# Application of Machine Learning and Multivariate Statistics to Predict Uniaxial Compressive Strength and Static Young’s Modulus Using Physical Properties under Different Thermal Conditions

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## Abstract

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_{s}) are fundamental parameters for the effective design of engineering structures in a rock mass environment. Determining these two parameters in the laboratory is time-consuming and costly, and the results may be inappropriate if the testing process is not properly executed. Therefore, most researchers prefer alternative methods to estimate these two parameters. This work evaluates the thermal effect on the physical, chemical, and mechanical properties of marble rock, and proposes a prediction model for UCS and E

_{S}using multi-linear regression (MLR), artificial neural networks (ANNs), random forest (RF), and k-nearest neighbor. The temperature (T), P-wave velocity (P

_{V}), porosity (η), density (ρ), and dynamic Young’s modulus (E

_{d}) were taken as input variables for the development of predictive models based on MLR, ANN, RF, and KNN. Moreover, the performance of the developed models was evaluated using the coefficient of determination (R

^{2}) and mean square error (MSE). The thermal effect results unveiled that, with increasing temperature, the UCS, E

_{S}, P

_{V}, and density decrease while the porosity increases. Furthermore, ES and UCS prediction models have an R

^{2}of 0.81 and 0.90 for MLR, respectively, and 0.85 and 0.95 for ANNs, respectively, while KNN and RF have given the R

^{2}value of 0.94 and 0.97 for both E

_{S}and UCS. It is observed from the statistical analysis that P-waves and temperature show a strong correlation under the thermal effect in the prediction model of UCS and E

_{S}. Based on predictive performance, the RF model is proposed as the best model for predicting UCS and E

_{S}under thermal conditions.

## 1. Introduction

_{S}used as input parameters in the effective design and rock mass behaviour analysis, it is essential to evaluate these parameters under high-temperature mechanics.

_{S}by both destructive and non-destructive methods [26]. The destructive testing for both parameters is time-consuming and expensive, and the core sampling needs high precision, while the obtained results can be ambiguous [27,28]. Therefore, researchers have focused their attentions on non-destructive techniques. Several studies have been conducted using various artificial intelligent (AI) techniques to predict rock’s strength and stiffness properties [29]. In this regard, Manouchehrian et al. [26] predicted UCS using texture as input variables based on ANN and multivariate statistics. Likewise, [26,30] used porosity (η), P

_{V}, and ρ as input variables and predicted UCS and E

_{S}based on ANNs and ANFIS. Abdi, Garavand, and Sahamieh [28] proposed the ANN and MLR methods for predictive modeling of E using η in %, dry density (γd), P-wave velocity (P

_{V}), and water absorption as input variables. It was found that the prediction performance of ANN is better than MLR. Dehghan et al. [31] predicted UCS and Es based on ANNs and MLR using P

_{V}, the point load index, the Schmidt hammer rebound number, and η as input variables. Some cutting-edge machine learning models are also adopted to predict UCS and E

_{s}. For example, Zhang et al. [32] proposed a beetle antennae search (BAS) algorithm-based RF model to accurately and effectively predict the UCS of lightweight self-compacting concrete (LWSCC). Matin et al. [33] used the RF model to select variables within several rock properties and indexes, namely porosity (η), water content, Is (50), p-wave velocity (P

_{V}), and rebound numbers (Rn), along with an effective model for the prediction of UCS and E based on the RF preferred variables. Suthar [34] appraised the potential of five modeling approaches, namely M5 model tree, RF, ANN, SVM, and Gaussian processes (GPs) for predicting the UCS of stabilized pond ashes with lime and lime sludge. Wang et al. [35] proposed an RF model to accurately predict the UCS of rocks from simple index tests. Matin et al. [33] predicted E using RF, and multivariate regression (MVR) and a generalized regression neural network (GRNN) were used for comparison. The results revealed that RF performed well compared to MVR and GRNN. Ren et al. [36] developed several ML algorithms, namely k-nearest neighbors (KNN), naive Bayes, RF, ANN, and SVM, to accurately predict rock’s UCS using ANN and SVM. Ghasemi et al. [37] evaluated the UCS and E of carbonate rocks by developing a tree-based approach. According to their findings, the applied method revealed highly accurate results. Saedi et al. [38] studied the prediction of the UCS and E of migmatite rocks by ANN, ANFIS, and multivariate regression (MVR). Shahani et al. [39] developed an XGBoost model to predict the UCS and E of intact sedimentary rock. Armaghani et al. [40] developed a hybrid model based on ANN and imperialist competitive algorithm (ICA) to predict UCS and E of granite rocks. Although, the above-discussed literature has provided useful insights into predicting the UCS and E by utilizing different machine learning approaches, there has been, to date, no significant study documented which consider the thermal effect on the physical and mechanical behaviors of rock. The temperature has a great effect on these physical and mechanical properties. Therefore, it is imperative to increase the performance of the proposed model and to explore the use of a new input variable, i.e., temperature, in predicting the UCS and E

_{s}.

