Estimation of uniaxial strength of rock: A comparison between Bayesian-optimized machine learning models

The uniaxial compressive strength (UCS) of rock measures its ability to resist compressive forces under unconfined loading. The laboratory determination of UCS is more time-consuming than the empirical and machine learning (ML) approaches. This investigation employs and compares the different ML approaches to estimate the UCS of rocks. Several statistical analyses introduce the Bayesian-optimized Gaussian process regression (GPR) as the most robust and reliable ML approach that estimates the UCS of rock with a confidence interval of ±75 MPa. Background The creep, permeability, porosity, fracture toughness, brittleness, ductility, hardness, Poisson’s ratio, Young’s modulus and strength (shear, tensile, uniaxial) properties demonstrate the behavior of rocks. The UCS, the most popular mechanical property, presents rock characteristics influenced by the formation process [1]. The UCS of rock is determined by laboratory testing, point load test, Schmidt hammer test, ultrasonic velocity and empirical methods. These methods are more lengthy and time-consuming than the ML approaches [2]. In recent years, the conventional and optimized models of random forest, artificial neural network, adaptive neuro-fuzzy inference system, extreme learning machine and extreme gradient boosting approaches have been utilized to estimate the UCS of rock [3]. The ML approaches are data-driven, and the prediction capabilities of approaches are significantly affected by the quality and quantity of the database [4]. The performance metrics, regression error characteristics (REC) curve, score analysis, uncertainty analysis, convergence and accuracy matrix ensure the prediction capabilities of the ML approaches. Methodology This investigation develops the Bayesian-optimized ensemble trees (ET), decision tree (DT), GPR and support vector machine (SVM) models to estimate the UCS of rocks. Further, it compares them with the multilinear regression (MLR) model. To develop the models and estimate the UCS of rocks, the P-wave velocity, Young’s modulus, density, mass and area properties of 131 rock specimens were utilized as features. The quality and reliability of the database were analyzed using variance analysis (ANOVA), Pearson’s product-moment correlation coefficient, multicollinearity and sensitivity analysis. The SVM, GPR, ET and DT models were developed, trained and tested using MATLAB R2020a. Thirteen metrics, including root mean square error (RMSE), mean absolute error (MAE) and performance index (PI), measured and analyzed each ML model. Based on comparing performance metrics, accuracy matrix, REC curve, uncertainty analysis and score analysis, the most reliable ML model was introduced to estimate the UCS of rock. Fig. 1 Illustration of (a) relationship between actual and estimated rock UCS and (b) error distribution plot. Results This investigation trained and tested each ML model using 104 and 27 rock specimens, respectively The performance comparison demonstrated that the GPR model achieved highest performance (R) in training (= 0.9297) and testing (= 0.9183) phase, followed by the ET, MLR, DT and SVM models (Fig. 1). The PI (training = 1.65, testing = 1.62), RMSE (training = 29.1618 MPa, testing = 27.1664 MPa) and MAE (training = 21.8895 MPa, testing = 23.2977 MPa) metrics also demonstrated the significant prediction capabilities of the GPR model because of its ability to model complex, nonlinear relationships with uncertainty quantification. The score analysis (Fig. 2), REC curve (area over the curve 4.51E-03 in training and 3.80E-03 in testing phase), and uncertainty (less uncertainty bandwidth) analysis presented the reliability and robustness of the GPR model over the ET, MLR, DT and SVM models due to its probabilistic predictions with confidence intervals and adapts well to small data sets. Its kernel-based approach enables capturing intricate spatial correlations, making it highly flexible and accurate. Discussion Mining, Metallurgy & Exploration, https://doi.org/10.1007/s42461-024-01168-y This study employed and analyzed the GPR, SVM, DT, ET and MLR models to identify the reliable robust ML model to estimate the UCS of rock. The multicollinearity parameter — that is, variance inflation factor (VIF) — demonstrated the effect of feature collinearity on the performance and accuracy of te SVM, DT, ET and MLR models in both phases, demonstrated in the convergence plot (Fig. 3). Moreover, the reliability of the GPR model was calculated by implementing a20, index of scatter (IOS) and index of agreement (IOA). It was observed that the GPR estimated rock UCS with a20 (training = 70.19, testing = 66.67), IOA (training = 0.8187, testing = 0.7783) and IOS (training = 0.2045, testing = 0.1872), which are close to the ideal values. In addition, the curve fitting was analyzed by computing the test to train RMSE ratio. It was observed that feature multicollinearity overfitted the MLR (= 1.1213), SVM (= 1.1113), ET (= 1.0950) and DT (1.1900) models. On the other hand, the GPR model predicted the UCS of rock with underfitting: that is, 0.9316 ≈ 1.00. Finally, the GPR model was identified as a robust reliability ML model to estimate the rock UCS. Conclusions Fig. 3 Illustration of convergence plot for each modelFig. 3 Illustration of convergence plot for each modelin estimating rock UCS. This study presents an optimal ML model for predicting rock’s uniaxial compressive strength (UCS) using five methods: MLR, GPR, SVM, DT and ET. The GPR model outperformed the others, achieving high accuracy and minimal residuals, followed by ET. The study also highlighted the positive influence of specimen area and Young’s modulus on GPR and ET performance. At the same time, MLR, SVM and DT performed poorly when these variables showed low sensitivity. Additionally, problematic multicollinearity adversely affected MLR, SVM and DT model accuracy. Future studies may explore other models like LSSVM, GEP and ELM for improved accuracy. Selected references Wang H, Zhang C, Zhou B, Xue S, Jia P, Zhu X (2023) Prediction of triaxial mechanical properties of rocks based on mesoscopic finite element numerical simulation and multi-objective machine learning. J King Saud Univ-Sci. 35(7):102846 Chen S, Zhang H, Wang L, Yuan C, Meng X, Yang G, Shen Y, Lu Y (2022) Experimental study on the impact disturbance damage of weakly cemented rock based on fractal characteristics and energy dissipation regulation. Theoret Appl Fract Mech 122:103665 Suo Y, Zhao Y, Fu X, He W, Pan Z (2023) Acoustic and mechanical tests of sandstone-shale composites in Songliao Basin and prediction of uniaxial compressive strength. Geoenergy Sci Eng 228:212034 Rezaei M, Asadizadeh M (2020) Predicting unconfined compressive strength of intact rock using new hybrid intelligent models. J Mining Environ 11(1):231–246

Estimation of uniaxial strength of rock: A comparison between Bayesian-optimized machine learning models

Authored by: Jitendra Khatti and Kamaldeep Singh Grover

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