Integrated approach of extreme learning machines and locally weighted linear regression for improved discharge coefficient prediction

Abstract

Abstract Accurate determination of the discharge coefficient (Cd) is essential for calculating discharge over side weirs. The current study aims to enhance the prediction accuracy of Cd for rectangular sharp-crested side weirs by addressing the limitation of the output layer of the Extreme learning machine (ELM). The output layer of ELM depends mainly on the linear system which limits its generalization capabilities. Therefore, this study uses Locally Weighted Linear Regression (LWLR) with radial basis kernel function instead of the linear system to effectively capture nonlinear relationships and enhance local data pattern recognition. The proposed model (ELM-LWLR) has been validated against classic multiple linear regression (MLR), ELM, LWLR, and Extreme Gradient Boosting (XGBoost). The quantitative results showed that the ELM-LWLR model has a superior performance, achieving higher prediction accuracy with a correlation coefficient of 0.968, and percentage bias (PBIAS) of -0.130%. Moreover, the accuracy of Cd prediction using the ELM-LWLR model improved by 37.21% compared to LWLR, 28.95% compared to XGBoost, 48.08% compared to ELM, and 64.94% compared to MLR. Additionally, sensitivity analysis identified the ratio of weir height to length and dimensionless length as critical factors affecting Cd estimation. Overall, the findings demonstrate that the ELM-LWLR model is a practical and robust tool for Cd modeling, offering significant advantages in cost reduction and enhanced hydraulic modeling for complex engineering applications.