Estimates of model parameters

Estimation of model parameters is frequently accomplished by the method of moments. For example, for the uniform distribution, the mean is... 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

This system of linear algebraic equations is easy to solve to find the estimates of model parameters b,. It can be rewritten in more general matrix notation ... 

The objective is to demonstrate the power of modern simulation packages in the estimation of model parameters. Here the parameters are estimated using SIMUSOLV running on VAX systems and on a PC using the ESL simulation package, the main features of which can be found on the last page of this book. 

The three described groups of methodologies are experimental ways leading to the estimation of model parameters for the description of the anaerobic processes according to the aerobic-anaerobic conceptual model (Table 6.6). The determination of the remaining kinetic and stoichiometric parameters in this model, however, requires a calibration procedure, where the results of the above three described methodologies are used. Table 6.7 shows typical values of such parameters determined by the three methodologies folio wed by a model calibration. 

The estimation of model parameters is an important activity in the design, evaluation, optimization, and control of a process. As discussed in previous chapters, process data do not satisfy process constraints exactly and they need to be rectified. The reconciled data are then used for estimating parameters in process models involving, in general, nonlinear differential and algebraic equations (Tjoa and Biegler, 1992). 

Estimates of modeled parameters of particular oceanic processes of the carbon cycle range widely. For instance, from the data of various authors the estimates of assimilation of carbon from the hydrosphere in the process of photosynthesis range from 10 GtC/yr to 155 GtC/yr. The value 127.8 GtC/yr is most widely used. However, because of large variations in these estimates, calculation of the C31 coefficient is fraught with great uncertainty therefore, specifying it requires numerical experiments using other, more accurate data. 

Estimates of Model Parameters. The reactor models for FFB, MAT and riser include important features for translating the MAT and FFB data to steady state riser performance. A series of key parameters specific to a given zeolite and matrix component are needed for a given catalyst. Such key parameters are intrinsic cracking anc( coking activities (kj, A ), activation energies and heats of reaction (Ej, AHj), coke deactivation rate (exponents nj), and axial dispersion in the FFB unit (DA). Other feedstock dependent parameters include the inhibition constants (kHAj), the coking constants (XAj), and the axial molar expansion factor (a). 

The estimation of model parameters can be reduced to the solution of an optimization problem. This means that once the objective is defined, a class of mathematical functions that best match the specified objectives must be found. 

Most industrial applications are boundary value problems. These include (1) the determination of reactor diameter and length while meeting specifications in residence time and maximum tubeskin temperature, (2) the determination of maximum feed rates subjected to a constraint in the tubeskin temperature and (3) the estimation of model parameters, such as reaction rate constants from yield data. Examples 1 and 2 will be discussed in detail later. 

Because estimation of model parameters, b0, bx,. .., bm uses m + 1 degrees of freedom, the remaining n-m- 1 degrees of freedom are used to estimate RMSEC. If the intercept b, is omitted from the calibration model, then the number of degrees of freedom for RMSEC is n-m. If the data has been mean-centered, the degrees of freedom remain n-m-1. Typically, RMSEC provides overly optimistic estimates of a calibration model s predictive ability for samples measured in the future. This is because a portion of the noise in the standards is inadvertently modeled by the estimated parameters. A better estimate of the calibration model s predictive ability may be obtained by the method of cross-validation with the calibration samples or from a separate set of validation samples. 

In accordance with least-squares method the error functional minimization of which will give the best estimations of model parameters has the following form ... 

Optimal choice of an additional event INEXT from a list of candidates, to obtain better estimates of model parameters and auxiliary functions and/or better discrimination among a set of candidate models. 

It was shown that the overall performance of the state estimation was improved with a constrained EKF, a time-varying process covariance matrix Q and the use of the proper estimator as a transient data reconciliation technique. In a further work, the proposals of this work will be also evaluated and compared through the MHE formulations proposed by [12]. Besides, an algorithm for automatic selection and estimation of model parameters proposed by [11] will be used to estimate the parameter covariance matrix. 

Collection of data that will allow more accurate estimates of model parameters. 

In estimating the range of ingested doses which could have resulted in a given biomarker concentration, there are three main sources of variability errors in model selection, errors in estimation of model parameters, and tme population variability (i.e., heterogeneity). The variability due to the first two sources can be reduced by the collection... 

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