An Interpretable Hybrid Shapley Additive Explanations Brown Bear Optimization Algorithm Predictive Models for End Bearing Capacity of Rock Socketed Piles

This study develops an interpretable hybrid predictive framework that combines SHapley Additive exPlanations (SHAP) with the Brown-Bear Optimization Algorithm (BBOA) to enhance the prediction of end-bearing capacity (qu) of rock-socketed piles. BBOA is employed to optimize hyperparameters of four machine learning (ML) models—Artificial Neural Networks (ANN), Support Vector Regression (SVR), Adaptive Neuro-Fuzzy Inference Systems (ANFIS), and Multivariate Adaptive Regression Splines (MARS)—leading to significant improvements in accuracy. A novel, interpretable MARS-based predictive formula is also introduced. The model uses pile diameter (B), embedment depths in soil (Hs) and rock (Hr), rock uniaxial compressive strength (σc), and Geological Strength Index (GSI) as inputs. Comprehensive evaluation using multiple statistical metrics (R², RMSE, MAE, U95) reveals that BBOA-ANFIS achieves the highest accuracy during training, while the optimized MARS model performs best in testing, with ANN consistently showing the weakest results. Compared to previous ML-based and empirical models, the proposed models, especially the MARS formula, demonstrate superior accuracy and reliability, with improvements of 11.3% in R², 16.6% in RMSE, and 19.2% in MAE over the best prior empirical model. Additionally, sensitivity analysis and SHAPE-based model interpretation identified key predictors, ranked by influence as σc > GSI > B > Hr > Hs, with σc being the most influential variable and Hs the least. This hybrid approach provides a robust and interpretable tool for accurate and cost-effective pile design in complex soil-rock environments.