# Performance of Statistical and Intelligent Methods in Estimating Rock Compressive Strength

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## Abstract

**:**

^{2}), and performance index (PI), the efficiency of the methods was evaluated. Analysis of model criteria using multiple linear regression allowed for the development of a user-friendly equation, which proved to have adequate accuracy. All intelligent methods (with R

^{2}> 90%) had excellent accuracy for estimating UCS. The percentage difference of the average of all six intelligent methods with the measured value was equal to +0.28%. By comparing the methods, the accuracy of the support vector machine with radial basis function in predicting UCS was (R

^{2}= 0.99 and PI = 1.92) and outperformed all the other methods investigated.

## 1. Introduction

## 2. Methodology

#### 2.1. Laboratory Tests

#### 2.2. Random Forest Algorithm (RFA)

#### 2.3. Gaussian Process Regression Based on Squared Exponential Kernel (GPR-SEK)

_{i}is the input vector with D dimension and y

_{i}is the target output. This set, consisting of two components, input and output, will be denoted as measured points. To simplify the problem, the inputs of the collection are aggregated at $X=\{{x}_{i},{x}_{2},\dots ,{x}_{n}\}$ matrix and the outputs are also combined at $Y=\{{y}_{i},{y}_{2},\dots ,{y}_{n}\}$ matrix. Regression based on the data set d creates a new input x* to arrive at the predicted distribution for the corresponding values of the measured y* data. The Gaussian process (GP) is a group of accidental parameters, a restricted number of which are combined with Gaussian distributions (GDs) [57]. The GP is a generalization of GD. The GD is actually scattered between accidental parameters, while GP represents scattering between functions. The f(x) GP is described using the m(x) average and covariance functions according to Equations (1) and (2).

#### 2.4. The SVM-RBF

#### 2.5. K Nearest Neighbor Algorithm (KNNA)

#### 2.6. ANFIS and FMP-ANN

#### 2.7. Performance Evaluation of Results

^{2}+ (VAF/100) − RMSE

_{min}is the minimum of the data, and X

_{max}is the maximum of the data.

## 3. Results and Discussion

#### 3.1. Geomechanical Properties of Samples

#### 3.2. Petrographic Features

#### 3.3. Influence of Independent Variables on the UCS

#### 3.4. Evaluation of Previous Emperical Relationships

^{2}. The results revealed that there is good compatibility between actual UCS and the estimated one using previous studies (Figure 4). A performance index (PI) was introduced by Yagiz et al. [85] for evaluating empirical equations and models. The value of this index is equal to two in the best case, and the lower it is, the lower the relationship performance. As can be seen, although the correlation coefficient is high, the performance index is negative, which indicates the poor performance of the previous researchers’ relationships in estimating the UCS of the studied rocks (Figure 4). For this reason, various researchers have emphasized that empirical relationships should be determined for each region [85].

#### 3.5. Multiple Linear Regression (MPLR)

#### 3.6. The Results of Modeling Using RFA and GPR-SEK Methods

^{2}equals 100%, all the observed values will be similar to the fitted values and all the data points will be on the fitted line [87].

#### 3.7. The FMP-ANN Results

#### 3.8. The KNNA Results

#### 3.9. Results of SVM Method for Estimating UCS

#### 3.10. Results of ANFIS Method for Estimating UCS

#### 3.11. Evaluation of the Used Methods

^{2}, MAPE%, RMSE, VAF, and PI), the SVM-RBF model displays greater precision than other models because the SVM uses the minimizing structural risk theorem and adapts the ability of the model to existing training data [103]. The number of input variables, number of samples, and training algorithm type also affect the accuracy of the methods [16,104,105]. Based on the correlation coefficient, all methods (R

^{2}> 90%) have excellent accuracy for estimating UCS.

## 4. Conclusions

^{2}and PI values were 0.99 and 1.92, respectively. The R

^{2}values of 98%, 98%, 97%, 98%, and 99% for forecasting the UCS were achieved using ANFIS, RFA, KNNA, GPR, and FMP-ANN, respectively. The number of samples and input variables had a significant impact on the performance of the methods. When the number of samples was small, the SVM method was more accurate. The percentage difference of the average of all six intelligent methods with the measured value was less than 1%, which indicates the superior capability of the intelligent methods in forecasting UCS.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Example samples for PWV, PLI, and UCS tests: (

