS | Uniaxial compressive strength |

Hdis | Horizontal distances | USBM | United states bureau of mines |

HDM | Hole diameter | Ve | Volume of extracted block |

HGS | Hunger games search | VOD | Velocity of detonation |

HHO | Harris hawks optimization | XGBoost | Extreme gradient boosting |

ICA | Imperialist competitive algorithm | ρe | Explosive density |

IS | Indian standard predictor | ρr | Rock density |

L–K | Langefors–Kihlstrom predictor | ɳ | Porosity |

LSSVM | Least-squares support vector machine | µ | Poisson ratio |

MARS | Multivariate adaptive regression splines |

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**Figure 8.**(

**a**) Measured versus Predicted PPV (USBM Method). (

**b**) Measured versus Predicted (L–K method). (

**c**) Measured versus Predicted (A–H). (

**d**) Measured versus Predicted (IS method). (

**e**) Measured versus Predicted (CMRI method).

No. | Mines | Company |
---|---|---|

1 | Chandan Coal Mine, Jharia | Bharat Cooking Coal Limited |

2 | Patherdih Coal Mine, Jharia | Bharat Cooking Coal Limited |

3 | Bera Coal Mine, Bastacola | Bharat Cooking Coal Limited |

4 | Golakdih Coal Mine, Bastacola | Bharat Cooking Coal Limited |

5 | Jogidih Coal Mine, Govindpur | Bharat Cooking Coal Limited |

6 | Dahibari Coal Mine, Chanch Victoria Area | Bharat Cooking Coal Limited |

7 | Gopalichuk Coal Mine, Pootkee Balihari Area | Bharat Cooking Coal Limited |

8 | Bagdigi Coal Mine, Lodna | Bharat Cooking Coal Limited |

9 | Tetulmari Coal Mine, Sijua Area | Bharat Cooking Coal Limited |

10 | Kujama Coal Mine, Bastacola | Bharat Cooking Coal Limited |

11 | Bhanora Coal Mine, Sripur area | Eastern Coalfields Limited |

12 | Magadh Coal Mine, Magadh Amrapali Area | Central Coalfields Limited |

13 | Pakri Barwadih Coal Mine, Barakagaon | National Thermal Power Corporation |

14 | Tasra Coal Mine, Jharia | Steel Authority of India Limited |

15 | Bermo Coal Mine, Bokaro | Damodar Valley Corporation |

16 | Jamuna Coal Mine, Jamuna and Kotma Area | South Eastern Coalfields Limited |

17 | Ramagundam-III Area Coal Mine, Peddapalli | Singareni Collieries Company Limited |

18 | Aditya Cement Limestone Mine, Shambhupura | M/S Ultratech Cement |

19 | Adhunik Cement Limestone Mine, Meghalaya | Adhunik Cement Limestone Mine |

20 | Manal Limestone Mine, Rajban | Cement Corporation of India Limited |

21 | Daroli Limestone Mine, Udaipur | Daroli Limestone Mines |

22 | SK2 Block Vikram Limestone Mine, Khor | Vikram Cement works |

23 | Karunda Limestone Mine, Chittorgarh | J K Cement |

24 | Malikhera Limestone Mine, Chittorgarh | J K Cement |

25 | Murlia Block Limestone Mine, Chandrapur | Murli Industries Limited |

26 | Jhamarkotra Rock Phosphate Mine, Udaipur | Rajasthan State Mines and Minerals Limited |

27 | Sanchali Calcite Mine, Udaipur | M/s Wollmine India Pvt. Limited |

28 | Guali Iron Ore Mine, Topadihi | M/s R. Sao |

29 | Narayanposhi Iron and Manganese Ore Mine Koria, Sundergarh | M/s Aryan Mining and Trending Corp. Limited |

30 | Balda Block Iron Ore Mine, Keonjhar | M/s Serajuddin and Company, Orissa |

31 | Banduhurang Opencast Uranium Mines | Uranium Corporation of India Limited |

32 | Obra Stone Mine (Dolomite quarry) | M/s B. Agarwal Stone Products Limited, Sonebhadra |

