# Investigating the Applications of Machine Learning Techniques to Predict the Rock Brittleness Index

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## Abstract

**:**

^{2}) of 0.971 provides higher performance capacity for prediction of the rock BI compared to KNN model with R

^{2}of 0.807 and ANN model with R

^{2}of 0.860. The results of this study suggest a practical use of the machine learning models in solving problems related to rock mechanics specially rock brittleness index.

## 1. Introduction

## 2. Methodology

#### 2.1. Models Developed

#### 2.2. Data and Case Study

^{3}/s. In order to excavate the tunnel, three different TBMs were used for about 35 km of the tunnel. The remainder of the tunnel was excavated using the drilling and blasting method. The geological units include granite, metamorphic and some sedimentary rocks; though, most of the rocks excavated with the abovementioned method is comprised of granite. Many geotechnical and geological investigations were conducted in the tunnel to collect rock block samples for testing. Finally, in multiple locations of TBMs site, more than 100 granite block samples were obtained from the tunnel face. A robust procedure from the International Society for Rock Mechanics [61] was followed for preparing the samples to test. Several lab tests were conducted on the samples, including density (in dry condition), Schmidt hammer rebound number (R

_{n}), uniaxial compression strength (σ

_{c}), tensile strength (σ

_{t}), point load index (Is

_{50}), and p-wave velocity (V

_{p}). In this study, the BI values were calculated according to the following equation [62]:

_{c}and σ

_{t}are the uniaxial compression strength and tensile strength, respectively.

^{3}, 40.5, 3.6 MPa, and 15.5 were obtained for V

_{p}, D, R

_{n}, Is

_{50}and BI, respectively. In the next section, modeling procedure in approximating BI as a function form of f (V

_{p}, D, R

_{n}, and Is

_{50}) and the obtained results will be presented in detail.

## 3. Modelling Process and Results

#### 3.1. Evaluation of the Developed Models

^{2}), root mean square error (RMSE), mean absolute error (MAE), variance account for (VAF), and a20-index. The formula that was used for calculating the mentioned performance indices are presented in Equations (2)–(6). This study also employed an easy to understand ranking system which ranked each model developed using the above-mentioned performance criteria for both training and testing stages. For each criteria, the ranking system first sorted the models based on their obtained values, then assigned the highest rank (5) to the best value and the lowest rank (1) to the worst value. Final rank of each model was calculated through summing the ranking values for both training and testing stages (Equation (6)):

^{2}, RMSE, and a20-index. The ANN also achieved the second-best rank for VAF. Based on these discussions and rank values, three models of RF, KNN, and ANN were selected to be discussed in more details in the following sections.

#### 3.2. Conqueror Models

#### 3.2.1. Random Forest Model

_{p}, D, R

_{n}, and Is

_{50}to predict the rock BI. The present study employed several parameters to develop the RF model. After trial and error procedure, the number of models to build was set as 100, the sample size was set as 0.95, the maximum number of nodes was set as 10,000, and maximum tree depth and minimum child node size were set as 10 and 2, respectively. Predicted BI values by RF, along with their actual values for training and testing datasets, are displayed in Figure 6. The obtained R

^{2}values of 0.89 and 0.75 for train and test stages of RF model, respectively revealed a high and suitable accuracy level of train and test stages. In addition, the RF model identified the importance of the input variables (Figure 7). As can be seen, the R

_{n}with importance of 0.37 was identified as the most important variable and followed by V

_{p}with importance of 0.35 and Is

_{50}with importance of 0.29. It is noteworthy to mention that the RF model did not consider D as an important factor.

#### 3.2.2. ANN Model

_{p}, D, R

_{n}, and Is

_{50}for predicting the rock BI. Here, several parameters have been used to develop the ANN model. The type of neural network model was multilayer perceptron. The study used “mean” as the default combing rule for our continuous target. Number of component models for boosting and/or bagging was set as 10. To avoid over-fitting, the over-fit prevention set was set as 30%. Different values were examined in order to determine the number of hidden neurons and in the final model, a number of 4 hidden neurons was used to predict BI. Figure 8 shows the suggested architecture of the ANN model with four input neurons, four hidden neurons and one output neuron in predicting BI of the rock. In addition, the predicted BI values by ANN, along with their actual values for train and test datasets, are displayed in Figure 9. According to obtained results of this section, R

^{2}values of 0.75 and 0.85 for train and test stages, respectively showed that the ANN model is able to provide acceptable level of accuracy specially in testing datasets for estimation of the BI. ANN is able to determine the importance values of the use inputs in the system (Figure 10). As a result, R

_{n}and V

_{p}are the most important and least important parameters on the BI which results of R

_{n}is same as the RF analysis part.

#### 3.2.3. KNN Model

^{2}of 0.81 and 0.84 for train and test stages, respectively, in fact, the KNN model is able to offer a balance range for these stages compared to RF and ANN. Figure 12 shows a suggested structure of the KNN predictive model in predicting BI.

_{n}was identified as the most important variable and followed by V

_{p}, D, and Is

_{50}, respectively. It should be noted that R

_{n}was introduced by all RF, ANN and KNN model as the most influential factor on rock BI.