**a**) device for measuring PWV, (

**b**) example samples after PLI test, and (

**c**) sample under UCS test.

**Figure 3.**Effect of (

**a**) point load index (PLI), (

**b**) water absorption by weight (WW), (

**c**) %porosity, (

**d**) density (D), and (

**e**) P−wave velocity (PWV) on the UCS.

**Figure 5.**Normal state of the residues of model 18 (

**a**) histogram of residuals, and (

**b**) normal probability plot.

**Figure 6.**Results of (

**a**) RFA for test data, (

**b**) RFA for all data, (

**c**) GPR-SEK for test data, and (

**d**) GPR-SEK for all data.

**Figure 8.**The FMP−ANN results for the optimum model (

**a**) error reduction trend and (

**b**) corelation cofficient between measured and predicted UCS.

**Figure 10.**Accuracy of predicted UCS using KNNA: (

**a**) correlation coefficient for test data and (

**b**) error histogram for all data.

**Figure 11.**Accuracy of predicted UCS using SVM-RBF: (

**a**) correlation coefficient and (

**b**) error histogram for all data.

**Figure 12.**Accuracy of predicted UCS using ANFIS: (

**a**) correlation coefficient and (

**b**) error histogram for all data.

**Figure 13.**Measured values versus predicted UCS using (

**a**) SVR−RBF, ANFIS, and RFA methods, and (

**b**) FMP−ANN, KNN, and GPR−SEK methods.

Equation | Reference | Lithology |
---|---|---|

UCS = 12.29PLI^{1.233} | Teymen and Mengüç [40] | Various Rocks |

UCS = −37.82 + (0.017PWV) | Salehin [41] | Sedimentary Rocks |

UCS = 0.043PWV − 136.8 | Aldeeky and Al Hattamleh [42] | Basalt Rocks |

UCS = 17.6PLI + 13.5 | Aliyu et al. [30] | Sedimentary Rocks |

UCS = 14.3PLI | Aladejare [8] | Sedimentary Rocks |

UCS = 9.95PWV^{(1.21)} | Kahraman [43] | Sedimentary rocks |

$\mathrm{UCS}=0.034\mathrm{PWV}-86.36$ | Wen et al. [7] | Limestone |

UCS = −5.10$\varphi $ + 110.79 | Edet [3] | Sandstone |

UCS = 0.025PWV − 8.619 | Azimian [29] | Limestone |

UCS = 6.6PWV^{1.6} | Uyanık et al. [44] | Sedimentary rocks |

UCS = 22.18PWV − 30.32 | Selcuk and Nar [31] | Various Rocks |

UCS = 0.041PWV − 15.40 | Abdi and Khanlari [4] | Sandstones |

UCS = 2.304(PWV)^{2.43} | Kılıç and Teyman [45] | Various Rocks |

UCS = 10 − 5D^{16.7} | Aladejare [8] | Sedimentary rocks |

Test | Standards and References | Descriptions |
---|---|---|

UCS | ISRM [47] | A constant loading rate of 0.7 MPa per second was applied to the samples. The amount of deformation was recorded using the corresponding gauge in the UCS test. |

Point load index (PLI) | ASTM D5731 [48] | This test was done on irregular and cylindrical samples. Then the PLI was calculated. |

Compressional wave velocity test | ASTM D2845 [49] | With a ½ MHz frequency |

Porosity ($\varphi $), density(D) and water absorption by weight (WW) | ISRM [47] | The total porosity ($\varphi $) of specimens was measured using the method of saturation and immersion way. Density was computed from the ratio of mass to sample volume. |

Petrography | Folk [50], Dunham [51] | For classifying the samples using thin section images. |

Properties | Density (g/cm^{3}) | PLI (MPa) | Water Absorption (%) | Porosity (%) | UCS (MPa) | Es (GPa) | PWV (km/s) | |
---|---|---|---|---|---|---|---|---|

Statistics | ||||||||

Average | 2.43 | 3.75 | 6.81 | 9.44 | 37.54 | 14.95 | 4.38 | |

Std. Dev. | 0.11 | 1.66 | 1.87 | 3.35 | 16.49 | 5.30 | 1.03 | |

Kurtosis | 0.13 | (0.58) | (0.50) | (0.41) | (0.58) | (0.51) | (0.38) | |

Skewness | (0.42) | 0.09 | 0.70 | 0.79 | (0.71) | (0.62) | (0.78) | |

Min. | 2.10 | 0.31 | 4.08 | 4.36 | 4.12 | 3.00 | 2.06 | |

Max. | 2.63 | 8.00 | 11.00 | 16.72 | 59.72 | 22.90 | 5.79 | |

Specimens | 65 | 65 | 65 | 65 | 65 | 65 | 65 |

Equation | R^{2} | RMSE (MPa) | MAPE% | VAF % | PI | DWS | ANOVA Results | Eq. No. |
---|---|---|---|---|---|---|---|---|

UCS = 5.03PWV − 1.735$\varphi $ + 2.667PLI | 0.88 | 1.10 | 1.08 | 87.85 | 0.66 | 1.93 | F-value = 79.37 p-value = 0.00 | (18) |

Term | Coefficients | T-Value | Significant Level (Sig.) | VIF (Variance Inflation Factor) |
---|---|---|---|---|