33 | Pachami Hatgacha Stone Mining, Birbhum | West Bengal Mineral Development and Trading Corporation Limited |

34 | Granite aggregate quarry, Setto, Benin republic | OKOUTA CARRIERES SA |

Parameters | Unit | Symbol | Category | Min | Max | Mean | Median | Sd. Dev |
---|---|---|---|---|---|---|---|---|

Hole diameter | mm | HDM | Input | 32 | 269 | 126.5 | 115 | 32.04 |

Hole depth | m | HD | Input | 0.7 | 13.5 | 6.59 | 6.2 | 2.28 |

Number of holes | - | NH | Input | 1 | 199 | 31.52 | 21 | 33.06 |

Burden | m | B | Input | 0.6 | 9 | 3.13 | 3 | 1.05 |

Spacing | m | S | Input | 0.6 | 10 | 4.04 | 3.5 | 1.52 |

stemming length | m | SL | Input | 0.5 | 7 | 3.04 | 3 | 0.94 |

Charge per hole | kg | CPH | Input | 0.17 | 400.75 | 39.2 | 32.14 | 36.49 |

Total charge | kg | TC | Input | 5.56 | 41294 | 1390.86 | 544.46 | 2767.81 |

Maximum charge per delay | kg | MCPD | Input | 2.19 | 2545.5 | 85.92 | 45.5 | 169.98 |

Monitoring distance | m | D | Input | 25 | 1500 | 321.36 | 293 | 185.45 |

Peak particle velocity | mm/s | PPV | Output | 0.22 | 43.59 | 3.37 | 2.44 | 3.12 |

Name | Equations |
---|---|

USBM | $PPV=K{\left(\mathit{D}/\sqrt{\mathit{M}\mathit{C}\mathit{P}\mathit{D}}\right)}^{-B}$ |

Langefors–Kihlstrom (L–K) | $PPV=K{\left(\sqrt{\mathit{M}\mathit{C}\mathit{P}\mathit{D}/{D}^{2/3}}\right)}^{B}$ |

Ambraseys–Hendron (A–H) | $PPV=K{\left(\frac{\sqrt[3]{\mathit{M}\mathit{C}\mathit{P}\mathit{D}}}{\mathit{D}}\right)}^{B}$ |

IS | $PPV=K{\left(\mathit{M}\mathit{C}\mathit{P}\mathit{D}/{D}^{2/3}\right)}^{B}$ |

CMRI | $PPV=n+K{\left(D/\sqrt{\mathit{M}\mathit{C}\mathit{P}\mathit{D}}\right)}^{-1}$ |

Parameters | Coefficients | Standard Error | t Stat | p-Value |
---|---|---|---|---|

Intercept | 1.9830 | 0.5470 | 3.6254 | 0.0003 |

HDM | 0.0171 | 0.0044 | 3.8961 | 0.0001 |

HD | 0.2007 | 0.0622 | 3.2266 | 0.0013 |

NH | −0.0053 | 0.0042 | −1.2416 | 0.2148 |

B | 0.0350 | 0.2057 | 0.1704 | 0.8648 |

S | 0.5605 | 0.1422 | 3.9411 | 0.0001 |

SL | −0.1010 | 0.1309 | −0.7714 | 0.4407 |

CPH | −0.0163 | 0.0048 | −3.4138 | 0.0007 |

TC | 0.0003 | 0.0001 | 4.2827 | 0.0000 |

MCPD | −0.000019 | 0.0009 | −0.0205 | 0.9837 |

D | 0.011698 | 0.0006 | −20.6706 | 0.0000 |

Name/References | Constant Coefficients | Performance Indices | |||||
---|---|---|---|---|---|---|---|

