# Predicting Rock Brittleness Using a Robust Evolutionary Programming Paradigm and Regression-Based Feature Selection Model

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## Abstract

**:**

_{A}= 0.9744) for modeling BI, followed by LWLR (R = 0.9490, RMSE = 0.6607, and I

_{A}= 0.9400), BRT (R = 0.9433, RMSE = 0.6875, and I

_{A}= 0.9324), and KStar (R = 0.9310, RMSE = 0.7933, and I

_{A}= 0.9095), respectively. In addition, the sensitivity analysis demonstrated that the dry density factor demonstrated the most effective prediction of BI.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.1.1. Field Investigation

_{p}), point load strength index (Is

_{50}), dry density (D), and Schmidt hammer rebound number (R

_{n}). In Figure 1 and Figure 2, BTS and UCS tests were conducted on the samples and their failures, respectively.

_{nor}is the normalized value and x

_{max}, x

_{min}, and x are the maximum, minimum, and original values of the modeling dataset, respectively.

#### 2.1.2. Feature Selection Process

^{2}, Mallows coefficient (Cp) [39], Akaike (AIC) [40], and Amemiya (PC) [41]) have been computed for choosing the best input combination [38] (see Table 2). The possible tree combination demonstrates that the last case includes all input parameters and has the highest R

^{2}(0.817) and lowest Mallows, Akaike, and Amemiya (MSE = 0.463, Cp = 5 AIC = −60.552, and PC = 0.21); as such, this case can be identified as the best combination for modeling BI. Thus, the functional relationship between the chosen features and target can be expressed as follows:

#### 2.2. Methods

#### 2.2.1. Linear Genetic Programming (LGP)

**Initialization**: Creating the initial population randomly (programs), and then calculating the fitness function of each program.**Main operators**:- (1)
**Tournament selection**: This operator randomly selects several individuals from the population. Two individuals with the best fitness functions are chosen from these individuals, and two others as the worst solutions [43].- (2)
**Crossover operator**: This operator is applied to combine some elements of the best solutions with each other to create two new solutions (individuals).- (3)
**Mutation operator**: Mutation is used to create two new individuals by transforming each of the best solutions.

**Elitist mechanism**: The worst solutions are replaced with transformed solutions based on this mechanism.

#### 2.2.2. Local Weighted Linear Regression (LWLR)

_{mk}is a dependent variable that can be calculated based on at least two independent variables (x

_{k}). α is the regression coefficient calculated by the least-squares (LS) method, M is the number of data, and ε is the random error.

_{o}is the observed data, and z is the data obtained from the model. The above equation can also be expressed in the form of the following matrix:

_{i}− x

_{k}) is the difference between point i and k [49]. It should be mentioned that the main setting parameter of LWLR model can be optimized by a trial and error procedure.

#### 2.2.3. KStar Model

#### 2.2.4. Bootstrap Aggregate (Bagged) Regression Tree (BRT)

## 3. Statistical Criteria for Evaluation of Models

- Correlation coefficient (R) can be expressed as

- 2.
- Root mean square error (RMSE) can be expressed as

- 3.
- Mean absolute percentage error is defined as

- 4.
- Scatter Index can be expressed as

- 5.
- Willmott’s agreement Index [49] can be expressed as

## 4. Results and Discussion

_{n}, V

_{p}, D, and Is

_{50}. Also, two lazy machine learning models (namely LWLR and KStar) and a tree decision-based model (BRT) were measured to evaluate the outcome of the LGP approach. Figure 6 depicts the regression tree constructed from the BRT model, in which the terminal nodes or leafs identify the response of prediction. Table 4 presents the modeling results obtained by all models in the training and testing phases. The quantitative results in the training phase indicate that the KStar model (R = 0.9984, RMSE = 0.0865, MAPE = 0.2564, and I

_{A}= 0.9992) is superior to the BRT (R = 0.9459, RMSE = 0.5297, and MAPE = 3.1569), LWLR (R = 0.9252, RMSE = 0.5960, and MAPE = 3.4088), and LGP (R = 0.9248, RMSE = 0.5867, and MAPE = 3.6279) models. Testing results show that the LGP approach exhibits the best efficiency for BI prediction by having the highest correlation coefficient (R = 0.9529) and lowest metrics error (RMSE = 0.4838 and MAPE = 3.2155), followed by LWLR (R = 0.9490, RMSE = 0.6607, and MAPE = 4.1549), BRT (R = 0.9433, RMSE = 0.6875, and MAPE = 4.3884), and KStar (R = 0.9310, RMSE = 0.9733, and MAPE = 5.0573), respectively. A scatter plot of each model, as a powerful graphical tool, is depicted in Figure 7 for comparison between predicted and observed values of BI. Careful examination of the scatters indicates that the LGP approach—due to the closest distribution of predicted points to the 1:1 line—demonstrates better performance than the other AI methods for whole data. The LWLR and BRT models, with acceptable accuracy and similar predictive performance, are ranked as the second and third best models, respectively. KStar, despite the remarkable performance in the training phase (R = 0.9984), is identified as the weakest method due to the highest dispersion of testing predicted points.