_{V,}ρ, η, and E

_{d}as input variables, the UCS and E

_{s}were predicted using different statistical and computational intelligence methods, including MLR, ANN, RNN, and RF. The results of this study will serve to help researchers to better understand the thermal effect on the physical and mechanical properties of rocks in a sweltering environment.

## 2. Regional Geological Setting

^{2}and belongs to the Alpurai Group metasediments, comprises marble, dolomite, and phyllites developed as a result of a high geothermal gradient associated with active crustal thickening and anatectic processes under the Barrovian metamorphic conditions between ca. 39 Ma and 28 Ma [41]. The total estimated marble in the district of Buner is 100 million tons. These marbles vary in color as well as in grain size [42]. The Nikani Ghar marble has a mainly fine to medium grain size. The marble individual bed thickness is 0.5–3.0 m, and the lateral extension (length) varies from 1.5–3.0 Km.

## 3. Experimental Design

#### 3.1. Rock Specimen

#### 3.2. Heating Procedure

#### 3.3. Samples Characterization

_{2}, Fe

_{2}O

_{3}, CaO, Al

_{2}O

_{3}, MgO, MnO, Na

_{2}O, and K

_{2}O and loss of ignition was assessed by X-ray fluorescence (XRF).

#### 3.4. Ultrasonic Test

#### 3.5. Universal Testing Machine (UTM)

_{S}, shear, and bulk modulus were determined for each predetermined temperature.

#### 3.6. Intelligent Models

#### 3.6.1. Multiple Linear Regression (MLR) Model

_{S}is based on five parameters, such as T, P

_{V,}ρ, η, E

_{d}, as shown in Table 1.

_{1}to X

_{n}, and b

_{1}to b

_{n}are the dependent variable, constant, independent variable, and partial regression coefficient, respectively [47,48].

#### 3.6.2. Artificial Neural Network (ANN) Model

#### ANN Code Compilation in MATLAB

_{V}, ρ, η, and E

_{d}) and two outputs (UCS and E

_{S}), as illustrated in Figure 3b. The dataset consisted of 60 data points in total. The data was separated into the following three sections: training (75%), testing (15%), and validation (15%).

#### 3.6.3. Random Forest Regression

_{1}, k

_{2}, etc., up to and including k

_{n}, denote the results of each individual DT.

#### 3.6.4. k-Nearest Neighbor

- It is straightforward to grasp and put into practice.
- When it is employed for classification and regression, it can learn non-linear decision boundaries, and it can also invent a highly flexible choice limit by adjusting the value of K. Both of these capabilities are available when it is applied.
- The KNN architecture does not have a step that is specifically dedicated to training.
- Since there is only one hyperparameter, which is denoted by the letter K, adjusting the other hyperparameters is quite simple.

^{th}dimension, and q stands for the order between the points x and y.

## 4. Experimental Results

#### 4.1. Physical Properties

#### 4.2. Micro Crack Analysis

#### 4.3. P-Wave Analysis

#### 4.4. Effect of Temperature on Stress–Strain Curve

_{S}. The temperature effect on the stress–strain behaviors is shown in Figure 9a. A gradual increase in temperature has decreased both the UCS and E

_{S}. Figure 9a shows the complete stress–strain curve at different temperatures. The pre-peak stress–strain is significantly influenced by temperature. Temperature variation illustrated a significant influence on stress–strain relations. The initial deformation of the non-linearity pattern increases in the stress–strain curve as the temperature increases. The stress–strain curves shape reveals that, as the temperature increases, the number of micro-cracks is increased and, as a result, the stress decreased. This is in agreement with the changes in material properties from brittle to ductile. The marble rock’s overall ductility increases with the increase in thermal heat, showing a strong agreement with the results of [15].

