Parameter estimation methods

The expression for the fugacity of a component j in a gas or liquid mixture, fj, based on the Trebble-Bishnoi EoS is available in the literature (Trebble and Bishnoi, 1988). This expression is given in Appendix 1. In addition the partial derivative, (dlnf/dx j>P, for a binary mixture is also provided. This expression is very useful in the parameter estimation methods that will be presented in this chapter. 

The complete data series is used to calculate the temperature response, but only certain parts of the experimental data are used to calculate the error. An example of a calibration run is given in Figure 53, the final calibrated TRNSYS model run is shown in Figure 54. Using the first part of the data (with constant heat flux) an estimate of ground thermal conductivity of 2.15 was obtained. Yavatzturk s method yielded an estimate of 2.18, while the estimate obtained with the TRNSYS parameter estimation method was 2.10. 

Model development and parameter estimation methods for PB-PK models are presented in Section III.B. 

At this point, no universal parameter estimation protocol has been adopted, although standard methods have been applied to individual models. The great flexibility that PB-PK models offer in terms of characterizing all types of drug transport makes routine parameter estimation methods unnecessary and unattractive. As with any modeling endeavor, the modeler should have a clear idea of which parameter estimation procedures will be employed so that experiments can be designed properly. 

From the resulting reactions a set of coupled differential equations can be derived describing the deactivation of P, L and PI and the reaction rate constants can be derived from storage stability data by the use of parameter estimation methods. The storage stability data give the concentration of P+PI (it is assumed that the inhibitor fully releases the protease during analysis due to fast dynamics and the extensive dilution in the assay) and L as a function of time. 

Signal processing may also involve parameter estimation methods (e.g., resolution of a spectral curve into the sum of Gaussian functions). Even in such cases, however, we may need non-parametric methods to approximate the position, height and half-width of the peaks, used as initial estimates in the parameter estimation procedure. 

Non-linear parameter estimation methods can also be used to estimate ka and kt. Two methods are recommended. Both use an organism-water two-compartment model (i.e., Equation (29)). The first method is the BIOFAC method (Blau and Algin 1978). This method is based on the condition that the chemical concentration in the water is constant throughout the duration of the experiment, such that Equation 29 can be integrated to give... 

The mathematical equations used, for the release of drug for other geometries, are shown in Table 6.4. With the exception of Equation (6.61d) for short times, one should use the nonlinear parameter estimation methods to determine the diffusivity of a drug. As shown in Table 6.4, it is recommended to use the approximation equations for long times and rearrange them into ... 

Table 2 Some Widely Used Parameter Estimation Methods for Characterization of Petroleum and its Fractions...

It is worth noting that the main difference in Figures 3 and 4 is the use of parameter estimation method for the calculation of the NRTL-SAC parameters, while for the UNIFAC model, the calculation is straightforward. Once the parameters (here, the segment numbers) are found, then they could be used for validation against other experimental data. 

The published work is divided between papers addressing applications, parameter estimation methods, and tests of hypotheses regarding mixture existence. Two NONMEM related papers described two subpopulations with respect to the clearance of intravenously administered cephalosporins (20, 21). The antianginal agent perhexilene, which is known to have polymorphic cytochrome P450-2D6... 

Witkowski and Rawlings (1990) point out that for the estimation technique outlined above, methods exist for the calculation of approximate confidence intervals of the estimated parameters (Bard 1974 Caracotsios 1986). These confidenee intervals provide a means by which to assess the parameter uneertainties. Obtaining parameter estimates with small confidenee intervals indicates that the available measurements contain enough information for parameter estimation. It has been shown that concentration and obscuration measurements are sufficient for estimating the four kinetic parameters corresponding to an isothermal batch crystallizer (Witkowski et al. 1990). A number of parameter estimation methods for batch crystallization are summarized by Tavare (1995). 

We revisit these data in Exercise 9.7 after we have developed more general parameter estimation methods in Chapter. 9.,... 

Given these expressions for the gradient and Hessian, we can construct a fairly efficient parameter estimation method for differential equation models using standard software for solving nonlinear optimization and solving differential equations with sensitivities. 

The typical for poly condensation this type value t = 60 min was chosen for all solvents, the value c is also the same for all solvents and therefore is included in the Eq. (26) constant coefficient and the parameter estimation method should be specially described. In paper [34] the values Q and for low-temperature polycondensation realization two conditions are adduced—with stirring (200 rpm) and without reactive mass Stirling. In the first case it is supposed, that stirring levels distinctions in rig value and then the simplest relationship was used for Q estimation [33] ... 

ARR residuals may be obtained from a DBG. If ARRs cannot be deduced from a DBG in closed symbolic form because of nonlinear implicit equations indicated by causal paths, ARR residuals are given implicitly. Their numerical values can be obtained by solving the entire DBG model in each step of the parameter estimation iteration. Derivatives of measured variables with respect to time that are needed in the evaluation of the DBG model are to be performed in discrete time. Once residuals are available for the time points of an observation window, the cost function can be built. If a gradient based parameter estimation method is used, the gradient of the least squares cost function can be obtained by using discrete derivatives. 

Finding the best fit involves calculation of appropriate values for the coefficients Aq and Aj in such a way that the net error between observed values of y and value of y predicted by straight line equation is the least. The least square parameter estimation method estimates... 

Velardi, S. A., Rasetto, V., Barresi, A. A., 2008. Dynamic Parameters Estimation Method advanced Manometric Temperature Measurement approach for freeze-drying... 

Moghaddass, R. and Zuo, M.J. 2011. A parameter estimation method for a multi-state deteriorating system with incomplete information. Proceedings of the Seventh International Conference on Mathematical Methods in Reliability Theory, Methods, and Applications (MMR 201 l) pp. 34-42. 

Ghasemi A., Yacout S. Quali M.S. 2010b. Parameter Estimation Methods for Condition-Based Maintenance With Indirect Observations, IEEE Transactions on Reliability 59(2) 426—429. 

The parameter estimation methods, the confidence lower bounds, and results of recent studies for comparing the methods are described in the following three sections. The proposed approach and its implementation are discussed later. 

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