Download Statistics Toolbox User's Guide

Nonlinear Regression Models
RSM is an empirical modeling approach using polynomials as local approximations to the true input/output relationship. This empirical approach is often
adequate for process improvement in an industrial setting.
In scientific applications there is usually relevant theory that allows us to
make a mechanistic model. Often such models are nonlinear in the unknown
parameters. Nonlinear models are more difficult to fit, requiring iterative
methods that start with an initial guess of the unknown parameters. Each iteration alters the current guess until the algorithm converges.
Mathematical Form
The Statistics Toolbox has functions for fitting nonlinear models of the form:
y = f ( X, β) + ε
where
• y is an n by 1 vector of observations
• f is any function of X and β
• X is an n by p matrix of input variables
• β is a p by 1 vector of unknown parameters to be estimated
• ε is an n by 1 vector of random disturbances
Nonlinear Modeling Example
The Hougen-Watson model (Bates and Watts 1988) for reaction kinetics is one
specific example of this type. The form of the model is:
β1 ⋅ x2 – x3 ⁄ β5
rate = -----------------------------------------------------------------------1 + β2 ⋅ x1 + β3 ⋅ x2 + β4 ⋅ x3
where β1, β2,...,β5 are the unknown parameters, and x1, x2, and x3 are the three
input variables. The three inputs are hydrogen, n-pentane, and isopentane. It
is easy to see that the parameters do not enter the model linearly.
1-65