_{S}, P

_{V}, and ρ inverse relation with the increase in temperature, while the strain, as well as η, shows a direct relationship, as indicated in Figure 9b,c. The value of UCS decreased at the temperature range of 25–200 °C, but showed an increase at 200–300 °C, which shows a resemblance to a previous study [70]. On the other hand, at temperatures above 300 °C, the UCS decreased again. The increase in η and decrease in P

_{V}are in strong agreement with [71]. Overall, the E

_{S}decreased with an increase in temperature, as shown in Figure 9d.

## 5. Prediction Models of UCS and ES

#### 5.1. Preliminary Data Analysis

_{V,}ρ, η, E

_{d}, UCS, and E

_{s}for machine learning and a statistical approach. The T, P

_{V,}ρ, η, E

_{d}are used as input for the prediction of UCS and E

_{s}. The statistical analysis of the inputs and outputs data is described in Table 4.

_{V}have a positive correlation, while other inputs and output have a negative correlation. Figure 10 and Figure 11 enable a researcher to easily understand the effect of inputs on output results of the predicted model. The greater the negative or positive relationship, the greater will be the importance in model efficiency.

#### 5.2. MLR Prediction Models

_{s}, respectively. These can be mathematically expressed using Equations (5) and (6), as follows:

_{s}is the static Young’s modulus (GPa), T is temperature (°C), E

_{d}is the dynamic Young’s modulus (GPa), η is porosity (%), ρ is density (gr/cm

^{3}), and P

_{V}is P-wave velocity (km/s).

#### Importance of Variable in MLR Models

^{2}) between the actual and predicted UCS (R

^{2}= 0.90), as shown in Figure 12a. In the UCS model, out of five independent variables, two variables are highly correlated with UCS, namely T and P

_{V}, and give a significant value less than (p = 0.05), while the other three parameters, namely ρ, η, and E

_{d}, have less significance because of their P-value is greater than 0.05. The E

_{S}model gives an effective coefficient of determination (R

^{2}= 0.817), as shown in Figure 12b. In this model, out of five parameters, the two parameters which are highly correlated with E

_{S}are P

_{V}with a significance value (p = 0.042), and T, with significance value (p = 0.036), while E

_{d}is worthless. The other two variables, porosity and density, have a significance level more than 0.05, namely η (0.693) and ρ (0.238). These models revealed that the P

_{V}and T have a dominant effect on both models of UCS and E

_{S}, while the other three parameters have shown no obvious significance in both models.

#### 5.3. Network Phases and Regression Model

_{S}models. The good regression is achieved in training and validation, and testing values between the predicted and measured values of UCS as shown in Figure 13b. In the case of E

_{S}, the regression values of the predicted and measured show high validation regression values, as presented in Figure 13a. The plot draws from the ANN model are shown in Figure 14a,b. A good R

^{2}value (0.95) between the predicted and measured UCS is found as shown in Figure 14a. Figure 14b shows a relatively lesser coefficient of determination value (0.85) between predicted E

_{S}and measured E

_{S}, as compared to predicted and measured UCS.

#### 5.3.1. Network Performance and Accuracy

_{S}, the MSE is evaluated separately. The optimum regression model is achieved through a lesser MSE value at 250 and 300 epochs for UCS and E

_{S}, respectively, as shown in Figure 15 and Figure 16. This also revealed the number of iteration and number of neuron play key role in the accuracy achievement of the model. that The neuron convergence analysis shows that the optimum regression and least MSE for UCS and E

_{S}are obtained on 5 and 7 neurons, respectively, as shown in Figure 17.

#### 5.3.2. Importance of Variable in ANN Models

_{S}is shown in Figure 18a,b. In Figure 18a, it seems that the E

_{d}, P

_{V}, and T show a strong relation with E

_{S}, while ρ and η show a weak relation to E

_{S}. The independent variable, such as T, P

_{V}, and E

_{d}, have a strong relation with UCS, the most important of which is temperature, as shown in Figure 18b. Moreover, ρ and η have a very low relation to UCS.

#### 5.4. Random Forest

^{2}= 0.97) for USC and E

_{S}, as shown in Figure 19.

#### 5.5. k-Nearest Neighbor

^{2}= 0.94), as can be shown in Figure 20. This is the case for both USC and E

_{S}.

## 6. A Comparative Evaluation of Statistics and Intelligent Techniques

^{2}, MAPE, RMSE, and VAF, were evaluated. An excellent model can be represented by performance indices as, R

^{2}= 1, MAPE = RMSE = 0, and VAF = 100%. The performance indices were calculated using Equations (7)–(10), as follows:

_{S}. On the basis of this performance indices, the RFR performed well.

