Obtaining Initial Parameter Estimates

One method to obtain initial estimates is linearization of the problem. For example, the 1-compartment model with bolus intravenous administration can be reformulated to a linear problem by taking the log-transform on both sides of the equation. This is a trivial example, but is often reported in texts on nonlinear regression. Another commonly used method is by eyeballing the data. For example, in an Emax model, Emax can be started at the maximal of all the observed effect values. EC50 can be started at the middle of the range of concentration values. Alternatively, the data can be plot- 

regressing Y against t (without intercept) an estimate of 03 can be obtained. 

Peeling, curve stripping, or the method of residuals is an old technique advocated in some of the classic textbooks on pharmacokinetics (Gibaldi and Perrier, 1982) as a method to obtain initial estimates. The method will not be described in detail the reader is referred to the references for details. Its use will be briefly illustrated. Consider the biexponential model 

For values of t greater than or equal to t, linear regression of Ln(Y) versus t will give an estimate of 03 and 04. The next step is to use Eq. (3.80) to predict Y for the values of t not included in the estimation of 03 and 04 (t t ) and then calculate the residuals, Y, 

The residuals can then be used to estimate the first term in Eq. (3.78), which after taking the natural log becomes 

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Using this crude grid, the minimum residual sum of squares can be found at a volume of distribution of 126 L and a clearance of 18 L/h, which was very near the theoretical values of 125 L and 19 L/h, respectively, used to simulate the data. Often a grid search, such as the one done here, is a good way to obtain initial parameter estimates. In fact, some software offer a grid search option if initial parameter estimates are unavailable. 

When estimates of k°, k, k", Ky, and K2 have been obtained, a calculated pH-rate curve is developed with Eq. (6-80). If the experimental points follow closely the calculated curve, it may be concluded that the data are consistent with the assumed rate equation. The constants may be considered adjustable parameters that are modified to achieve the best possible fit, and one approach is to use these initial parameter estimates in an iterative nonlinear regression program. The dissociation constants K and K2 derived from kinetic data should be in reasonable agreement with the dissociation constants obtained (under the same experimental conditions) by other means. 

In the example above, the sums-of-squares surface is transformed to that shown by Fig. 10. The best point estimates of the parameters of kQ and E are thus more readily obtained than k0 and E the initial-parameter estimates are less critical, and the estimation routine converges more rapidly to the minimum. Although the correlation between the parameter estimates has been reduced by this reparameterization, the size of the confidence region of the original parameters k0 and E will not change. 

In addition to these problems, there exist the set of conditions under which the user must manually set all initial parameter estimates. Manual constraints on the parameters may often be the only way to obtain a proper convergence if the true, bounded region of the solution is known by the user, or if one has some specific knowledge of the correct starting values. This may be particularly true if there are additional parameters with carplex functional forms such as phase angle. Automated paramater setting which takes into account some of these problems could lead to more consistent results and require less user expertise. 

Initial parameter estimates were obtained from the PCA versus time data. The baseline value (120 s) was obtained from the intercept on the effect axis. This value is the ratio Km/kd. From the intercept and slope, Alin was calculated to be 3.5 s h 1. The plasma concentration at the time of the trough of the effect corresponded approximately with the EC50 value. Thus, IC50=0.35mg 1 k = 0.3 A-1, n = 3.5,... 

Fig. 5.59. Concentration-time curves obtained by HPLC experimental data after volume correction (symbols) together with data obtained by parameter estimation according to a Runge-Kutta simulation (curve). In methanol/water (3 1) initially at 4.4 x 10 mol/1 solution the quantum yields were determined to 0.4, = 0.2 and = 0.03 for irradiation at 313...

In metabolism and nutrition, where each experiment has been designed to be complete and population analysis is used to fill in missing values and to incorporate relative uncertainties into the estimation, a good procedure would be to first examine in detail those individual studies which are the most complete. This will familiarize the user with the behavior of the model, produce initial estimates for the system parameters, provide a chance to verify that these values are reasonable, and allow the use of tools for the identifiability of individual experiments (Jacquez and Perry, 1990). After this exercise has been complete, all experiments, including those that are incomplete, can be pooled for population analysis and testing the effects of covariates. If required, the final step would be to use the estimated distributions to obtain Bayesian parameter estimates for the individual experiments. This procedure should yield the most appropriate estimates for the incomplete experiments. 

The above simple methods of estimating y/zc depend only on the determination of Xo and yo from the impedance complex plane arc or on the use of a few points in the admittance plane. Althongh they are often adequate for initial investigation, it is worth mentioning a relatively simple alternative procedure which can be used to test the appropriateness of Eqs (5) and (6) and obtain the parameter estimates of interest. Consider the point Z on the arc of Fignre 1.3.2, a point marking a specific value of Z. It follows from the figure and Eq. (5) that Z - = (Bo - BJ/f = u and Bq -... 