^{2}ranged between 0.851 and 0.932. In another study, Koopialipoor et al. [25] predicted BI through a combination of ANN and firefly algorithm, yielding prediction results with an R

^{2}of 0.896. In the present study, BI has been predicted with better performance (R

^{2}of 0.953) from the LGP model. This indicates the effectiveness of the model proposed in this study compared to aforementioned models used in the literature. According to the objectives of this study, the uncertainty of the data has not been investigated. Given great importance, uncertainty of data and results of machine learning-based methods could be considered as the subject of future research. Also, the models presented in the current study generally suffer from a lack of laboratory data. Therefore, in the future, it is necessary to examine the accuracy of presented methods with a greater number of datasets.

## 5. Sensitivity Analysis

_{p}(R = 0.9163 and RMSE = 0.7944) ranks second, followed by Is

_{50}(R = 0.9169 and RMSE = 0.7861) and R

_{n}(R = 0.9273 and RMSE = 0.6959). A spider plot based on the six statistical criteria for all combining inputs is displayed in Figure 12. According to this figure, the combination with eliminating the dry density variable (i.e., all-dry density), showing the lowest R and I

_{A}and highest RMSE and MAPE, has the greatest impact on the accuracy of predicting BI. It should be mentioned that some feature selection methods such as Boruta-random forest can be utilized to specify the influential parameters, which has great ability to capture the non-linear interaction between the predictors and target. This aim can be considered as an alternative of classical sensitivity analysis.

## 6. Conclusions

_{p}, Is

_{50}, D, and R

_{n}) and BI as the output parameter. In the modeling processes, 64 and 21 datasets, respectively, were used for training and testing phases. Finally, the models’ accuracy was compared using several statistical criteria such as R and RSME. The findings of this study can be summarized as follows:

- Based on the results, all developed models’ performance capacity was suitable and acceptable. Accordingly, all proposed models can be used with confidence for future research on predictions of other issues in the field of rock mechanics.
- Among the proposed models, the KStar (R = 0.9984 and RMSE = 0.0865) model predicted BI with the best performance in the training phase, while the best performance for the testing phase was achieved by the LGP (R = 0.9529 and RMSE = 0.4838) model. In addition, both LWLR (R = 0.9490 and RMSE = 0.6607) and BRT (R = 0.9433 and RMSE = 0.6875), ranking second and third, respectively, lead to desired results for modeling BI values.
- The authors recommend increasing the accuracy of BI modeling as a possible future study, examining the ensemble of stacked models to integrate the advantages of standalone data-driven models.
- Sensitivity analysis demonstrated that dry density (D) was the most influential parameter with respect to BI.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Failure of a sample under a BTS test [3].

**Figure 10.**Box plots for the relative deviation (%) distribution of all predictive models in testing and training.

**Figure 11.**The cumulative frequency percentage versus the relative absolute error (%) for LWLR, BRT, KStar, and LGP models for the testing dataset.

Parameters | ${\mathbf{R}}_{\mathbf{n}}$ | V_{p} (m/s) | Dry Density (g/cm^{3}) | Is_{50} (MPa) | BI |
---|---|---|---|---|---|

Minimum | 20 | 2910 | 2.38 | 0.8722 | 10.12 |

Maximum | 59 | 7943 | 2.75 | 6.59 | 16.75 |

Mean | 37.16 | 4975 | 2.536 | 3.441 | 12.61 |

Std. Deviation | 10.12 | 1199 | 0.079 | 1.118 | 1.554 |

Range | 39 | 5033 | 0.37 | 5.718 | 6.626 |

Skewness | 0.3951 | 0.2449 | 0.1161 | 0.1294 | 0.7339 |

Kurtosis | −0.76 | −0.605 | −0.3473 | 0.3369 | 0.2216 |

Number of Variables | Variables | MSE | R^{2} | Adjusted R^{2} | Mallows’ Cp | Akaike’s AIC | Amemiya’s PC |
---|---|---|---|---|---|---|---|