## 4. Validation of the Selected Models

^{2}results (0.971, 0.860, and 0.807) and VAF results of (96.852, 85.633, and 80.642 %) were obtained for RF, ANN and KNN models, respectively in validation stage which indicate that RF model is better than the other 2 models. In terms of system error, RF model with RMSE of 0.62 and MAE of 0.46 received lower amount of error compared to ANN and KNN models. Figure 14 shows the measured and predicted BI values for the RF, ANN, and KNN models in validation phase. In addition, Figure 15 depicts predicted BI values by RF, ANN and KNN together with their measured BI for all 15 data samples assigned for validation stage. As it can be seen from these two figures, the BI values by RF model are closer to the measured BI values in comparison with the KNN and ANN models.

## 5. Discussion and Conclusions

_{n}as the most important factor for predicting the BI. The KNN and ANN considered D as an important predictor, while the RF did not. This can be explained by the fact that the data of D diverged from the average value. It also showed that the RF is intolerable to the dispersion of data points in a data series around the mean.

^{2}of 0.971 is more capable to predict rock BI compared to KNN model with R

^{2}of 0.807 and ANN model with R

^{2}of 0.860. This indicates that all models can be used for similar conditions in the future. More specifically, this research suggests to use RF and KNN models (or each of them) by the other researcher or designers in order to predict rock BI in design stage of geotechnical projects.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 15.**Predicted BI values by RF, ANN and KNN together with their measured BI for all 15 data samples.

**Table 1.**The Range, Mean, Unit, Category and Symbol of Inputs and Output Parameters in Predicting BI of the Rock Samples.

Parameter | Symbol | Unit | Category | Min | Max | Mean |
---|---|---|---|---|---|---|

P-wave velocity | V_{p} | m/s | Input | 2870 | 7702 | 5491.6 |

Density | D | g/cm^{3} | Input | 2.37 | 2.79 | 2.59 |

Schmidt hammer rebound number | R_{n} | - | Input | 20 | 61 | 40.5 |

Point load strength | Is_{50} | MPa | Input | 0.89 | 7.1 | 3.6 |

Brittleness index | BI | - | Output | 8.90 | 24.01 | 15.5 |

Performance Index | RF | CHAID | KNN | SVM | ANN | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

TR | TE | TR | TE | TR | TE | TR | TE | TR | TE | |||||||||||

Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | |

R^{2} | 0.89 | 5 | 0.75 | 2 | 0.77 | 3 | 0.74 | 1 | 0.81 | 4 | 0.84 | 4 | 0.73 | 1 | 0.84 | 3 | 0.75 | 2 | 0.85 | 5 |

RMSE | 1.08 | 5 | 1.75 | 2 | 1.39 | 3 | 1.8 | 1 | 1.33 | 4 | 1.44 | 3 | 1.57 | 1 | 1.42 | 4 | 1.50 | 2 | 1.39 | 5 |

VAF (%) | 86.84 | 5 | 80.5 | 2 | 78.31 | 3 | 75.1 | 1 | 80.62 | 4 | 86.1 | 3 | 72.62 | 1 | 87.5 | 5 | 74.60 | 2 | 87 | 4 |

MAE | 0.91 | 5 | 1.36 | 1 | 1.14 | 4 | 1.35 | 2 | 1.16 | 3 | 1.15 | 4 | 1.32 | 1 | 1.06 | 5 | 1.29 | 2 | 1.16 | 3 |

a20-index | 1.00 | 5 | 0.91 | 2 | 0.95 | 2 | 0.84 | 1 | 0.99 | 4 | 0.97 | 4 | 0.91 | 1 | 0.96 | 3 | 0.97 | 3 | 0.97 | 5 |

Sum of the ranks | TR | 25 | TE | 9 | TR | 15 | TE | 6 | TR | 19 | TE | 18 | TR | 5 | TE | 20 | TR | 11 | TE | 22 |

Final rank | 34 | 21 | 37 | 25 | 33 |

^{2}= 1; Perfect RMSE = 0; Perfect VAF = 100%; Perfect MAE = 0; a20-index = 1. Training dataset = TR; Testing dataset = TE

Performance Index | RF | ANN | KNN |
---|---|---|---|

R^{2} | 0.971 | 0.860 | 0.807 |

RMSE | 0.62 | 1.22 | 1.42 |

VAF (%) | 96.852 | 85.633 | 80.64 |

MAE | 0.46 | 0.99 | 1.14 |

a20-index | 1.00 | 1.00 | 1.00 |

^{2}= 1; Perfect RMSE = 0; Perfect VAF = 100%; Perfect MAE = 0; a20-index = 1. Training dataset = TR; Testing dataset = TE.

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**MDPI and ACS Style**

Sun, D.; Lonbani, M.; Askarian, B.; Jahed Armaghani, D.; Tarinejad, R.; Thai Pham, B.; Huynh, V.V.
Investigating the Applications of Machine Learning Techniques to Predict the Rock Brittleness Index. *Appl. Sci.* **2020**, *10*, 1691.
https://doi.org/10.3390/app10051691

**AMA Style**

Sun D, Lonbani M, Askarian B, Jahed Armaghani D, Tarinejad R, Thai Pham B, Huynh VV.
Investigating the Applications of Machine Learning Techniques to Predict the Rock Brittleness Index. *Applied Sciences*. 2020; 10(5):1691.
https://doi.org/10.3390/app10051691

**Chicago/Turabian Style**

Sun, Deliang, Mahshid Lonbani, Behnam Askarian, Danial Jahed Armaghani, Reza Tarinejad, Binh Thai Pham, and Van Van Huynh.
2020. "Investigating the Applications of Machine Learning Techniques to Predict the Rock Brittleness Index" *Applied Sciences* 10, no. 5: 1691.
https://doi.org/10.3390/app10051691