## 7. Discussion

_{S}based on statistical (MLR) and intelligent models (ANNs, RFR, and KNN). The accuracy and performance of models are satisfactory on the basis of MSE, MAPE, VAF, and R

^{2}. The MSE, MAPE, VAF of the MLR is greater than that of the intelligent models. The intelligent models have shown a better prediction performance than the statistical model due to its MSE, MAPE, VAF values and high R

^{2}value. The MSE, MAPE, VAF, R

^{2}values of the MLR are fixed, while the MSE, MAPE, VAF and R

^{2}of the intelligent model are varied. It depends on the neuron optimization in the hidden layer for ANN and the hyperparameters. The MSE, MAPE, and VAF of a prediction model can improve through trial and error methods using an intelligent model. The intelligent model’s optimization needs an expert person who can know the tuning the of hyperparameters number, and how to fine-tune hyperparameters to obtain more reliable results.

_{S}respectively are better, as shown in Figure 14a,b, Figure 19a,b, and Figure 20a,b, than MLR. In this research work, ANN gives 5% for UCS and 4% for E

_{S}, RFR give 7% for UCS and 16% for E

_{S}, and gives 4% for UCS and 13% for E

_{S}, meaning that the intelligent model is more accurate than the MLR. Furthermore, RFR give 7% for UCS and 16% for E

_{S}, which is more accurate than the statistical model and has 4-5% high accuracy than ANN and KNN. The models are based on limited data and only valid to a specific area. The models can extend to a generalized form in the future to take a large amount of data on different rocks. Temperature and P-wave velocity are strongly correlated in both models. The three other input parameters play a worthless role in the equation. This work result is strongly supported by existing research [35]. It suggested P

_{V}, ρ, and η as input variables and, after prediction, revealed that only P-wave velocity has a strong correlation, and the other input variables have a worthless contribution. This study is based on thermal effect; therefore, the temperature is considered as an input parameter that has a strong influence on the mechanical and physical properties of rock. Furthermore, in these modes, the temperature and P-wave velocity both have a strong correlation with output, and the other three independent parameters are meaningless in the model. The performance of this model is better than in the model developed by Torabi-Kaveh, Naseri, Saneie, and Sarshari [26].

_{S}. The variable performance in both MLR and the intelligent model shows that the T and P

_{V}played an active role in the prediction models, while the ρ, η, and E

_{d}have a less active predictive role.

## 8. Conclusions

- The physical and mechanical properties are greatly affected by the increase in temperature from 200–600 °C due to the generation and propagation of micro-cracks. The porosity is increased, while P
_{V}is decreased with the increase in temperature. The strength properties i.e., UCS and E_{S}, of rock also decrease with the increasing temperature; - The behavior of the stress–strain curve is changed from brittle to ductile when the temperature is increased;
- The MLR predictive models for UCS and E
_{S}give a performance coefficient of 0.90% and 0.81%, respectively. The intelligent models i.e., ANN, RFR, and KNN for UCS and E_{S}give a performance coefficient, revealing that the model for UCS is (5–7%) and E_{S}is (4–16%) better than the statistical model (MRL models). - The model’s important feature revealed that the temperature and P
_{V}have a significant role in prediction models; - Based on comparative analysis of MLR, ANN, RFR, and KNN, it has been proposed that RFR model is suitable for use in the prediction of UCS and E
_{S}under thermal treatment.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic flowsheet of samples preparation and sample testing, as follows: (

**a**) core bit machine for core extraction, (

**b**) furnace for heating sample, (

**c**) universal testing machine (UTM) for UCS, (

**d**) cylindrical core before cutting and polishing, (

**e**) cylindrical core after cutting and polishing, (

**f**) core sample under compression in UTM, (

**g**) core cutting & polishing instrument (

**h**) the PUNDIT for P-waves, and (

**i**) core samples after failure.

**Figure 3.**(

**a**) Flowchart of ANN for the UCS and E

_{S}prediction model. (

**b**) The architecture topology of ANN for UCS and E

_{S}.

**Figure 5.**(

**a**) The XRD peak at a different temperature and (

**b**) the XRF analysis at different temperature ranges.

**Figure 7.**Optical microscopy at different temperatures. (

**a**) SEM images for crack propagation; (

**b**) Micrograph of the thin section.

**Figure 9.**(

**a**) Stress–strain curve, (

**b**) porosity and PV curve, (

**c**) UCS and strain curve and (

**d**) all static and dynamic moduli at a different temperature.

**Figure 12.**(

**a**) Relationship between the predicted and actual UCS, (

**b**) Relationship between the predicted and actual E

_{S}.

