Process Modeling with Multiresponse Data

Multiresponse experimentation is important in studies of complex systems and of systems observed by multiple methods. Chemical engineers and chemists use multiresponse experiments to study chemical reactions, mixtures, separation, and mixing processes similar data structures occur widely in science and engineering. In this chapter we study methods for investigating process models with multiresponse data. Bayes theorem now yields more general methods than those of Chapter 6, and Jeffreys rule, discussed in Chapter 5, takes increased importance. 

The methods of Chapter 6 are not appropriate for multiresponse investigations unless the responses have known relative precisions and independent, unbiased normal distributions of error. These restrictions come from the error model in Eq. (6.1-2). Single-response models were treated under these assumptions by Gauss (1809, 1823) and less completely by Legendre (1805), co-discoverer of the method of least squares. Aitken (1935) generalized weighted least squares to multiple responses with a specified error covariance matrix his method was extended to nonlinear parameter estimation by Bard and Lapidus (1968) and Bard (1974). However, least squares is not suitable for multiresponse problems unless information is given about the error covariance matrix we may consider such applications at another time. 

Bayes theorem (Bayes 1763 Box and Tiao 1973, 1992) permits estimation of the error covariance matrix S from a multiresponse data set, along with the parameter vector 0 of a predictive model. It is also possible, under further assumptions, to shorten the calculations by estimating 6 and I separately, as we do in the computer package GREGPLUS provided in Athena. We can then analyze the goodness of fit, the precision of estimation of parameters and functions of them, the relative probabilities of alternative models, and the choice of additional experiments to improve a chosen information measure. This chapter summarizes these procedures and their implementation in GREGPLUS details and examples are given in Appendix C. 

Jeffreys (1961) advanced Bayesian theory by giving an unprejudiced prior density p(6, S) for suitably differentiable models. His result, given in Chapter 5 and used below, is fundamental in Bayesian estimation. 

Box and Draper (1965) took another major step by deriving a posterior density function p 6 Y), averaged over S, for estimating a parameter vector 6 from a full matrix Y of multiresponse observations. The errors in the observations were assumed to be normally distributed with an unknown m X m covariance matrix S. Michael Box and Norman Draper (1972) gave a corresponding function for a data matrix Y of discrete blocks of responses and applied that function to design of multiresponse experiments. 

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Chapter 7 Process Modeling with Multiresponse Data... 

Process Modeling with Multiresponse Data and substitution into Eq. (A.4) gives... 

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