# Appraisal of Different Artificial Intelligence Techniques for the Prediction of Marble Strength

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## Abstract

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^{2}), Root Mean Square Error (RMSE), Mean Square Error (MSE), and Mean Absolute Error (MAE). The results show that the ANN model had the best performance indicators, with values of 0.9995, 0.2634, 0.0694, and 0.1642 for R

^{2}, RMSE, MSE, and MAE, respectively. However, the XG Boost algorithm model performance was also excellent and comparable to the ANN model. Therefore, these two models were proposed for predicting UCS effectively. The outcomes of this research provide a theoretical foundation for field professionals in predicting the strength parameters of rock for the effective and sustainable design of engineering structures

## 1. Introduction

^{2}0.99 than the MLRM. Manouchehrian et al. [48] used texture as an input variable in ANN and multivariate modeling for the prediction of UCS. They predicted UCS values effectively, and the performance of the ANNs was better compared to the multivariate statistics. Similarly, Torabi-Kaveh, Naseri, Abdi, Garavand [49,50] used various input variables such as porosity (η), P-wave (P

_{V}), and density (ρ) to predict UCS and E

_{S}using ANNs and ANFIS. Dehghan et al. [1] used ANNs and MLR to predict UCS and the Static Young modulus (Es) based on the P

_{V}, point load index, Schmidt hammer rebound number, and η as input variables. Due to the advancement in artificial intelligence, some of the latest and most effective algorithms have been developed to predict UCS. In this regard, Zhang et al. [51] created a Random Forest (RF) model based on the beetle antennae search (BAS) method for estimating the UCS of lightweight self-compacting concrete (LWSSC) with high precision and efficiency. Matin et al. [52] used just a few rock parameters and indices such as porosity (η), water content (Wc), Is (50), P-wave velocity (P

_{V}), and rebound numbers (Rns) that were determined using an RF model. Based on these variables, an effective model for predicting UCS was created. Suthar [53] used the M5 model tree, RF, ANN, Support Vector Machine (SVM), and Gaussian processes to forecast the UCS of stabilized pond ashes with lime and lime sludge (GPs). Wang et al. [54] developed an effective model for predicting UCS based on RF-selected variables. Ren et al. [55] created k-nearest neighbors (KNNs), naive Bayes, RF, ANN, and (SVM) as machine-learning (ML) methods to precisely predict rock UCS using multiple input parameters. Ghasemi et al. [56] built a tree-based method for predicting the UCS and Young Modulus (E) of carbonate rocks. They found that the applied method gives promising results. Saedi et al. [57] predicted UCS and E of migmatite rocks using ANNs and multivariate regression (MVR). They found that ANN and ANFIS show better prediction performance. Shahani et al. [58] suggested an XGBoost model for predicting the UCS and E of sedimentary rock that is still intact. To estimate the UCS and E of various rocks, Armaghani et al. [59] created a hybrid model based on an ANN and an imperialist competitive algorithm (ICA). This literature review provides insight into the use of different artificial intelligence techniques for predicting UCS. While ANNs have shown promising results, their prediction performance is still a matter of debate. Moreover, the literature highlights that there has been no analysis of input variables using statistical methods to identify the most suitable input variables, which could enhance the prediction performance of both artificial intelligence and statistical techniques. In addition, the latest artificial intelligence techniques, such as the XG Boost algorithm, Random Forest (RF), Elastic Net (EN), Lasso, and Ridge, have not been adequately explored for effective prediction of UCS. Therefore, accurate prediction of rock UCS is vital for ensuring the safe and efficient stability analysis of engineered structures in a rock mass environment.

## 2. Materials and Methods

#### 2.1. Design of Experimental Works

#### 2.2. Predictive Models

**XG Boost Algorithm**

_{j}denotes a jth tree prediction in the formula [63]. Figure 2 shows the XGBoost model structure [64].

**Random Forest**

**Support Vector Machine**

_{i}). The output neuron receives an addition of the buried layer’s neuron linear outputs. The output neuron is biased [70].

**Lasso Regression**

**Ridge Regression**

**Elastic Net**

**Artificial Neural Networks (ANNs)**

- $w={w}_{1},{w}_{2},{w}_{3},{w}_{4},\dots \dots ,{w}_{n}$;
- $x={x}_{1},{x}_{2},{x}_{3},{x}_{4},\dots \dots ,{x}_{n}$.

_{Calculated}) minus estimated values (V

_{Predicted}) of the network”. By increasing or decreasing the neuron’s weight, it is possible to reduce the error in this network to some extent. Equation (7) represents the inaccuracy of networks in their mathematical form.