Training | Testing | ||||||

K | B | n | RMSE | R^{2} | RMSE | R^{2} | |

USBM | 66.676 | 0.902 | - | 2.369 | 0.467 | 0.918 | 0.630 |

L-K | 1.567 | 0.220 | - | 2.370 | 0.062 | 1.405 | 0.096 |

A-H | 211.910 | 1.034 | - | 2.328 | 0.513 | 0.855 | 0.673 |

IS | 2.313 | 0.346 | - | 3.213 | 0.150 | 1.324 | 0.223 |

CMRI | 85.482 | - | 0.478 | 2.318 | 0.471 | 0.890 | 0.622 |

ccp_alpha | Training | Testing | ||
---|---|---|---|---|

RMSE | R^{2} | RMSE | R^{2} | |

0.001 | 0.016 | 0.890 | 1.135 | 0.733 |

0.002 | 0.145 | 0.881 | 1.235 | 0.707 |

0.003 | 0.169 | 0.860 | 1.262 | 0.701 |

0.004 | 0.263 | 0.858 | 1.278 | 0.693 |

0.005 | 0.384 | 0.854 | 1.284 | 0.688 |

0.006 | 0.454 | 0.851 | 1.284 | 0.680 |

0.007 | 0.484 | 0.844 | 1.284 | 0.690 |

0.008 | 0.483 | 0.845 | 1.270 | 0.690 |

0.009 | 0.531 | 0.834 | 1.170 | 0.680 |

0.01 | 0.524 | 0.834 | 1.139 | 0.744 |

0.011 | 0.536 | 0.833 | 1.141 | 0.742 |

0.012 | 0.550 | 0.813 | 1.268 | 0.694 |

0.013 | 0.584 | 0.823 | 1.273 | 0.692 |

0.014 | 0.599 | 0.821 | 1.212 | 0.716 |

0.015 | 0.623 | 0.816 | 1.258 | 0.698 |

0.016 | 0.664 | 0.809 | 1.280 | 0.689 |

0.017 | 0.681 | 0.805 | 1.283 | 0.689 |

0.018 | 0.690 | 0.804 | 1.291 | 0.704 |

0.019 | 0.690 | 0.804 | 1.243 | 0.704 |

0.02 | 0.709 | 0.800 | 1.247 | 0.702 |

0.021 | 0.730 | 0.796 | 1.294 | 0.684 |

0.022 | 0.740 | 0.794 | 1.295 | 0.683 |

0.023 | 0.740 | 0.794 | 1.247 | 0.702 |

0.024 | 0.773 | 0.787 | 1.295 | 0.684 |

0.025 | 0.796 | 0.782 | 1.308 | 0.678 |

c | Training | Testing | c | Training | Testing | ||||
---|---|---|---|---|---|---|---|---|---|

RMSE | R^{2} | RMSE | R^{2} | RMSE | R^{2} | RMSE | R^{2} | ||

1 | 2.2819 | 0.4879 | 1.619 | 0.6739 | 120 | 0.7502 | 0.9446 | 1.0437 | 0.8644 |

2 | 2.0996 | 0.5664 | 1.4207 | 0.7489 | 128 | 0.7367 | 0.9462 | 1.0424 | 0.8648 |

4 | 1.8943 | 0.647 | 1.2273 | 0.8126 | 130 | 0.7367 | 0.9466 | 1.0423 | 0.8648 |

8 | 1.6598 | 0.729 | 1.0635 | 0.8592 | 140 | 0.7247 | 0.9483 | 1.0441 | 0.8643 |

10 | 1.5724 | 0.7568 | 1.0266 | 0.8688 | 150 | 0.7169 | 0.9494 | 1.0488 | 0.8631 |

16 | 1.3689 | 0.8157 | 0.9896 | 0.8781 | 160 | 0.7054 | 0.951 | 1.0555 | 0.8614 |

20 | 1.25 | 0.8452 | 0.9966 | 0.8764 | 170 | 0.6945 | 0.9525 | 1.0608 | 0.86 |

30 | 1.0378 | 0.894 | 0.9967 | 0.8764 | 180 | 0.6844 | 0.9539 | 1.0642 | 0.8591 |

32 | 1.0047 | 0.9007 | 0.9981 | 0.876 | 190 | 0.6749 | 0.9551 | 1.0665 | 0.8584 |

40 | 0.9122 | 0.9181 | 0.9931 | 0.8773 | 200 | 0.6666 | 0.9562 | 1.0685 | 0.8579 |