2 | V_{p}/D | 0.652 | 0.736 | 0.730 | 36.387 | −33.419 | 0.276 |

3 | ${\mathrm{V}}_{\mathrm{p}},\mathrm{D},{\mathrm{I}}_{\mathrm{S}50}$ | 0.530 | 0.788 | 0.781 | 15.611 | −50.109 | 0.227 |

4 | ${\mathrm{R}}_{\mathrm{n}},{\mathrm{V}}_{\mathrm{p}},\mathrm{D},{\mathrm{I}}_{\mathrm{S}50}$ | 0.463 | 0.817 | 0.808 | 5.000 | −60.552 | 0.201 |

Models | Setting of Parameter | |
---|---|---|

LGP | Function set | +, −, ×, ÷, √, power, sin, cos |

Population size | 300 | |

Mutation frequency % | 85 | |

Crossover frequency % | 50 | |

Number of replication | 10 | |

Block mutation rate % | 20 | |

Instruction mutation rate % | 20 | |

Instruction data mutation rate % | 60 | |

Homologous crossover % | 90 | |

Program size | 64–256 | |

LWLR | • µ = 4 | |

KStar | • Global blend = 30 | |

BRT | • Function: “Bag”, Learning cycles = 50, MinLeafSize = 1 |

Metrics | LGP | K-Star | BRT | LWLR | |
---|---|---|---|---|---|

Training | R | 0.9248 | 0.9984 | 0.9459 | 0.9252 |

RMSE | 0.5867 | 0.0865 | 0.5297 | 0.5960 | |

MAPE% | 3.6279 | 0.2564 | 3.1569 | 3.4088 | |

SI | 0.0463 | 0.0068 | 0.0418 | 0.0470 | |

I_{A} | 0.9560 | 0.9992 | 0.9628 | 0.9531 | |

St.D | 1.3339 | 1.5195 | 1.2640 | 1.2828 | |

Testing | R | 0.9529 | 0.9310 | 0.9433 | 0.9490 |

RMSE | 0.4838 | 0.7933 | 0.6875 | 0.6607 | |

MAPE% | 3.2155 | 5.0573 | 4.3884 | 4.1549 | |

SI | 0.0389 | 0.0638 | 0.0553 | 0.0532 | |

I_{A} | 0.9744 | 0.9095 | 0.9324 | 0.9400 | |

St.D | 1.5059 | 1.0861 | 1.1116 | 1.1686 |

Metric | All-R_{n} | All-V_{p} | All-Dry Density | All-Is_{50} | All |
---|---|---|---|---|---|

R | 0.9273 | 0.9163 | 0.9081 | 0.9169 | 0.9433 |

RMSE | 0.6959 | 0.7944 | 0.8027 | 0.7861 | 0.6875 |

MAPE | 4.4592 | 5.1695 | 5.4642 | 5.0433 | 4.3884 |

SI | 0.0560 | 0.0639 | 0.0646 | 0.0633 | 0.0553 |

I_{A} | 0.9318 | 0.9018 | 0.9004 | 0.9049 | 0.9324 |

St.D | 1.6277 | 1.6277 | 1.6277 | 1.6277 | 1.6277 |

Rank | 4.0000 | 3.0000 | 1.0000 | 2.0000 | - |

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Jamei, M.; Mohammed, A.S.; Ahmadianfar, I.; Sabri, M.M.S.; Karbasi, M.; Hasanipanah, M.
Predicting Rock Brittleness Using a Robust Evolutionary Programming Paradigm and Regression-Based Feature Selection Model. *Appl. Sci.* **2022**, *12*, 7101.
https://doi.org/10.3390/app12147101

**AMA Style**

Jamei M, Mohammed AS, Ahmadianfar I, Sabri MMS, Karbasi M, Hasanipanah M.
Predicting Rock Brittleness Using a Robust Evolutionary Programming Paradigm and Regression-Based Feature Selection Model. *Applied Sciences*. 2022; 12(14):7101.
https://doi.org/10.3390/app12147101

**Chicago/Turabian Style**

Jamei, Mehdi, Ahmed Salih Mohammed, Iman Ahmadianfar, Mohanad Muayad Sabri Sabri, Masoud Karbasi, and Mahdi Hasanipanah.
2022. "Predicting Rock Brittleness Using a Robust Evolutionary Programming Paradigm and Regression-Based Feature Selection Model" *Applied Sciences* 12, no. 14: 7101.
https://doi.org/10.3390/app12147101