An intelligent approach for estimating aeration efficiency in stepped cascades: optimized support vector regression models and mutual information theory

Abstract

Soft computing (SC) methods have increasingly been used to solve complex hydraulic engineering problems, especially those characterized by high uncertainty. SC approaches have previously proved to be an accurate tool for predicting the aeration efficiency coefficient (E20) in hydraulic structures such as weirs and flumes. In this study, the performance of the standalone support vector regression (SVR) algorithm and three of its hybrid versions, support vector regression–firefly algorithm (SVR-FA), support vector regression–grasshopper optimization algorithm (SVR-GOA), and support vector regression–artificial bee colony (SVR-ABC), is assessed forthe prediction of E20in stepped cascades.Mutualinformationtheoryis usedto constructinput variable combinations for prediction, including the parameters unit discharge (q), the total number of steps (N), step height (h), chute overall length (L), and chute inclination (a). Entropy indicators, such as maximum likelihood, Jeffrey, Laplace, Schurmann–Grassberger, and minimax, are computed to quantify the epistemic uncertainty associated with the models. Four indices—correlation coefficient (R), Nash– Sutcliffe efficiency (NSE), root mean square error (RMSE), and mean absolute error (MAE)—are employed for evaluating the models’ prediction performance. The models’ outputs reveal that the SVR-FA model (with R ¼ 0:947; NSE ¼ 0:888; RMSE ¼ 0:048 and MAE ¼ 0:027 in testing phase) has the best performance among all the models considered. The input variable combination,including q,N, h, andL, providesthe best predictions withthe SVR, SVR-FA, and SVR-GOA models. Fromthe uncertainty analysis, the SVR-FA model shows the closest entropy values to the observed ones (3.630 vs. 3.628 for the ‘‘classic’’ entropy method and 3.647 vs. 3.643 on average for the Bayesian entropy method). This study proves that SC algorithms can be highly accurate in simulating aeration efficiency in stepped cascades and provide a valid alternative to the traditional empirical equation.