**Code Development for ANNs using MATLAB**

#### 2.3. Data Analysis for Selecting the Most Appropriate Input Variables

#### 2.4. Performance Indicator

^{2}, Root Mean Square Error (RMSE), Mean Square Error (MSE), and Mean Absolute Error (MAE) were used to evaluate the prediction performance of the Artificial Neural Network (ANN), XG Boost algorithm, Random Forest Regression (RFR), Elastic Net (EN), Lasso, Support Vector Machine (SVM), and Ridge models. The following formulas as mentioned in Equations (9)–(12) were used [43,64]:

- RSS = sum of the square of the residual;
- TSS = total sum of the square.

^{th}observation;

^{th}observation;

## 3. Analysis of Results

#### 3.1. Model Hyperparameter Optimization

- In order to train the data set, the training data set needs to be divided into k folds.
- The (k˗-1) fold is used for training out of all k folds.
- The remaining last k-fold is used for validation.
- In order to train the model with specific hyperparameters, training data (k-1 folds) are used, and validation data are used as 1-fold. For each fold, the model’s performance is recorded.
- K-fold cross-validation refers to the process of repeating the steps above until each k-fold is used for validation purposes. That is why this process is known as “K-fold cross-validation”.
- After calculating each model score for each model in step d, the mean and standard deviation of the model performance are computed.
- It is necessary to repeat steps b to f for different values of the hyperparameters.
- The hyperparameters associated with the best mean and standard deviation of the model scores are then selected.
- Using the entire training data set, the model is trained, and its performance is evaluated on the basis of the test data set.

#### 3.2. Prediction of UCS using Artificial Neural Networks

^{2}value of 0.9995 between the predicted and measured UCS.

**Ridge Regression**

^{2}= 0.9790).

**Elastic Net**

^{2}= 0.9755).

**Lasso Regression**

^{2}= 0.9755).

**Support Vector Machine**

^{2}= 0.9573).

**Random Forest**

^{2}= 0.9949).

**XG Boost Algorithm**

^{2}= 0.9990).

^{2}, MAE, MSE, and RMSE for the ANN model were 0.999, 0.1428, 0.0782, and 0.2796, respectively, on the training data set, while they were 0.9995, 0.6420,0.0694, and 0.2634 on testing data, respectively, which shows that the performance of the ANN model is greater than all the other predictive models. For the XG Boost Regressor, the value of the performance indicator R

^{2}was 0.9989, MAE was 0.5694, MSE was 0.0782, and RMSE was 0.2796 for the training data set, while for the testing data set, the R

^{2}was 0.9990, MAE was 0.1145, MSE was 0.0694, and RMSE was 0.4162. For the Random Forest Regression, the performance indicator R

^{2}was 0.9943, MAE was 0.7176, MSE was 1.3294, and RMSE was 1.1530 for the training data set, while for the testing data set, the R

^{2}was 0.9949, MAE was 0.3555, MSE was 0.6584, and RMSE was 0.8114. For the Lasso, the performance indicators R

^{2}, MAE, MSE, and RMSE were 0.9887, 1.367, 3.0666, and 1.7512 for the training data set, respectively, while for the testing data set, the R

^{2}was 0.9755, MAE was 1.8918, MSE was 3.5788, and RMSE was 1.2555. For the Ridge model, the R

^{2}, MAE, MSE, and RMSE were 0.9876, 1.3906, 3.0492, and 1.7462, respectively, for the training data set, while for the testing data, the performance indicators R

^{2}, MAE, MSE, and RMSE were 0.979, 1.2149, 3.001, and 1.7347, respectively. For the Elastic net model, the performance indicator R

^{2}was 0.9887, MAE was 1.3751, MSE was 3.2071, and RMSE was 1.7908 for the training data set, while for the testing data set, the R

^{2}was 0.9755, MAE was 1.241, MSE was 3.6308, and RMSE was 1.9055. Similarly, for the Support Vector Machine, the R

^{2}, MAE, MSE, and RMSE were 0.9826, 9.4444, 187.2607, and 13.68, respectively, for the training data set, and for the testing data set, the value of R

^{2}, MAE, MSE, and RMSE were 0.9573, 6.5449, 111.4614, and 10.5575, respectively. According to Table 2, the ANN model had values of 0.9995, 0.2634, 0.0694, and 0.1642 for R

^{2}, RMSE, MSE, and MAE, respectively. This highlights that the ANNs model’s performance was better than that of any other prediction model. However, the hierarchy of the mentioned predictive models in terms of their efficacy based on the performance indicators in predicting the UCS can be ANN > XG Boost Regressor > SVR > Random Forest Regressor > Lasso > Elastic Net > Ridge.