50 | 0.8694 | 0.9256 | 0.9964 | 0.8764 | 300 | 0.6064 | 0.9638 | 1.0894 | 0.8523 |

60 | 0.8445 | 0.9298 | 1.0073 | 0.8737 | 500 | 0.5691 | 0.9681 | 1.1134 | 0.8457 |

64 | 0.8366 | 0.9311 | 1.0097 | 0.8731 | 1000 | 0.533 | 0.972 | 1.166 | 0.8308 |

70 | 0.8267 | 0.9327 | 1.0146 | 0.8719 | 5000 | 0.4513 | 0.9799 | 1.3085 | 0.7869 |

80 | 0.8115 | 0.9352 | 1.0289 | 0.8682 | 10000 | 0.4189 | 0.9827 | 1.3823 | 0.7622 |

90 | 0.7951 | 0.9378 | 1.0344 | 0.8668 | 50000 | 0.347 | 0.9881 | 2.0686 | 0.4677 |

100 | 0.7789 | 0.94033 | 1.0394 | 0.8656 | 100000 | 0.3258 | 0.9895 | 2.786 | 0.3681 |

110 | 0.7643 | 0.9425 | 1.0431 | 0.8646 |

**Table 9.**Performance metrics of different MARS models with varying values of minispan alpha and endspan of alpha.

Minispan/Endspan Alpha | Training | Testing | ||
---|---|---|---|---|

RMSE | R^{2} | RMSE | R^{2} | |

0.01 | 0.413 | 0.935 | 0.601 | 0.875 |

0.05 | 0.463 | 0.927 | 0.227 | 0.951 |

0.1 | 0.439 | 0.931 | 0.476 | 0.905 |

0.15 | 0.440 | 0.931 | 0.523 | 0.894 |

0.2 | 0.440 | 0.931 | 0.523 | 0.894 |

0.25 | 0.468 | 0.926 | 0.559 | 0.886 |

0.3 | 0.468 | 0.926 | 0.559 | 0.886 |

0.35 | 0.468 | 0.926 | 0.559 | 0.886 |

0.4 | 0.468 | 0.926 | 0.559 | 0.886 |

0.45 | 0.440 | 0.931 | 0.499 | 0.899 |

0.5 | 0.440 | 0.931 | 0.499 | 0.899 |

0.55 | 0.440 | 0.931 | 0.499 | 0.899 |

0.6 | 0.425 | 0.933 | 0.577 | 0.881 |

0.65 | 0.457 | 0.928 | 0.494 | 0.901 |

0.7 | 0.430 | 0.932 | 0.599 | 0.876 |

0.75 | 0.456 | 0.928 | 0.583 | 0.880 |

0.8 | 0.456 | 0.928 | 0.583 | 0.880 |

0.85 | 0.456 | 0.928 | 0.583 | 0.880 |

0.9 | 0.488 | 0.923 | 0.538 | 0.891 |

0.95 | 0.488 | 0.923 | 0.538 | 0.891 |

1 | 0.334 | 0.947 | 0.628 | 0.869 |

Basis Function $\mathit{B}\mathit{F}\left(\mathit{x}\right)$ | Co-Efficient $\left({\mathit{\beta}}_{\mathit{n}}\right)$ | Basis Function $\mathit{B}\mathit{F}\left(\mathit{x}\right)$ | Co-Efficient $\left({\mathit{\beta}}_{\mathit{n}}\right)$ |
---|---|---|---|

Intercept (${\beta}_{0}$) | 1.960120000 | BF17 = h(145−NH)*B*h(341−D) | −0.000145193 |

BF1 = h(D−341) | −0.002770310 | BF18 = SL*TC*h(1130−TC) | −0.000002270 |

BF2 = h(S−7.5)*h(341−D) | 0.402890000 | BF19 = MCPD*h(408−D)*h(156.25−CPH) | 0.000001079 |

BF3 = h(10000−TC)*h(341−D) | 0.000003126 | BF20 = h(NH−145)*B*h(12.25−HD) | 0.001295930 |

BF4 = D*h(341−D) | −0.000149723 | BF21 = h(HDM−260)*h(CPH−156.25) | 0.001648200 |

