Models fitted to data are used extensively in chemical engineering for a variety of purposes, including simulation, design and control. In any of these contexts it is important to assess the uncertainties in the estimated parameters and in any functions of these parameters, including predictions from the fitted model. Profiling M simulation, design, control and testing of hypotheses of underlying phenomena. A model is usually "calibrated" by fitting it to experimental or process data. Before using a fitted model, it is important to evaluate the uncertainty in any statistics of interest derived from the model. Such statistics may include the parameters themselves or functions of parameters, such as model predictions of yields and compositions. Occasionally, a model may be used to estimate some quantity or process characteristic which cannot be directly measured. For example, in process control the cross-over frequency may be of interest.For the purpose of illustration, consider the series reaction:where chemical species A is consumed to produce species B which is subsequently consumed in a reaction producing species C. Let y , , y z and y3 be the molar concentrations of chemical species A, B and C, respectively, present at time t. Given the initial conditions: y , = 1 and y z = y , = 0 at t = 0, and assuming first order kinetics, the model for the molar concentration of species B can be written as:
-02at which the concentration of B reaches a maximum. Note that t,, is a function of the parameters of the model (1). But to make judicious use of the estimates of the parameters ( O , , 0,) or of this function of parameters (tma), the uncertainty in each of those estimates must also be quantified, for it is this uncertainty which prescribes how much trust should be placed in the estimate.For models which are linear in the parameters, and for which the random errors in the observations are normally and independently distributed (the usual least squares assumptions), uncertainty in the parameters or in any linear combination of the parameters can be readily calculated. The results are analytic and exact. This uncertainty may be expressed via confidence intervals, or confidence regions for the parameters and confidence intervals for linear functions of the parameters. (The term "inference result" will be used to refer to confidence intervals, confidence regions, likelihood intervals or likelihood regions). For models which are nonlinear in the parameters, hereafter called nonlinear models, inference results for the parameters, or for any function of the parameters, are complex functions of the distribution of the random errors associated with the measured responses, the structure of the model, the parameterization of the model, and the design of the experiment (Donaldson and Schnabel, 1987;Bates and Watts, 1988). Exact inference results are generally unavailable for nonlinear models, except for special cases (see for example: Williams,
