Initial parameter estimates

Subroutine VLDTA2. VLDTA2 loads the binary vapor-liquid equilibrium data to be correlated. If the data are in units other than those used internally, the correct conversions are made here. This subroutine also reads the estimated standard deviations for the measured variables and the initial parameter estimates. All input data are printed for verification. 

Second card FORMAT(8F10.2), control variables for the regression. This program uses a Newton-Raphson type iteration which is susceptible to convergence problems with poor initial parameter estimates. Therefore, several features are implemented which help control oscillations, prevent divergence, and determine when convergence has been achieved. These features are controlled by the parameters on this card. The default values are the result of considerable experience and are adequate for the majority of situations. However, convergence may be enhanced in some cases with user supplied values. 

I. The next card gives the initial parameter estimates. 

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. 

A method is described for fitting the Cole-Cole phenomenological equation to isochronal mechanical relaxation scans. The basic parameters in the equation are the unrelaxed and relaxed moduli, a width parameter and the central relaxation time. The first three are given linear temperature coefficients and the latter can have WLF or Arrhenius behavior. A set of these parameters is determined for each relaxation in the specimen by means of nonlinear least squares optimization of the fit of the equation to the data. An interactive front-end is present in the fitting routine to aid in initial parameter estimation for the iterative fitting process. The use of the determined parameters in assisting in the interpretation of relaxation processes is discussed. 

Figure 5. Calculated curves from initial parameter estimates of Figure 4.

A brief survey of the sums-of-squares surface, such as shown in this figure, can eliminate many computer hours of fruitless iteration with repeated initial parameter estimates. Other problems related to estimation from Figs. 7 and 8 can be reduced by reparameterization, as discussed in Section III,B. 

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. 

Automated initial parameter estimation may not be accurate and then the user will be required to intercede with manual estimation. 

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. 

MATRIX OF WEIGHTING COEFFICIENTS ONLY FOR N1=2 i INITIAL PARAMETER ESTIMATES THRESHOLD ON RELATIVE STEF LENGTH MAXIMUM NUMBER OF ITERATIONS... 

This method, first introduced by Isaac Newton and better formulated in the actual form by Joseph Raphson, is the simplest second-order algorithm. The basic idea is to use a quadratic approximation to the objective function around the initial parameter estimate and, then, to adjust the parameters in order to minimize the quadratic approximation until their values converge. 

The initial parameter estimations of pXmax and ks can be based on a simplification of Equation 21, by disregarding the inhibition term for the substrate. Thus, Equation 21 becomes Equation 80. 

This technique has been extended to the fitting of two-dimensional NMR spectra.90 In the cores of large molecules such as peptides and proteins, well-chosen areas need to be selected in order to avoid exorbitant computation time. Initial parameter estimates corresponding to the appropriate portion of the multidimensional x2 surface are of crucial importance to ensure convergence to the global minimum corresponding to the correct parameter set. 

Wong-Sandler mixing rule is extremely sensitive to the initial parameters estimation, such that with the UNIQUAC method it was not capable to converge in most of the mixtures ... 

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,... 

The simplex optimization is a very simple and robust technique for optimizing any function of a moderate number of variables. Only the function values for different variable sets are needed. In this case, the function to be optimized is the penalty function, and the variables to vary are the force field parameters. To initialize a simplex optimization of N parameters, one must first select Ai + 1 linearly independent trial sets. A very simple way to achieve this is to start with the initial parameter estimate and then to vary each parameter in turn by a small amount, yielding N new trial sets. This is illustrated for a two-parameter case in Fig. 7. With two parameters, the shape of the simplex is a triangle, with three parameters a tetrahedron, and so on. 

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. 

Initial parameter estimates for the first-order rate constant were arbitrarily set equal to 2/h and inhalational bioavailability was set equal to 0.9. After fitting the model, the estimate of ka was 660 per hour. Hence, in essence, all the drug reached the systemic compartment almost immediately. The model was modified, removing the dosing compartment, and allowing the drug to be treated as an intravenous model with variable bioavailability (Model 3 in Fig. 5.5). Under this model the bioavailability of the inhaled dose was 62.6 4.3%. 

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