Parameters, estimation

The flux parameters are usually estimated from the tracer studies data by minimization of the deviations between experimental and modeled labeling data corresponding to the optimized set of fluxes. In general, the isotopomer balance equations are non-linear and numerical routines are used for their so- 

In case, the network is overdetermined a least square approach is possible. The information obtained by MS from tracer studies can be combined with metabolite balancing for the estimation of flux parameters [25, 27]. Measurement errors can be included to predict uncertainties for the obtained flux parameters [23]. 

The analysis of two moments offers to estimate only two characteristic parameters that characterize a peak but not the whole peak shape. By fitting of either analytical equations (Section 6.5.3.3) or simulation results to the complete peak observed this drawback may be overcome. It also allows a direct comparison between the calculated and measured concentration profiles. 

As suitable analytical solutions are not available for most of the column models, simulation-based parameter estimation using simulation software can be recommended as a powerful tool. This method is very versatile in terms of the number and complexity of models that can be handled. Estimation tasks can be solved, for example, with the gEST tool included in the gPROMS program package (PS Enterprise, UK). An additional advantage of the simulation-based approach is the consistency of the obtained data, if the same models and simulation tools are used for subsequent process analysis and optimization. 

The fitting procedure results in a set of model parameters, minimizing the difference between theoretical and measured concentration profiles. A discussion of statistical-based objective functions and optimization procedures is beyond the scope of this book. For further information see, for example, Lapidus (1962), Bams (1994), Korns (2000), and Press et al. (2002, http //www.nr.com/). 

Objective functions O are often based on least squares methods. They can be, for example, a function of the absolute or relative squared error of Wp measured concentration values, Cexp, and the theoretical values, Ctheo- 

In general, objective functions can capture more than one set of experimental data. 

Chemical engineers develop mathematical models of systems of interest that usually include parameters. This chapter describes methodology that can be used to determine these parameters, which usually appear in a nonlinear manner. For example, the material balance equation for the reaction 

Equation (9.3) shows that the model equation for C (f) depends on jin a nonlinear manner (i.e., appears in the exponential term in equation (9.3)). Be-quette [1] (p. 458) shows how to use experimental data to determine both and 

C (0) t y using the least squares method for a linearized form of equation (9.3). That is, taking the natural logarithm of both sides of equation (9.3) yields 

In equation (9.4), the dependent variable InC (f) depends linearly on the parameter and the parameter In (0). Bequette rewrites equation (9.4) as 

System Identification of the Core Model 5.2.4.1 Parameter Estimation 

The system identification step in the core-box modeling framework has two major sub-steps parameter estimation and model quality analysis. The parameter estimation step is usually solved as an optimization problem that minimizes a cost function that depends on the model s parameters. One choice of cost function is the sum of squares of the residuals, Si(t p) = yi(t) — yl(t p). However, one usually needs to put different weights, up (t), on the different samples, and additional information that is not part of the time-series is often added as extra terms k(p). These extra terms are large if the extra information is violated by the model, and small otherwise. A general least-squares cost function, Vp(p), is thus of the form 

After the cost function has been specified, a unique model is - in principle -determined by the parameters that minimize it  

For a given model, the following questions arise. How many and which parameters can be estimated how are they estimated What is the precision of parameter estimation What is the quality of the fit of the model to the data Furthermore, several models are generally checked against experimental data, posing the question of comparison of models. 

It has been shown that a sensitivity analysis (Section 2.5.4) could answer the first question, i.e. what are the number and the identity of model parameters to be estimated. From now on, parameters al5 

flp will be assumed to be the determining ones and p will be defined as the smallest number of parameters that can be used to parameterize the problem. 

The most immediate goal of scientific or industrial experimentation is to find relationships among manipulated and observed variables, or to validate such relationships coming from some underlying theory. A mathematical description almost invariably involves estimating the values of some unknown parameters to best match the available body of experimental observations. 

The simplest mathematical description or model of a system is the function 

To begin with a relatively simple problem we will assume that the independent variables can be manipulated or observed error—free, and only the dependent variable is corrupted by measurement errors. Thus the outcome of the i-th 

Our basic assumption is that the response function f(x,p) is a correct one and the random quantity represents the measurement error. It is then 

Parameter estimation is rooted in several scientific areas with their own preferences and approaches. While linear estimation theory is a nice chapter of mathematical statistics (refs. 1-3), practical considerations are equally important in nonlinear parameter estimation. As emphasised by Bard (ref. 4), in spite of its statistical basis, nonlinear estimation is mainly a variety of computational algorithms which perform well on a class of problems but may fail on some others. In addition, most statistical tests and estimates of 

In any modeling procedure, values of the predicted rate must approximate the values of the observed rate before an adequate model has been established. A simple indication of the comparative shapes of the predicted and 

Parameter values producing predicted-model results near the observed results have frequently been selected by minimizing S  

