GPS-X Technical Reference

387 Optimizer
APPENDIX A: MAXIMUM LIKELIHOOD METHOD
Introduction
In this appendix the maximum likelihood method is discussed in more detail. First a
general introduction on parameter estimation is given based on material found in
Bard (1974). This is followed by a detailed discussion of the maximum likelihood
method as implemented in GPS-X. Finally, a brief description of the sum of squares
objective function is given.
Parameter Estimation
Parameter estimation is the procedure of fitting a mathematical model of a process to
measured data by calculating optimal estimates of the model parameters. Parameter
estimation differs from simple curve fitting in that the criterion used to judge the best fit
is not arbitrary but is based on statistical considerations. In addition, the model structure
is based on theoretical principles and the model parameters often have physical
significance. In curve fitting, the choice of model structure is more arbitrary and is often
chosen to simplify the computations. The aim of parameter estimation is to not only fit a
model to data but to calculate parameter values that are good estimates of the true values
of the physical quantities.
Parameter estimation techniques can be applied to empirical models, but the statistical
properties of the estimates may not be as meaningful in a physical sense. In addition,
empirical models are not as well suited for extrapolation as mechanistic models.
Parameter estimation is an important step in the development of mathematical process
models. Process models contain parameters with physical significance that may vary
significantly from plant to plant. To develop process models that can be used for
predictive purposes, it is important to estimate the unknown process parameters using
measured process data. Using literature values for the unknown parameters will often
result in a model that is not very useful for predicting actual plant behavior.
Maximum Likelihood Method
GPS-X uses the maximum likelihood method for parameter estimation. In the maximum
likelihood method, the optimal parameter estimates are obtained by maximizing the joint
probability density function of the measurements. This joint probability density function
is a function of the parameters and is known as the likelihood function. The form of the
likelihood function depends on the structure of the experimental error.
GPS-X Technical Reference