Parameter estimation techniques

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. 

In this section three applications of the parameter estimation technique to problems in polymer science involving sequence distribution data are described. These problems are of varying degrees of difficulty and each serves to point out different aspects of the method. 

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-... 

We determined the reaction parameters using the optimal parameter estimation technique with the experimentally obtained copolymer yield and norbomene composition data. Based on the literature report, we assume that k = 3 [5]. Fig. 1 shows that the estimated rate constant values depend on the norbomene block length. Note that the reaction rate constant... 

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

An advantage over the line source method and other parameter estimation techniques is that the estimate can be made directly on the measured return temperature. Using the derived average heat extraction or injection rate may... 

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. 

This parameter-estimation technique has also been extended to the multiple-response case (D3). Just as was seen in the multiple-response... 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

Approaches based on parameter estimation assume that the faults lead to detectable changes of physical system parameters. Therefore, FD can be pursued by comparing the estimates of the system parameters with the nominal values obtained in healthy conditions. The operative procedure, originally established in [23], requires an accurate model of the process (including a reliable nominal estimate of the model parameters) and the determination of the relationship between model parameters and physical parameters. Then, an online estimation of the process parameters is performed on the basis of available measures. This approach, of course, might reveal ineffective when the parameter estimation technique requires solution to a nonlinear optimization problem. In such cases, reduced-order or simplified mathematical models may be used, at the expense of accuracy and robustness. Moreover, fault isolation could be difficult to achieve, since model parameters cannot always be converted back into corresponding physical parameters, and thus the influence of each physical parameters on the residuals could not be easily determined. 

Vink J. P. M., Nbrtershauser P., Richter O., Diekkriiger B., and Groen K. P. (1994) Modelling the microbial breakdown of pesticides in soil using a parameter estimation technique. Pestic. Sci. 40, 285-292. 

Because there are only two equations and four unknowns, values for the individual cannot be solved for. Only C and C2 can be obtained by nonlinear parameter estimation techniques. C] and C2 are imponant parameters, C2 is the retardation time (101) and C is the equilibrium die stress. 

Inadequacy of the parameter estimation technique used (especially with the large number of parameters involved in a relatively rigorous kinetic model), in addition to the different experimental errors involved especially in the rigorous calibration of the gas chromatographs used. 

More sophisticated techniques for the estimation of growth kinetics involve the use of the entire desupersaturation curve with parameter estimation techniques (Qiu and Rasmussen 1990). The combination of the desupersaturation curve and the crystal size distribution can be used to estimate both growth and nucleation... 

Having established the importance of reactivity ratios, it falls to the researcher to have to estimate their values. Given the number of statistical tools and computational devices available over the past several decades, one would expect this to be straightforward. However, there has been resistance to using proper parameter estimation techniques and the reader is advised to exercise caution when using reactivity ratios found in the literature [117]. A good practice is to consider reevaluating these from... 

The procedure recommended by Hochman and McCord [87] is to use two tracer probes to directly measure the bottom-to-top residence time, x-y, by the first appearance from an impulse injection at z = 0, and the top-to-bottom residence time, i2, with an impulse injection at z = L. Then, the recycle parameter, r, can be found from ft, = V/F g, Eq. (k) and the crossflow parameter, k-pO , from [Pg.637]

Constant distribution coefficient obtained by means of parameter estimation techniques using Aspen Tech simulation software. 

J. Santolaria, J.J. Aguilar, J.A. Yague, J. Pastor, Kinematic parameter estimation technique for calibration and repeatability improvement of articulated arm coordinate measuring machines,... 

RusseU, N.M. and Ekerdt, J.G. 1996. Nonlinear parameter estimation technique for kinetic analysis of thermal desorption data. Surf. Sci. 364 199-218. 

Chiu and co-workers [44] measured the cylindrical orthotropic thermal conductivity of spiral woven fabric composites using a mathematical model that they had devised previously. A parameter estimation technique was used to evaluate the thermal properties of spiral woven fabric composites to verify the predictability of the mathematical model. Good agreement was found between the temperatures measured in a transient heat conduction experiment and those calculated using the prediction equations formulated by the estimated parameters. 

---