This method of least squares is not only intuitively desirable, but also provides estimates having desirable properties, if certain assumptions are met (D4). For models that are linear in the parameters, that is, models of the form 

The equations represented by Eq. (26) are termed the normal equations they provide parameter estimates possessing the minimum distance from the 

Many kinetic equations can be suitably linearized to the form of Eq. (20). For example, Eq. (1) can be transformed logarithmically, or Eq. (2) can be transformed reciprocally. Two equations proposed for describing pentane-isomerization data (Cl, Jl) are the single site 

Having set up a model to describe the dynamics of the system, a very important first step is to compare the numerical solution of the model with any experimental results or observations. In the first stages, this comparison might be simply a check on the qualitative behaviour of a reactor model as compared to experiment. Such questions might be answered as Does the model confirm the experimentally found observations that product selectivity increases with temperature and that increasing flow rate decreases the reaction conversion  

Following the first preliminary comparison, a next step could be to find a set of parameters, that give the best or optimal fit to the experimental data. This can be done by a manual, trial-and-error procedure or by using a more sophisticated mathematical technique which is aimed at finding those values for the system parameters that minimise the difference between values given by the model and those obtained by experiment. Such techniques are general, but are illustrated here with special reference to the dynamic behaviour of chemical reactors. 

Algebraic equations Steady state of CSTR with first-order kinetics. Algebraic solution and optimisation (least squares. Draper and Smith, 1981). Steady state of CSTR with complex kinetics. Numerical solution and optimisation (least squares or likelihood function). 

Differential equations Batch reactor with first-order kinetics. Analytical or numerical solution with analytical or numerical parameter optimisation (least squares or likelihood). Batch reactor with complex kinetics. Numerical integration and parameter optimisation (least squares or likelihood). 

The mass transfer coefficients in the crossflow cell were estimated from independent measurements of dissolution of a plate of benzoic acid into water at two different crossflow rates 50 L h and 120 L h , at 30 °C. Mass transfer coefficients for docosane and TOABr were estimated based on the experimentally measured benzoic add mass transfer coefficients values and the Chilton-Colbum mass transfer coeffident correlation. Details of the procedure applied are described elsewhere [32]. 

The molar volumes of toluene and docosane were taken from the hterature [25]. The molar volume of TOABr was estimated based on Fedors method [45]. 

The mass transfer coeffidents calculated for docosane and TOABr, using the Chilton-Colbum correlation, are presented in Tab. 4.2. 

Although all of the mass transfer coeffident data available in the hterature are for aqueous systems and most are for ultrafiltration [46—49], the values are in the same order of magnitude as those obtained in these experiments -10 m s , indicating that the technique used gives reasonable estimates. 

The activity coeffidents for docosane and tofuene were calculated applying the modified UNIFAC method [50]. From these resufts it was possible to develop a simple algebraic function describing the activity coefficient as a function of mole fraction of docosane and tofuene, respectivefy  

DigiElch s kinetic compiler is an extension of the compiler used by DigiSim in order to address the wider spectra of experiments possible to simulate with DigiElch. 

Electro Chemical Compiler [80], see also Errata [98], is the kinetic compiler component of the EChem-i-i- project [80] and enables the modelling of homogeneous and surface reactions and heterogeneous electron transfers as well as adsorption processes. Eollowing the EChem-n- philosophy, Ecco is separated from the subsequent simulation procedure and can therefore be combined with any numerical method. This ensures a high reuseability of the kinetic compiler. As part of the EChem-H- project, the source code of Ecco is available [87] under the GPL [86], 

One motivation of performing simulations is the interpretation of experimental data, e.g. voltammograms or chronoamperometric data, by estimation of physical parameters. This can be achieved by fitting simulated curves to experimental ones in a nonlinear regression analysis process. The user provides a model to the simulation program and an objective function is then minimized by systematic variation of the model parameter. The best fit is achieved when a global minimum of the object 

By its nature, the downhill simplex lacks statistical information about the estimated parameter. This information is obtained from the shape or gradients of the X hypersurface at the minimum. A suitable way to overcome this situation is to fit 

Finally, during the procedure of minimizing the object function (17.1), simulations are repeated perhaps a thousand times with systematic variation of the model parameters. Therefore the importance of the efficiency of the simulation method should not be underestimated. 

FIGURE 11.6 Predicted and experimental data of cumulative weight percent of dimensionless curves at LHSV = 0.33 h and 9.8 MPa. ( ) Feedstock, (A) 420°C. 

FIG U RE 11.7 Temperature dependence of parameters of the continuous kinetic model. 

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Figure 4-14. Predicted liquid-liquid equilibria for a typical type-II system shows good agreement with experimental data, using parameters estimated from binary data alone.

UNIQUAC equation with binary parameters estimated by supplementing binary VLE data with ternary tie-line data. 

