An adaptive hybrid modeling approach integrating elastic net and Kriging for activated sludge treatment

When solving complex nonlinear problems, constructing a Kriging model by all variables is prone to overfitting, resulting in reduced prediction accuracy. To enhance the generalization capability of the Kriging model, this study proposes a hybrid modeling method that combines elastic net (EN) with Kriging, named as KrigNet. This method first determines the optimal value of the penalty coefficient λ by optimizing the K-fold cross-validation error, and then obtains the EN regression coefficient vector corresponding to λ. Subsequently, EN regularization is employed to shrink the regression coefficients of less influential variables to zero, thereby simplifying the regression function vector of Kriging. Finally, the weights for EN and Kriging are computed by the leave-one-out cross-validation error, and a weighted average of the predictions from both models is taken to obtain the final KrigNet model. Through evaluation and comparison based on benchmark functions and a case on water quality prediction of activated sludge treatment (AST), the KrigNet demonstrates superior generalization capability and higher prediction accuracy.