Estimation techniques, model parameter

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. 

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. 

As described in Sec. 12.3, the moments of the impulse response can be used to characterize the RTD curve. This technique is also useful here for estimating the model parameter, D , although better techniques will be described below for the latter. It is found for a closed system that... 

Estimate the model parameters using graphical techniques and the basall dataset. Then compare the model and experimental response data for both datasets. 

In Fig. 16.26, three sets of computations are employed estimation of the model parameters, calculation of the controller settings, and implementation of the controller output in a feedback loop. Most realtime parameter estimation techniques require that an external forcing signal occasionally be introduced to allow accurate estimation of model parameters (Hang et al., 1993). Such a pertubation signal can be deliberately introduced through the set point or added to the controller output. 

The classification methods illustrated in the previous subsection are only a part, although relevant, of the methods proposed in the literature for discriminant classification. Indeed, to cope with the different degrees of complexity that real-world classification problems involve (various degrees of class separability, requiring a corresponding level of nonlinearity in the model statistical assumption which may not always hold insufficient number of observations to estimate the model parameters and so on). As a complete description of all the discriminant techniques goes beyond the scope of the present text, the reader is referred to specific literature covering these other methods in more detail [39-42]. 

We have presented applications of a parameter estimation technique based on Monte Carlo simulation to problems in polymer science involving sequence distribution data. In comparison to approaches involving analytic functions, Monte Carlo simulation often leads to a simpler solution of a model particularly when the process being modelled involves a prominent stochastic coit onent. 

Classic parameter estimation techniques involve using experimental data to estimate all parameters at once. This allows an estimate of central tendency and a confidence interval for each parameter, but it also allows determination of a matrix of covariances between parameters. To determine parameters and confidence intervals at some level, the requirements for data increase more than proportionally with the number of parameters in the model. Above some number of parameters, simultaneous estimation becomes impractical, and the experiments required to generate the data become impossible or unethical. For models at this level of complexity parameters and covariances can be estimated for each subsection of the model. This assumes that the covariance between parameters in different subsections is zero. This is unsatisfactory to some practitioners, and this (and the complexity of such models and the difficulty and cost of building them) has been a criticism of highly parameterized PBPK and PBPD models. An alternate view assumes that decisions will be made that should be informed by as much information about the system as possible, that the assumption of zero covariance between parameters in differ-... 

In the minds of all authors who favour the estimation of flashpoints based on a theoretical model rather than experimental results this approach was temporary and only supposed to be used during the period used by commissions of experts to lay down a standard technique for the determination of flashpoints. As has already been seen, it is less likely that this method will be used in the near future. This is the reason why we think estimation techniques have to be part of the priority tools of risk analysis in work on chemical risk prevention. Why is such work on estimation important We will see later that flashpoint is the cruciai parameter in order to establish the ievel of fire hazard of a substance. 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary. In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package SIMUSOLV. Beeause of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

DATA ANALYSIS USING A NUMERICAL MODEL AND PARAMETER ESTIMATION TECHNIQUES... 

The Geothermal Response Test as developed by us and others has proven important to obtain accurate information on ground thermal properties for Borehole Heat Exchanger design. In addition to the classical line source approach used for the analysis of the response data, parameter estimation techniques employing a numerical model to calculate the temperature response of the borehole have been developed. The main use of these models has been to obtain estimates in the case of non-constant heat flux. Also, the parameter estimation approach allows the inclusion of additional parameters such as heat capacity or shank spacing, to be estimated as well. 

Estimation methods for tissue-to-blood partition coefficients (i.e., Rt) have been the most prolific, no doubt due to the need for this parameter in most organ models. Both in vitro and in vivo parameter estimation techniques are available. 

The CAT model considers passive absorption, saturable absorption, degradation, and transit in the human small intestine. However, the absorption and degradation kinetics are the only model parameters that need to be determined to estimate the fraction of dose absorbed and to simulate intestinal absorption kinetics. Degradation kinetics may be determined in vitro and absorption parameters can also be determined using human intestinal perfusion techniques [85] therefore, it may be feasible to predict intestinal absorption kinetics based on in vitro degradation and in vivo perfusion data. Nevertheless, considering the complexity of oral drug absorption, such a prediction is only an approximation. 

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