## 4. Discussion

^{2}) of 0.9995, indicating a strong correlation between predicted and measured UCS values. This suggests that the ANN model can be used to predict UCS values accurately. Among the other models, the Random Forest Regression (RFR) performed well with an R

^{2}value of 0.9949. This suggests that RFR can also be used as an alternative method for predicting UCS values. The XG Boost algorithm also performed well, with an R

^{2}value of 0.9990, which is similar to the ANN model. The Ridge regression, Elastic Net, and Lasso regression models also showed good performance with R

^{2}values ranging from 0.9887 to 0.9886. However, their performance was slightly lower than that of the ANN, XG Boost, SVM, and RFR models. Overall, the analysis suggests that the ANN model, followed by XG Boost, SVM, and RFR are the best models for predicting UCS values, while Ridge regression, Elastic Net, and Lasso regression are also good alternatives. The SVM model may not be the best option for predicting UCS values. This study considers a small data set due to limited resources. In future studies, the authors will use the application of infrared radiation (IR) technology and AI together to avoid such a large parameter determination in the field as used in this study training. The IR and AI together will make the prediction more reliable and applicable. Moreover, in the future, the given 70 sample data set can be increased using the harmony search optimization algorithm [80].

## 5. Conclusions

^{2}), Root Mean Square Error (RMSE), Mean Square Error (MSE), and Mean Absolute Error (MAE). The results show that the performance indicators for the ANN were 0.9995, 0.2634, 0.0694, and 0.1642, respectively. The comparative analysis based on the performance indicators revealed that the ANN model has greater prediction efficacy compared to the other AI models; however, the ANN model gives approximately a similar performance as the XG Boost Regressor model. Furthermore, it was noticed that SVM, RFR, Ridge, Lasso, and Elastic Net models give acceptable prediction performance; however, they are less effective in performance than the ANN and XG Boost Regressor models when predicting UCS. Therefore, the ANN and XG Boost Regressor are recommended to be used as the most effective predictive models for the prediction of UCS. Since this research work was conducted using a limited number of rock samples, it would be beneficial to extend the data set in order to refine the findings. Additionally, since this study was focused on marble only, it would be necessary to carry out further fine-tuning of the models before applying them to any other type of rock mass environment to ensure the best possible results. Further research needs to be carried out to explore the applications of the various AI techniques for the effective prediction of the UCS. The outcomes of this research will provide a theoretical foundation for field professionals in the prediction of the strength parameters of rock for an effective and sustainable design of engineering structures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**A**) Afghanistan marble, (

**B**) Mardan (spin kala) marble, (

**C**) Chitral marble, (

**D**) Mohmand marble, (

**E**) Bunir Bumbpoha, (

**F**) Chitral marble, and (

**G**) Mohmand (super white) marble. (

**H**) Core drilling, (

**I**) core cutting, (

**J**) triaxial testing machine, (

**K**) Schmidt hammer apparatus, (

**L**) slake durability apparatus, (

**M**) sample dipped for 24 h, (

**N**) desiccator used to cool sample, (

**O**) volumetric cylinder used to find volume, (

**P**) sampling weight, (

**Q**) core samples, (

**R**) oven for drying samples, (

**S**) core failure after UCS testing, and (

**T**) sampling using a Schmidt hammer.

**Figure 4.**General Architecture of a Support Vector Machine [70].

**Figure 10.**ANN phases of training, validation, and testing and the regression coefficient of determination for UCS.

**Figure 12.**Ridge regression model with the coefficient of determination between the measured and predicted UCS.

**Figure 13.**Elastic Net model with the coefficient of determination between the measured and predicted UCS.

**Figure 14.**Lasso regression with the coefficient of determination between the measured and predicted UCS.

**Figure 15.**Support Vector Machine model with the coefficient of determination between the measured and predicted UCS.

**Figure 16.**Random Forest Regression with the coefficient of determination between the measured and predicted UCS.

**Figure 17.**XG Boost Model with the coefficient of determination between the measured and predicted UCS.

S.No | Input and Output | N total | Mean | Standard Deviation | Sum | Min | Median | Max |
---|---|---|---|---|---|---|---|---|

1 | bulk density (g/mL) | 70.00 | 2.73 | 0.27 | 191.34 | 2.12 | 2.69 | 3.53 |

2 | dry density (g/mL) | 70.00 | 2.67 | 0.24 | 187.16 | 2.12 | 2.65 | 3.61 |

3 | moisture content (MC (%)) | 70.00 | 0.36 | 0.19 | 25.46 | 0.00 | 0.35 | 0.99 |

