Estimating Parameter Values

The only drawback in using this method is that any numerical errors introduced in the estimation of the time derivatives of the state variables have a direct effect on the estimated parameter values. Furthermore, by this approach we can not readily calculate confidence intervals for the unknown parameters. This method is the standard procedure used by the General Algebraic Modeling System (GAMS) for the estimation of parameters in ODE models when all state variables are observed. 

Table 7.1 Estimated parameter values with short cut methods for different values of the smoothing parameter (s/N) in IMSL routine CSSMH...

The above model equation now satisfies all the criteria for least squares estimation. As an initial guess for k we can use our estimated parameter values when it was assumed that no correlation was present. Of course, in the second step we have to include p in the list of the parameters to be determined. 

When the Gauss-Newton method is used to estimate the unknown parameters, we linearize the model equations and at each iteration we solve the corresponding linear least squares problem. As a result, the estimated parameter values have linear least squares properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

It was shown by Englezos et al. (1998) that use of the entire database can be a stringent test of the correlational ability of the EoS and/or the mixing rules. An additional benefit of using all types of phase equilibrium data in the parameter estimation database is the fact that the statistical properties of the estimated parameter values are usually improved in terms of their standard deviation. 

First the values for the parameter vector k=[p(0), p(l), C9 ]T were obtained by using the Na2Si03 data and minimizing Equation 15.1. The estimated parameter values are shown in Table 15.3 together with their standard deviation. 

Table 16.1 Catalytic Oxidation of 3-Hexanol Estimated Parameter Values and Standard Deviations...

The LS objective function was found to be 0.7604x10"9. This value is almost three orders of magnitude smaller than the one found earlier at a local optimum. The estimated parameter values were At=22.672, A2=132.4, A3=585320, Ej=l3899, E2=2439.6 and E3=13506 where parameters A, and E were estimated back from Ai and E. With this reparameterization we were able to lessen the ill-conditioning of the problem since the condition number of matrix A was now 5.6x108. 

The model-calculated reaction rates are compared to the experimental data in Table 16.11 where it can be seen that the match is quite satisfactory. Based on the six estimated parameter values, the kinetic constants (k, k2 and k3) were computed at each temperature and they are shown in Table 16.12. 

Table 16.27 HPA Hydrogenation Estimated Parameter Values from the Data Collected at 333 K and 353 K...

Table 17.4 Enzyme Kinetics Estimated Parameter Values...

Figures 18.13, through 18.17 show the experimental data and the calculations based on model I for the low temperature oxidation at 50, 75, 100, 125 and 150TZ of a North Bodo oil sands bitumen with a 5% oxygen gas. As seen, there is generally good agreement between the experimental data and the results obtained by the simple three pseudo-component model at all temperatures except the run at 125 TT. The only drawback of the model is that it cannot calculate the HO/LO split. The estimated parameter values for model I and N are shown in Table 18.2. The observed large standard deviations in the parameter estimates is rather typical for Arrhenius type expressions.

In Figures 18.18, 18.19 and 18.20 the experimental data and the calculations based on model I are shown for the high temperature cracking at 360, 397 and 420 T of an Athabasca oil sands bitumen (Drum 20). Similar results are seen in Figures 18.21, 18.22 and 18.23 for another Athabasca oil sands bitumen (Drum 433). The estimated parameter values for model I are shown in Table 18.3 for Drums 20 and 433. 

Table 18.2 Estimated Parameter Values for Models I and Nfor the Low Temperature Oxidation of North Bodo Oil Sands Bitumen...

The validity of the computer prediction must be checked. After agreeing sufficiently well with available knowledge, experiments must then be designed to further check its validity and to estimate parameter values. Steps (1) to (4) will often need to be revised at frequent intervals. 

Figure 4.17 Profiles of (a) Fe(II), (b) Fe(III) and (c) pH in columns of reduced soil exposed to O2 at one end for different times. Points are experimentally measured lines are predicted using the model described in the text with independently estimated parameter values (Kirk and Solivas, 1994). Reproduced by permission of Blackwell Publishing...

Equation 7.1 is one of the most important relationships in the area of experimental design. As we have seen in this chapter, the precision of estimated parameter values is contained in the variance-covariance matrix V the smaller the elements of V, the more precise will be the parameter estimates. As we shall see in Chapter 11, the precision of estimating the response surface is also directly related to V the smaller the elements of V, the less fuzzy will be our view of the estimated surface. 

Another approach for the determination of the kinetic parameters is to use the SAS NLIN (NonLINear regression) procedure (SAS, 1985) which produces weighted least-squares estimates of the parameters of nonlinear models. The advantages of this technique are that (1) it does not require linearization of the Michaelis-Menten equation, (2) it can be used for complicated multiparameter models, and (3) the estimated parameter values are reliable because it produces weighted least-squares estimates. 

When the relative volumes are known and the diffusion coefficients in the capsule core and capsule membrane can be estimated a priori in single component adsorption, the parameter to work with is the effective diffusivity in the adsorbent pore (Dn). Then, with the above estimated parameter values, the parameters of competitive adsorption are the maximum concentration at the solid phase of the adsorbent (CsmT), and the equilibrium constants of the target product (KS1) and byproduct (KS2)-... 

A posteriori identifiability is linked to the theory of optimization in mathematics because one normally uses a software package that has an optimization (data-fitting) capability in order to estimate parameter values for a multicompartmental model from a set of pharmacokinetic data. One obtains an estimate for the parameter values, an estimate for their errors, and a value for the correlation (or covariance) matrix. The details of optimization and how to deal with the output from an optimization routine are beyond the scope of this chapter, and the interested reader is referred to Cobelli et al. (12). The point to be made here is that the output from these routines is crucial in assessing the goodness-of-fit — that is, how well the model performs when compared to the data — since inferences about a drug s pharmacokinetics will be made from these parameter values. 

We have shown that the values of oxyacids of the type HOX, where X is [M(OH) 0 ], are well described by equation 23 with q representing the charge on the X group. This result is in accord with equation 41. We conclude that, in general, ionic substituents can be described by simply adding the term a q to the equation used to estimate parameter values for uncharged groups. Thus ... 

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