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This short note describes a relatively simple methodology, procedure or approach to increase the performance of already installed industrial models used for optimization, control, simulation and/or monitoring purposes. The method is called Excess or X-Model Regression (XMR) where the concept of “excess modeling” or an X-model is taken from the field of thermodynamics to describe the departure or residual behaviour of real (non-ideal) gases and liquids from their ideal state (Kyle, 1999; Poling et. al., 2001; Smith et. al., 2001). It has also been applied to model the non-ideal or nonlinear behaviour of blending motor gasoline octanes with its synergistic and antagonistic interactional effects (Muller, 1992).
The fundamental idea of XMR is to calibrate, train, fit or estimate, using actual data and multiple linear regression (MLR) or ordinary least squares (OLS), the deviations of the measured responses from the existing model responses. The existing model may be a glass, grey or black-box model (known or unknown, linear or nonlinear, implicit/open or explicit/closed) depending on the use of the model. That is, for optimization and control the model structure and parameters are available given that derivative information is required although for simulation and monitoring, the model may only be observed through the dependent output variables given the necessary independent input variables.