**Figure 13.**(

**a**) The ANN phases of training, validation, and testing, and the regression coefficient for UCS, (

**b**) The ANN phases of training, validation, and testing, and the regression coefficient for E

_{S}.

**Figure 14.**(

**a**) The ANN scatter plot between the predicted and measured UCS and (

**b**) scatter plot between the predicted and actual E

_{S}.

**Figure 15.**

**The**UCS neural network performance for the selected network. (

**a**) 50, (

**b**) 80, (

**c**) 250, and (

**d**) 600.

**Figure 16.**The E

_{S}neural network performance for the selected network. (

**a**) 50, (

**b**) 100, (

**c**) 300, and (

**d**) 500.

**Figure 17.**The optimum performance of network under different number of neurons; (

**a**) UCS and (

**b**) Es.

**Figure 18.**Independent variable importance chart from ANN model; (

**a**) UCS, (

**b**) E

_{S}. Key is as follows: ρ density; η porosity; T: Temperature; Pv: p-wave velocity, and E

_{d}: dynamic Young’s modulus.

Parameter | Minimum | Maximum | Mean | Std. Deviation | Mean Std. Error |
---|---|---|---|---|---|

T | 25 | 600 | 328.12 | 168.77 | 21.09 |

E_{d} | 22 | 82 | 43.08 | 14.91 | 1.86 |

N | 9 | 29 | 16.80 | 5.21 | 0.65 |

Ρ | 3 | 3 | 2.69 | 0.02 | 0.01 |

P_{V} | 3 | 6 | 4.22 | 0.65 | 0.08 |

UCS | 63 | 115 | 84.72 | 18.72 | 2.34 |

E_{s} | 8 | 66 | 24.77 | 15.41 | 1.93 |

Temperature (°C) | SiO_{2}(%) | TiO_{2}(%) | Al_{2}O_{3}(%) | Fe_{2}O_{3}(%) | MnO (%) | MgO (%) | CaO (%) | Na_{2}O(%) | K_{2}O(%) | P_{2}O_{5}(%) | LoI (%) |
---|---|---|---|---|---|---|---|---|---|---|---|

25 | 0.405 | 0 | 0.352 | 0.121 | 0.012 | 0.373 | 53.892 | 2.552 | 0.012 | 0.000 | 42.28 |

200 | 0.404 | 0 | 1.5 | 1 | 0.014 | 2.17 | 52.89 | 2.2 | 0.012 | 0.000 | 41.81 |

400 | 0.5 | 0 | 2.45 | 2.3 | 0.34 | 2.37 | 50 | 3.1 | 0.71 | 0.000 | 38.23 |

600 | 0.51 | 0 | 3.1 | 2.6 | 0.4 | 2.80 | 48.9 | 3.3 | 0.82 | 0.000 | 37.58 |

S. No | T (°C) | P_{V}(km/s) | ρ (gr/cm ^{3}) | n (%) | Dynamic Moduli | Static Moduli | Strain | UCS (MPa) | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

E_{d}(GPa) | K_{d}(GPa) | G_{d}(GPa) | E_{s}(GPa) | K_{s}(GPa) | G_{s}(GPa) | ԑ_{p}(10 ^{−3}) | |||||||

1 | 25 | 5.49 | 2.711 | 12.15 | 77.83 | 48.304 | 35.086 | 61.62 | 42.792 | 24.453 | 1.83 | 113 | |

2 | 200 | 5.03 | 2.707 | 12.56 | 47.73 | 28.393 | 19.577 | 24.5 | 17.229 | 9.699 | 4.16 | 105 | |

3 | 250 | 4.98 | 2.698 | 15.79 | 44.32 | 27.403 | 18.018 | 22.32 | 15.966 | 8.808 | 4.25 | 107 | |

4 | 300 | 4.87 | 2.695 | 16.02 | 41.90 | 26.642 | 16.932 | 20.9 | 15.145 | 8.228 | 4.30 | 109 | |

5 | 350 | 4.78 | 2.689 | 16.51 | 41.37 | 25.370 | 15.041 | 18.32 | 13.754 | 7.167 | 4.36 | 80 | |