The initial estimates for the pre-exponential factors and the activation energies are based on literature [4]. The initial formaldehyde concentrations were given by the experimentalists. For numerical reasons it is better to have the parameters of the same order of magnitude. To obtain this we take the natural logarithm of the pre-exponential factors, and the activation energies are scaled by a factor 1/1000, E = Ei/1000. The scaled initial parameter estimates are listed in the second column of Table 8.2. 

Using the basic Kalman filter as given in Theorem 5.5 and the initial parameter estimate, 6q, obtain the innovations and error covariances for t=l,..m. 

The proposed model function shown in Figure 9.44 is the model described in the referenced source for the data set. The function definitions and initial parameter estimations are shown in Listing 9.12 along with the evaluated model parameters with standard errors printed in the selected output listing. The model has four parameters, but the data range covers a sufficient range of the functional dependency to obtain good estimates of all the parameters. 

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. 

Obtaining Kinetic Samples for Reactive Extrusion. To develop and test kinetic models, homogeneous samples with a well defined temperature-time history are required. Temperature history does not necessarily need to be isothermal. In fact, well defined nonisothermal histories can provide very good test data for models. However, isothermal data is very desirable at the initial stages of model building to simplify both model selection and parameter estimation problems. 

The structure of such models can be exploited in reducing the dimensionality of the nonlinear parameter estimation problem since, the conditionally linear parameters, kl5 can be obtained by linear least squares in one step and without the need for initial estimates. Further details are provided in Chapter 8 where we exploit the structure of the model either to reduce the dimensionality of the nonlinear regression problem or to arrive at consistent initial guesses for any iterative parameter search algorithm. 

Step 1. Perform a series of initial experiments (based on a factorial design) to obtain initial estimates for the parameters and their covariance matrix. 

In this problem you are asked to determine the unknown parameters using the dominant zeros and poles of the original system as an initial guess. LJ optimization procedure can be used to obtain the best parameter estimates. 

The MO concentrations versus time profiles were fitted to second order polynomial equations and the parameters estimated by nonlinear regression analysis. The initial rates of reactions were obtained by taking the derivative at t=0. The reaction is first order with respect to hydrogen pressure changing to zero order dependence above about 3.45 MPa hydrogen pressure. This was attributed to saturation of the catalyst sites. Experiments were conducted in which HPLC grade MIBK was added to the initial reactant mixture, there was no evidence of product inhibition. 

In this section the sequential approach discussed in Chapter 6 will be extended to parameter estimation. An initial estimate of 0 can be obtained from a minimal data set for which the number of observations is equal to the number of components of 0. Let yo denote an observation at time t = to if 9 has n components, the minimal data set is symbolized by the vector... 

Different runs were performed by changing the noise level on the measurements as well as initial value of the parameter. Only a single set of experimental data was considered in our calculations. Table 2 gives a summary of the results, that is, the final value of the parameter obtained from the application of the algorithm, as a function of the initial value and the measurement noise. We can clearly see the effect of the noise level on the parameter estimator s accuracy, as well as its effect on the number of iterations. 

By using only simple hand calculations, the single-site model has been rejected and the dual-site model has been shown to represent adequately both the initial-rate and the high-conversion data. No replicate runs were available to allow a lack-of-fit test. In fact this entire analysis has been conducted using only 18 conversion-space-time points. Additional discussion of the method and parameter estimates for the proposed dual-site model are presented elsewhere (K5). Note that we have obtained the same result as available through the use of nonintrinsic parameters. 

Structural information on aromatic donor molecule binding was obtained initially by using H NMR relaxation measurements to give distances from the heme iron atom to protons of the bound molecule. For example, indole-3-propionic acid, a structural homologue of the plant hormone indole-3-acetic acid, was found to bind approximately 9-10 A from the heme iron atom and at a particular angle to the heme plane (234). The disadvantage of this method is that the orientation with respect to the polypeptide chain cannot be defined. Other donor molecules examined include 4-methylphenol (p-cresol) (235), 3-hydroxyphenol (resorcinol), 2-methoxy-4-methylphenol and benzhydroxamic acid (236), methyl 2-pyridyl sulfide and methylp-tolyl sulfide (237), and L-tyrosine and D-tyrosine (238). Distance constraints of between 8.4 and 12.0 A have been reported (235-238). Aromatic donor proton to heme iron distances of 6 A reported earlier for aminotriazole and 3-hydroxyphenol (resorcinol) are too short because of an inappropriate estimate of the molecular correlation time (239), a parameter required for the calculations. Distance information for a series of aromatic phenols and amines bound to Mn(III)-substituted HRP C has been published (240). 

Most standard chemical engineering tests on kinetics [see those of Car-berry (50), Smith (57), Froment and Bischoff (19), and Hill (52)], omitting such considerations, proceed directly to comprehensive treatment of the subject of parameter estimation in heterogeneous catalysis in terms of rate equations based on LHHW models for simple overall reactions, as discussed earlier. The data used consist of overall reaction velocities obtained under varying conditions of temperature, pressure, and concentrations of reacting species. There seems to be no presentation of a systematic method for initial consideration of the possible mechanisms to be modeled. Details of the methodology for discrimination and parameter estimation among models chosen have been discussed by Bart (55) from a mathematical standpoint. 

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