Constant | −32.1 | −1.34 | 0.187 | - |

PWV | 5.03 | 2.44 | 0.018 | 7.58 |

D | 21.4 | 1.82 | 0.074 | 3.02 |

WW | 0.281 | 0.35 | 0.728 | 3.81 |

$\varphi $ | −1.735 | −3.97 | 0.000 | 3.64 |

PLI | 2.667 | 3.05 | 0.003 | 3.77 |

References | Neuron Numbers Calculated for This Study | Equations |
---|---|---|

Hecht-Nielsen [90] | $\le $3 | ≤$2\ast {N}_{i}+1$ |

Hush [91] | 3 | $3{N}_{i}$ |

Ripley [92] | 3 | $({N}_{i}+{N}_{0})/2$ |

Paola [93] | 11 | $\frac{2+{N}_{i}\ast {N}_{0}+0.5{N}_{0}\ast \left({N}_{0}^{2}+{N}_{i}\right)-3}{({N}_{i}+{N}_{0})}$ |

Wang [94] | 1 | $2{N}_{i}$/3 |

Kaastra and Boyd [95] | 2 | $\sqrt{{N}_{0}\ast {N}_{i}}$ |

Kanellopoulos and Wilkinson [96] | 1 | $2{N}_{i}$ |

_{0}and N

_{i}are the numbers of input and output neurons, respectively.

**Table 7.**The most important kernel functions for solving engineering problems [102].

Function | Description | Kernel Function Type |
---|---|---|

$k\left({x}_{i},{x}_{j}\right)={({x}_{i}.{x}_{j}+1)}^{d}$ | This kernel is widely used in image processing, where d is the degree of the polynomial. | Polynomial kernel (PK) |

$k\left({x}_{i},{x}_{j}\right)=\mathit{exp}(-\gamma \Vert {x}_{i}-{x}_{j}{\Vert}^{2})$ | This kernel is used for general purposes. It is used when there is no prior knowledge about the data. In $\gamma >0$ condition, $\gamma =1/2{\sigma}^{2}$ parameter is used. | Radial basis function (RBF) |

$k\left({x}_{i},{x}_{j}\right)={x}_{i}.{x}_{j}$ | - | Linear kernel (LK) |

Kernel Function | Optimal Values of Parameters | Test Period | Train Period | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\epsilon}$ | t | d | $\mathit{\sigma}$ | c | RMSE | R^{2} | PI | MAPE | RMSE | R^{2} | PI | MAPE | |

PK | 1.72 | 280.01 | 4 | - | 12.12 | 0.08 | 0.97 | 1.87 | 2.86 | 0.07 | 0.98 | 1.84 | 2.81 |

RBF | 0.02 | - | - | 1.10 | 27 | 0.06 | 0.99 | 1.90 | 2.82 | 0.06 | 0.99 | 1.90 | 2.80 |

LK | 0.45 | - | - | - | 0.90 | 0.09 | 0.96 | 1.83 | 2.84 | 0.09 | 0.97 | 1.81 |

FIS Generation Method | GENFIS4 |
---|---|

Influence radius | 0.60 |

Number of epochs | 500 |

Error goal | 0.00 |

Type | Sugeno |

Rules | 4 |

Number of membership functions (MFs) | 6 |

Input MF type | Gauss MF |

Output MF type | Linear |

APPROACHES | MAPE% | R^{2} | RMSE | VAF% | PI |
---|---|---|---|---|---|

RFA | 9.27 | 0.98 | 0.09 | 97.63 | 1.87 |

SVM-RBF | 2.83 | 0.99 | 0.06 | 98.96 | 1.92 |

ANFIS | 2.98 | 0.98 | 0.09 | 97.86 | 1.87 |

KNNA | 8.44 | 0.97 | 0.11 | 97.25 | 1.83 |

GPR-SEK | 6.63 | 0.98 | 0.09 | 97.45 | 1.86 |

FMP-ANN | 4.66 | 0.99 | 0.24 | 98.36 | 1.73 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Altalbawy, F.M.A.; Gasmalla, T.A.S.; Al-Khafaji, A.H.D.; Iraji, A.; Syah, R.B.Y.; Nehdi, M.L.
Performance of Statistical and Intelligent Methods in Estimating Rock Compressive Strength. *Sustainability* **2023**, *15*, 5642.
https://doi.org/10.3390/su15075642

**AMA Style**

Zhang X, Altalbawy FMA, Gasmalla TAS, Al-Khafaji AHD, Iraji A, Syah RBY, Nehdi ML.
Performance of Statistical and Intelligent Methods in Estimating Rock Compressive Strength. *Sustainability*. 2023; 15(7):5642.
https://doi.org/10.3390/su15075642

**Chicago/Turabian Style**

Zhang, Xuesong, Farag M. A. Altalbawy, Tahani A. S. Gasmalla, Ali Hussein Demin Al-Khafaji, Amin Iraji, Rahmad B. Y. Syah, and Moncef L. Nehdi.
2023. "Performance of Statistical and Intelligent Methods in Estimating Rock Compressive Strength" *Sustainability* 15, no. 7: 5642.
https://doi.org/10.3390/su15075642