6 | 400 | 4.67 | 2.685 | 17.09 | 40.75 | 24.287 | 13.524 | 16.32 | 12.477 | 6.365 | 4.47 | 73 | |

7 | 500 | 4.48 | 2.681 | 19.78 | 40.00 | 23.682 | 12.744 | 15.34 | 12.117 | 5.950 | 4.75 | 69 | |

8 | 600 | 4.35 | 2.68 | 24.49 | 39.13 | 22.404 | 11.091 | 13.24 | 10.712 | 5.116 | 5.80 | 63 |

Model | Variables | Coefficient | Std. Error | T | p-Value | R^{2} |
---|---|---|---|---|---|---|

UCS | C | 186.017 | 95.212 | 1.954 | 0.056 | 0.905 |

T(°C) | −0.116 | 0.013 | −9.053 | 0.038 | ||

E_{d} (GPa) | −0.232 | 0.133 | −1.747 | 0.086 | ||

η (%) | 0.078 | 0.216 | 0.359 | 0.721 | ||

ρ (gr/cm^{3}_{)} | −24.410 | 36.162 | −0.675 | 0.502 | ||

P_{V} (km/sec) | 2.694 | 2.898 | 0.930 | 0.356 | ||

E_{S} | C | −164.932 | 108.981 | −1.513 | 0.136 | 0.817 |

T(°C) | 0.003 | 0.015 | 0.235 | 0.036 | ||

E_{d} (GPa) | 0.657 | 0.152 | 4.325 | 0.000 | ||

η (%) | −0.098 | 0.248 | −0.397 | 0.693 | ||

ρ (gr/cm^{3}_{)} | 49.318 | 41.391 | 1.192 | 0.238 | ||

P_{V} (km/sec) | 6.890 | 3.317 | 2.078 | 0.042 |

Parameters | Values | Details |
---|---|---|

n_estimators | 100 | Number of trees in RFR |

max_depth | 12 | Maximum depth of tree |

random_state | 32 | Random state |

Parameters | Values | Descriptions |
---|---|---|

n_neighbors | 5 | Number neighbors |

Metric | Minkowski | The distance metric to use |

Predicted Paramter | Models | R^{2} | MSE | MAPE (%) | VAF (%) |
---|---|---|---|---|---|

UCS | MLR | 0.90 | 23.15 | 31.53 | 90.23 |

ANN | 0.94 | 0.14 | 1.18 | 94.23 | |

RFR | 0.97 | 2.04 | 0.25 | 97.22 | |

KNN | 0.94 | 3.02 | 0.94 | 94.01 | |

E_{S} | MLR | 0.81 | 27.15 | 34.53 | 81.02 |

ANN | 0.86 | 0.54 | 2.18 | 86.03 | |

RFR | 0.97 | 2.04 | 0.25 | 97.22 | |

KNN | 0.94 | 3.02 | 0.94 | 94.23 |

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## Share and Cite

**MDPI and ACS Style**

Khan, N.M.; Cao, K.; Yuan, Q.; Bin Mohd Hashim, M.H.; Rehman, H.; Hussain, S.; Emad, M.Z.; Ullah, B.; Shah, K.S.; Khan, S.
Application of Machine Learning and Multivariate Statistics to Predict Uniaxial Compressive Strength and Static Young’s Modulus Using Physical Properties under Different Thermal Conditions. *Sustainability* **2022**, *14*, 9901.
https://doi.org/10.3390/su14169901

**AMA Style**

Khan NM, Cao K, Yuan Q, Bin Mohd Hashim MH, Rehman H, Hussain S, Emad MZ, Ullah B, Shah KS, Khan S.
Application of Machine Learning and Multivariate Statistics to Predict Uniaxial Compressive Strength and Static Young’s Modulus Using Physical Properties under Different Thermal Conditions. *Sustainability*. 2022; 14(16):9901.
https://doi.org/10.3390/su14169901

**Chicago/Turabian Style**

Khan, Naseer Muhammad, Kewang Cao, Qiupeng Yuan, Mohd Hazizan Bin Mohd Hashim, Hafeezur Rehman, Sajjad Hussain, Muhammad Zaka Emad, Barkat Ullah, Kausar Sultan Shah, and Sajid Khan.
2022. "Application of Machine Learning and Multivariate Statistics to Predict Uniaxial Compressive Strength and Static Young’s Modulus Using Physical Properties under Different Thermal Conditions" *Sustainability* 14, no. 16: 9901.
https://doi.org/10.3390/su14169901