4 | water absorption (%) | 70.00 | 0.36 | 0.24 | 25.28 | 0.00 | 0.34 | 1.20 |

5 | slake durability index (Id_{2}) | 70.00 | 97.08 | 3.21 | 6795.85 | 83.24 | 98.25 | 99.11 |

6 | rebound number (R) | 70.00 | 45.88 | 6.31 | 3211.57 | 34.70 | 44.82 | 64.14 |

7 | porosity (η) | 70.00 | 0.36 | 0.24 | 25.28 | 0.00 | 0.34 | 1.20 |

8 | void ratio (e) | 70.00 | 1.15 | 3.03 | 80.25 | 0.00 | 0.0034 | 0.012 |

9 | P-wave (km/s) | 70.00 | 4.74 | 0.20 | 331.52 | 4.43 | 4.70 | 5.49 |

10 | S-wave (km/s) | 70.00 | 3.02 | 0.01 | 211.14 | 2.98 | 3.02 | 3.03 |

11 | UCS (Mpa) | 70.00 | 52.17 | 12.10 | 3651.59 | 34.89 | 49.51 | 93.76 |

Output | Model | Parameters |
---|---|---|

UCS (Mpa) | Artificial Neural Network | Neuron = 48 |

XG Boost Regressor | learning_rate = 0.01, max_depth = 3, n_estimators = 100 | |

Support Vector Machine | n_split = 10, n_repeats = 5, random state = 42, C = 1 function = SVR (kernal ‘rbf’) | |

Random Forest Regression | n_split = 10, n_repeats = 5, random state = 42, max_depth = 3 | |

Lasso | Alpha = 0.01, n_split = 10, n_repeats = 5, random state = 42 | |

Elastic Net | Alpha = 0.01, l1_ratio = 0.95, n_split = 10, n_repeats = 5, random state = 42 | |

Ridge | Alpha = 0.1, n_split = 10, n_repeats = 5, random state = 42 |

S.no | Models | Training Accuracy | Testing Accuracy | ||||||
---|---|---|---|---|---|---|---|---|---|

R^{2} | MAE | MSE | RMSE | R^{2} | MAE | MSE | RMSE | ||

1 | Artificial Neural Network | 0.9990 | 0.1428 | 0.0782 | 0.2796 | 0.9995 | 0.1642 | 0.0694 | 0.2634 |

2 | XG Boost Regressor | 0.9989 | 0.5694 | 0.8664 | 0.9308 | 0.9990 | 0.1145 | 0.1732 | 0.4162 |

3 | Support Vector Machine | 0.9987 | 0.3649 | 0.3022 | 0.5498 | 0.9983 | 0.2891 | 0.2595 | 0.5094 |

4 | Random Forest Regression | 0.9943 | 0.7176 | 1.3294 | 1.1530 | 0.9949 | 0.3555 | 0.6584 | 0.8114 |

5 | Lasso | 0.9887 | 1.3670 | 3.0666 | 1.7512 | 0.9755 | 1.8918 | 3.5788 | 1.2555 |

6 | Elastic Net | 0.9887 | 1.3751 | 3.2071 | 1.7908 | 0.9755 | 1.2410 | 3.6308 | 1.9055 |

7 | Ridge | 0.9876 | 1.3906 | 3.0492 | 1.7462 | 0.9790 | 1.2149 | 3.0010 | 1.7347 |

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## Share and Cite

**MDPI and ACS Style**

Jan, M.S.; Hussain, S.; e Zahra, R.; Emad, M.Z.; Khan, N.M.; Rehman, Z.U.; Cao, K.; Alarifi, S.S.; Raza, S.; Sherin, S.;
et al. Appraisal of Different Artificial Intelligence Techniques for the Prediction of Marble Strength. *Sustainability* **2023**, *15*, 8835.
https://doi.org/10.3390/su15118835

**AMA Style**

Jan MS, Hussain S, e Zahra R, Emad MZ, Khan NM, Rehman ZU, Cao K, Alarifi SS, Raza S, Sherin S,
et al. Appraisal of Different Artificial Intelligence Techniques for the Prediction of Marble Strength. *Sustainability*. 2023; 15(11):8835.
https://doi.org/10.3390/su15118835

**Chicago/Turabian Style**

Jan, Muhammad Saqib, Sajjad Hussain, Rida e Zahra, Muhammad Zaka Emad, Naseer Muhammad Khan, Zahid Ur Rehman, Kewang Cao, Saad S. Alarifi, Salim Raza, Saira Sherin,
and et al. 2023. "Appraisal of Different Artificial Intelligence Techniques for the Prediction of Marble Strength" *Sustainability* 15, no. 11: 8835.
https://doi.org/10.3390/su15118835