Enhanced ELM Model Approach to Mitigate Multicollinearity of Lagged Independent Variables of ARMA Process

Abstract

This paper presents a regularized Extreme Learning Machine (ELM) framework for identifying nonlinear dynamic systems affected by multicollinearity, with application to a Hammerstein-structured model of a Continuous Stirred Tank Reactor (CSTR). The model architecture employs a single-hidden-layer feedforward network (SLFN) for the static nonlinear block, and an autoregressive linear dynamic block whose order is determined using a Lipschitz quotient criterion. Traditional ELM models are known to suffer from instability when lagged input variables are highly correlated, a common occurrence in block-oriented system identification. To address this, the study investigates enhanced variants of ELM incorporating regularization, namely Ridge-ELM and Liu-ELM, which introduce biasing parameters to improve numerical stability and generalization. The proposed regularized ELM variants are evaluated against traditional ELM using performance metrics such as Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and the coefficient of determination (R²). Results show that Ridge-ELM and Liu-ELM significantly reduce parameter variance and improve predictive performance on datasets. Additionally, confidence intervals and condition number analysis demonstrate improved robustness in the presence of multicollinearity. Cross-validation is used to tune hyperparameters, and the Diebold-Mariano test confirms that the improvements are statistically significant. This approach offers a computationally efficient, scalable solution for robust nonlinear system identification in multivariate chemical processes and beyond