Unfortunately, good binary data are often not available, and no model, including the modified UNIQUAC equation, is entirely adequate. Therefore, we require a calculation method which allows utilization of some ternary data in the parameter estimation such that the ternary system is well represented. A method toward that end is described in the next section. 

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. 

Figure 4-16. Representation of ternary liquid-liquid equilibria using the UNIQUAC equation is improved by incorporating ternary tie-line data into binary-parameter estimation. Representation of binary VLB shows small loss of accuracy. ---- Binary...

Many well-known models can predict ternary LLE for type-II systems, using parameters estimated from binary data alone. Unfortunately, similar predictions for type-I LLE systems are not nearly as good. In most cases, representation of type-I systems requires that some ternary information be used in determining optimum binary parameter. 

While many methods for parameter estimation have been proposed, experience has shown some to be more effective than others. Since most phenomenological models are nonlinear in their adjustable parameters, the best estimates of these parameters can be obtained from a formalized method which properly treats the statistical behavior of the errors associated with all experimental observations. For reliable process-design calculations, we require not only estimates of the parameters but also a measure of the errors in the parameters and an indication of the accuracy of the data. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

The method used here is based on a general application of the maximum-likelihood principle. A rigorous discussion is given by Bard (1974) on nonlinear-parameter estimation based on the maximum-likelihood principle. The most important feature of this method is that it attempts properly to account for all measurement errors. A discussion of the background of this method and details of its implementation are given by Anderson et al. (1978). 

When there is significant random error in all the variables, as in this example, the maximum-likelihood method can lead to better parameter estimates than those obtained by other methods. When Barker s method was used to estimate the van Laar parameters for the acetone-methanol system from these data, it was estimated that = 0.960 and A j = 0.633, compared with A 2 0.857 and A2- = 0.681 using the method of maximum likelihood. Barker s method uses only the P-T-x data and assumes that the T and x measurements are error free. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

Bard, Y., Nonlinear Parameter Estimation, Academic Press, New York (1974). ... 

Beck, J. V., Arnold, K. J., "Parameter Estimation in Engineering and Science," John Wiley Sons, New York (1977). ... 

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. 

Measurements have been made in a static laboratory set-up. A simulation model for generating supplementary data has been developed and verified. A statistical data treatment method has been applied to estimate tracer concentration from detector measurements. Accuracy in parameter estimation in the range of 5-10% has been obtained. 

After having proved the principles a dynamic test facility has been constructed. In this facility it is possible to inject 3 tracers in a flownng liquid consisting of air, oil and water. By changing the relative amounts of the different components it is possible to explore the phase diagram and asses the limits for the measurement principle. Experiments have confirmed the accuracy in parameter estimation to be below 10%, which is considered quite satisfactorily for practical applications. The method will be tested on site at an offshore installation this summer. 

Keywords, protein folding, tertiary structure, potential energy surface, global optimization, empirical potential, residue potential, surface potential, parameter estimation, density estimation, cluster analysis, quadratic programming... 

Thus, a list of 1 5 descriptors was calculated for these purposes, as described below. The partition coefficient log P (calculated by a method based on the Gho.sc/Crip-pen approach [11]) (see also Chapter X, Section 1.1 in the Handbook) was calculated because it affects the solubility dramatically [17, 18]. All the other descriptors were calculated with the program PETRA (Parameter Estimation for the Treatment of Reactivity Applications) [28. ... 

This coding is performed in three steps (cf Chapter 8) First the 3D coordinates of the atoms arc calculated using the structure generator CORINA (COoRdlNAtes). Subsequently the program PETRA (Parameter Estimation for the Treatment of Reactivity Applications) is applied for calculating physicochemical properties such as charge distribution and polarizability. The 3D information and the physicochemical atomic properties are then used to code the molecule. 

Parameter Estimation. WeibuU parameters can be estimated using the usual statistical procedures however, a computer is needed to solve readily the equations. A computer program based on the maximum likelihood method is presented in Reference 22. Graphical estimation can be made on WeibuU paper without the aid of a computer however, the results caimot be expected to be as accurate and consistent. 

Example 7. In order to illustrate graphical parameter estimation, five faUure times are considered 24,000 km, 39,000 km, 52,000 km, 64,000 km, and 82,000 km. These times-to-faUure were obtained by placing five items on test and allowing them to go to faUure. 

Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. 

Input information includes the known values of T and [Xi], as well as the equation-of-state and G -expression parameters. Estimates are also needed of P and [yj], the quantities to be evaluated, and these require some preliminaiy calciilations ... 

Hofmann, Tndustrial process kinetics and parameter estimation , in ACS Advances in Chemlstiy, 109, 519-534 (1972) "Kinetic data analysis and parameter estimation , in de Lasa, ed.. Chemical Reactor De.sign and Technology, Martinus Nijhoff, 1986, pp. 69-105. 

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