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Show p=Ax+b Assuming a linear system, Equation 1 can be solved analytically via matrix inversion. Linear regression can model linear processes or the linear component of nonlinear processes. However, most plants exhibit nonlinear behavior, and, therefore, most linear models are not as accurate as nonlinear regression models. NONLINEAR REGRESSION MODELS Nonlinear regression techniques model the output as a nonlinear function of the input variables: p = f (x) where f is a nonlinear function. Nonlinear regression is a well-studied discipline, with polynomial regression perhaps being the most common traditional form. Polynomial regression, however, has many disadvantages including "overfitting" the data, that is, the data used in model development may be fit perfectly, but any new data point may yield an output very far from the true value. The newest and most sophisticated modeling software allows the use of both firstprinciples information and advanced nonlinear regression techniques. The key to the modeling methodology is based on major advances in nonlinear regression that have occurred over the past decade which facilitate the building of high-dimensional nonlinear models without the problems of overfitting. These new regression methods are computationally intensive, and their development was facilitated by the widespread availability of powerful computers. The underlying mathematics of these regression models and several different regression algorithms have been reviewed in hundreds of papers and books on the subject [References 2-8], and several authors [References 9- 10] have proven that these regression structures are capable of approximating any continuous multivariate function with very weak restrictions to arbitrary accuracy, a property called "universal approximation" . 6 |