GREGPLUS

Comment (1) is consistent with normal practice in parameter estimation. Use of the detailed p(0 ) in nonlinear parameter estimation would take much extra effort, since fourth derivatives of the expectation model would then appear. Comment (2) is implemented in Chapters 6 and 7 and in the package GREGPLUS. 

The following chapters and the package GREGPLUS apply these principles to practical models and various data structures. Least squares, multiresponse estimation, model discrimination, and process function estimation are presented there as special forms of Bayesian estimation. 

The permitted region of 6 can take various forms. Our package GREGPLUS uses a rectangular region... 

Uniform weighting (known as simple least squares) is appropriate wJien the expected variances of the observations are equal and is commonly used when these values are unknown. GREGPLUS uses this w eighting when called with JWT = 0 the values MwijJ = 1 are then provided automatically. 

The remaining sections of this chapter are implemented in the package GREGPLUS, described and demonstrated in Appendix C. 

The derivatives F r are called the first-order parametric sensitivities of the model. Their direct computation via Newton s method is implemented in Subroutines DDAPLUS (Appendix B) and PDAPLUS. Finite-difference approximations are also provided as options in GREGPLUS to produce the matrix A in either the Gauss-Newton or the full Newton form these approximations are treated in Problems 6.B and 6.C. 

Difficulties of this sort can be avoided by minimizing S 6) of Eq. (6.3-7) over a trust region of 6 in each iteration. A line search can be used to adjust the correction vector A6 and the trust-region dimensions for the next iteration. Such a method is outlined below and used in GREGPLUS. 

GREGPLUS calculates a F.Ui,U2) for this test whenever replicates are identified in the input see Examples 6.1, 6.2, and Appendix C. The user can then decide whether the reported probability a is large enough for acceptance of the model. 

GREGPLUS for all members of the estimated parameter set de- Several examples of these are given in Appendix C. 

The model with the largest posterior probability is normally preferred, but before acceptance it should be tested for goodness of fit, as in Section 6.5. GREGPLUS automatically performs this test and summarizes the results for all the candidate models if the variance cr been specified. 

Before accepting a model, one should test its goodness of fit, as outlined in Section 6.5 and implemented in GREGPLUS. Examples of model discrimination and criticism are given in Appendix C. 

The determinant criterion and the trace criterion are well regarded, and both are provided as options in GREGPLUS. Example C.l applies the full determinant criterion to a three-parameter model. 

The exponent A is set at 2 by GREGPLUS to emphasize discrimination initially as Hill, Hunter and Wichern did. 

Fitting these models to the data by nonlinear least squares yields the results in Table 6.2. Model 1 fits the data better than Model 2 and has a posterior probability share of 0.987. The posterior probability shares H Mj Y) are calculated here from Eq. (6.6-33) with p(A/i) = p M2) and normalization to a sum of 1. Analyses of variance for both models are given in Table 6.3 there is a significant lack of fit for Model 2. These results are as expected, since the data were constructed from Model 1 with simulated random errors. All these calculations were done with GREGPLUS. 

Each model fj was fitted to the data by weighted least squares, using our program GREGPLUS and the nonnegativity constraints 9iA > 0. The data for all temperatures were analyzed simultaneously under these constraints, as advocated by Blakemore and Hoerl (1963) and emphasized by Pritchard and Bacon (1975). The resulting nonlinear parameter estimation calculations were readily handled by GREGPLUS. 

With Hr defined as above, the rounding error in the difference quotient is dominated by the subtraction in the numerator of Eq. (6.B-3). GREGPLUS uses the estimate 6mS 6 ) for the rounding error of each numerator term, thereby obtaining the root-mean-square estimate... 

This result is used by GREGPLUS for those parameters that are marked with nonzero DEL(r) values to request divided-difference computations of parametric sensitivities. The same expression is used in the multiresponse levels of GREGPLUS, where negative values of S(6) can occur. 

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. 

GREGPLUS reports this matrix near the end of the estimation from data of Type 1. 

These four formulas need nonsingular block matrices Vb(0), obtainable by proper choice of response variables for each block. GREGPLUS uses subroutines from LINPACK (Dongarra et ah, 1979) to choose the working... 

GREGPLUS treats full or block-rectangular data structures, using the objective function S 6) = —2 np 6 y). This gives... 

GREGPLUS selects the working response variables for each application by least squares, using a modification of LINPACK Subroutine DCHDC (Dongarra et al., 1979) that forms the Cholesky decomposition of the matrix selected responses linearly independent, a... 

For inquisitive readers, we give next the detailed expressions used by GREGPLUS to compute 0-derivatives of Eqs. (7.2-1) and (7.2-3). Other... 

The terms in the second line of Eq. (7.2-14) vanish if the model is linear in 6 and are unimportant if the data are well fitted. They also are computationally expensive. GREGPLUS omits these terms this parallels the Gauss-Newton approximation in the normal equations of Chapter 6. 

A common choice is q = 0.05, which gives U[a/2) — 1.96, GREGPLUS computes this interval for each estimated parameter. 

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