Constrained Parameter Estimation

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

The above constrained parameter estimation problem becomes much more challenging if the location where the constraint must be satisfied, (xo,yo), is not known a priori. This situation arises naturally in the estimation of binary interaction parameters in cubic equations of state (see Chapter 14). Furthermore, the above development can be readily extended to several constraints by introducing an equal number of Lagrange multipliers. 

Most of the constrained parameter estimation problems belong to this case. Based on scientific considerations, we arrive quite often at constraints that the parameters of the mathematical model should satisfy. Most of the time these are of the form,... 

Should A be singular, this method still gives a basis inverse, namely, the inverse of the submatrix of A in which the pivots were found. A version of this algorithm using strictly positive, diagonal pivots is used for constrained parameter estimation in Subroutine GREGPLUS, as described in Sections 6.4 and 7.3 there the determinant of the basis submatrix is obtained directly as the product of the pivotal divisors. Basis inverse matrices also prove useful in Sections 6.6 and 7.5 for interval estimation of parameters and related functions. 

The implicit LS, ML and Constrained LS (CLS) estimation methods are now used to synthesize a systematic approach for the parameter estimation problem when no prior knowledge regarding the adequacy of the thermodynamic model is available. Given the availability of methods to estimate the interaction parameters in equations of state there is a need to follow a systematic and computationally efficient approach to deal with all possible cases that could be encountered during the regression of binary VLE data. The following step by step systematic approach is proposed (Englezos et al. 1993)... 

If incorrect phase behavior is predicted by the EOS then constrained least squares (CLS) estimation should be performed and new parameter estimates be obtained. Subsequently, the phase behavior should be computed again and if the fit is found to be acceptable for the intended applications, then the CLS estimates should suffice. This was found to be the case for the carbon dioxide-n-hexane system presented later in this chapter. 

Figure 14.8 The stability function calculated with interaction parameters from simplified constrained LS estimation.

Figure 14.10 The stability function calculated with interaction parameters from constrained LS estimation [reprinted from Computers Chemical Engineering with permission from Elsevier Science],...

Parameter Estimation and Optimization Using Constrained Derivatives... 

PBPK and classical pharmacokinetic models both have valid applications in lead risk assessment. Both approaches can incorporate capacity-limited or nonlinear kinetic behavior in parameter estimates. An advantage of classical pharmacokinetic models is that, because the kinetic characteristics of the compartments of which they are composed are not constrained, a best possible fit to empirical data can be arrived at by varying the values of the parameters (O Flaherty 1987). However, such models are not readily extrapolated to other species because the parameters do not have precise physiological correlates. Compartmental models developed to date also do not simulate changes in bone metabolism, tissue volumes, blood flow rates, and enzyme activities associated with pregnancy, adverse nutritional states, aging, or osteoporotic diseases. Therefore, extrapolation of classical compartmental model simulations... 

Tjoa and Biegler (1991) used this formulation within a simultaneous strategy for data reconciliation and gross error detection on nonlinear systems. Albuquerque and Biegler (1996) used the same approach within the context of solving an error-in-all-variable-parameter estimation problem constrained by differential and algebraic equations. 

Kiers, H. and Smilde, A.K., Constrained three-mode factor analysis as a tool for parameter estimation with second-order instrumental data, J. Chemom., 12,125-147, 1998. 

The values of the estimated parameters and their confidence intervals (Cl) are presented in Table 1. From these results, a ratio R of fco-DHB p-DHB to fco-DHB cOj can be established and used to constrain the estimation of parameters in the complete reaction system. 

GREGPLUS in selecting pivots in the normal-equation matrix A for each constrained minimization of S 6). GREGPLUS does not judge a parameter estimable unless its test divisor exceeds ADTOL at pivoting time. 

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