Enhancing shear strength predictions of rocks using a hierarchical ensemble model

Shear strength (SS) parameters are essential for understanding the mechanical behavior of materials, particularly in geotechnical engineering and rock mechanics. This study proposes a novel hierarchical ensemble model (HEM) to predict SS parameters: cohesion ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> ) and angle of internal friction ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math> ). The HEM addresses the limitations of traditional machine learning models. Its performance was validated using leave-one-out cross-validation (LOOCV) and out-of-bag (OOB) evaluation methods. The model's accuracy was assessed with R-squared correlation (R²), absolute average relative error percentage (AAREP), Taylor diagrams, and quantile-quantile plots. The computational results demonstrated that the proposed HEM outperforms previous studies using the same database. The model predicted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> with R² values of 0.93 and 0.979, respectively. The AAREP values were 1.96% for φ and 4.7% for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> . These results indicate that the HEM significantly improves the prediction quality of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> , and exhibits strong generalization capability. Sensitivity analysis revealed that σ_3maxσ3max (maximum principal stress) had the greatest impact on modeling both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> . According to uncertainty analysis, the LOOCV and OOB had the widest uncertainty bands for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> parameters, respectively.