BF5 = TC*h(10000−TC)*h(341−D) | 0.000000001 | BF22 = h(408−D) | −0.023060800 |

BF6 = B*h(341−D) | 0.018892700 | BF23 = D*D*h(341−D) | 0.000000465 |

BF7 = MCPD*h(7.5−S)*h(341−D) | −0.000095901 | BF24 = h(S−7.5)*HDM*h(12.25−HD) | −0.002203750 |

BF8 = MCPD*B*h(341−D) | −0.000123622 | BF25 = h(7.5−S)*HDM*h(12.25−HD) | −0.000569359 |

BF9 = S*h(10000−TC)*h(341−D) | 0.000001058 | BF26 = HDM*B*h(341−D) | 0.000183443 |

BF10 = h(408−D)*h(156.25−CPH) | −0.000181658 | BF27 = SL*HDM*h(12.25−HD) | 0.000580015 |

BF11 = HD*h(408−D)*h(156.25−CPH) | 0.000017127 | BF28 = MCPD*h(341−D) | 0.000683227 |

BF12 = h(145−NH)*h(408−D)*h(156.25−CPH) | 0.000002181 | BF29 = SL*B*h(341−D) | −0.002187630 |

BF13 = TC*h(1130−TC) | 0.000008631 | BF30 = B*B*h(341−D) | −0.003844060 |

BF14 = HDM*h(1130−TC) | −0.000025563 | BF31 = HDM*h(10000−TC)*h(341−D) | −0.000000028 |

BF15 = HDM*HDM*h(1130−TC) | 0.000000146 | BF32 = HDM*D*h(341−D) | −0.000001313 |

BF16 = h(NH−145)*B*h(341−D) | −0.001396160 | $PPV={\beta}_{0}+{\displaystyle \sum}_{n=1}^{n=N}{\beta}_{n}BF\left(x\right)$ | |

Resulting expression |

Model | Training | Testing | ||
---|---|---|---|---|

RMSE | R^{2} | RMSE | R^{2} | |

USBM | 2.369 | 0.489 | 0.918 | 0.630 |

Langefors–Kihlstrom | 2.370 | 0.001 | 1.405 | 0.096 |

Ambraseys–Hendron | 2.328 | 0.493 | 0.855 | 0.673 |

ISI | 3.213 | 0.144 | 1.324 | 0.223 |

CMRI predictor | 2.318 | 0.482 | 0.890 | 0.621 |

MLR | 2.503 | 0.384 | 1.095 | 0.400 |

CART | 0.524 | 0.834 | 1.138 | 0.744 |

SVR | 1.005 | 0.90 | 0.998 | 0.876 |

MARS | 0.463 | 0.927 | 0.227 | 0.951 |

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**MDPI and ACS Style**

Komadja, G.C.; Rana, A.; Glodji, L.A.; Anye, V.; Jadaun, G.; Onwualu, P.A.; Sawmliana, C.
Assessing Ground Vibration Caused by Rock Blasting in Surface Mines Using Machine-Learning Approaches: A Comparison of CART, SVR and MARS. *Sustainability* **2022**, *14*, 11060.
https://doi.org/10.3390/su141711060

**AMA Style**

Komadja GC, Rana A, Glodji LA, Anye V, Jadaun G, Onwualu PA, Sawmliana C.
Assessing Ground Vibration Caused by Rock Blasting in Surface Mines Using Machine-Learning Approaches: A Comparison of CART, SVR and MARS. *Sustainability*. 2022; 14(17):11060.
https://doi.org/10.3390/su141711060

**Chicago/Turabian Style**

Komadja, Gbétoglo Charles, Aditya Rana, Luc Adissin Glodji, Vitalis Anye, Gajendra Jadaun, Peter Azikiwe Onwualu, and Chhangte Sawmliana.
2022. "Assessing Ground Vibration Caused by Rock Blasting in Surface Mines Using Machine-Learning Approaches: A Comparison of CART, SVR and MARS" *Sustainability* 14, no. 17: 11060.
https://doi.org/10.3390/su141711060