Parameter estimation graphical methods

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. 

The parameters A, B, and C are dependent on the particular nature of the gas. Katz developed a simple method for gas mixtures that takes the composition of the gas into account [942,1086]. Furthermore, a graphic method is available that permits the estimation of the hydrate-forming temperatures at pressures for natural gas containing up to 50% hydrogen sulfide [129]. 

The attached worksheet from MathCad ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) is used for computing the statistical parameters and graphics discussed in Chapters 58 through 61, in references [b-l-b-4]. It is recommended that the statistics incorporated into this series of Worksheets be used for evaluations of goodness of fit statistics such as the correlation coefficient, the coefficient of determination, the standard error of estimate and the useful range of calibration standards used in method development. If you would like this Worksheet sent to you, please request this by e-mail from the authors. 

Chan (Chapter 6) presents a simple graphical method for estimating the free energy of EDL formation at the oxide-water interface with an amphoteric model for the acidity of surface groups. Subject to the assumptions of the EDL model, the graphical method allows a comparison of the magnitudes of the chemical and coulombic components of surface reactions. The analysis also illustrates the relationship between model parameter values and the deviation of surface potential from the Nernst equation. 

The selection of a solvent for a new separation problem, even today, is a matter of trial and error. The application of theory (2) with the additional application of the solubility parameters (6J-65) makes it possible to estimate the composition of appropriate solvent mixtures for the separation of relatively simple compounds. In order to calculate the necessary solvent strength, however, a set of experimental data concerning the behavior of the sample components, the adsorbent, and the elution strength of the eluents with the specific adsorbent are necessary. Others (J5) recommend a graphical method as a time-saving alternative to bi th calculation and the trial-and-error approach to obtain a first approximation of the eluent composition appropriate for the separation of a givin sample. 

Simple, graphical methods for testing the fit of rate data to Equation 9 and for estimating the kinetic parameters k2 and KR involve using linearized forms of Equation 9. By far the most widely used linear form is that of Kitz and Wilson (14). Taking reciprocals of both sides of Equation 9, we obtain... 

The graphical methods for the estimation of the protein binding parameters are limited to protein systems, which have only 1 class of binding sites. If the protein concentration P is known the product n KA can be calculated by using equation 4. The slope of the regression line is equal to n KA P. 

Where the number of points is sufficiently large, the limits of error of the position of plotted points can be inferred from their scatter. Thus an upper bound and a lower bound can be drawn, and the lines of lintiting slope drawn so as to lie within these bounds. Since the theory of least squares can be applied not only to yield the equation for the best straight line but also to estimate the uncertainties in the parameters entering into the equation (see Chapter XXI), such graphical methods are justifiable only for rough estimates. In either case, the possibility of systematic error should be kept in ntind. 

Graphical methods suffice for approximate parameter estimation when the models considered are simple and the parameters are few. Least squares gives closer estimates, especially for multiparameter models. 

Some simple reaction kinetics are amenable to analytical solutions and graphical linearized analysis to calculate the kinetic parameters from rate data. More complex systems require numerical solution of nonlinear systems of differential and algebraic equations coupled with nonlinear parameter estimation or regression methods. 

The Bird-Carreau model employs the use of four empirical constants (ai, a2, Ai, and A2) and a zero shear limiting viscosity (770) of the solutions. The constants a, az, Ai, and A2, can be obtained by two different methods one method is using a computer program which can combine least square method and the method of steepest descent analysis for determining parameters for the nonlinear mathematical models (Carreau etal, 1968). Another way is to estimate by a graphic method as illustrated in Fig. 20 two constants, Q i and A], are obtained from a logarithmic plot of 77 vs y, and the other two constants, az and A2, are obtained from a logarithmic plot of 77 vs w. 

A useful graphical method for the estimation of kinetic parameters in substrate inhibition was described by Marmasse (1963). However, with substrate inhibition plots, even after a successful graphical analysis is completed, one should always fit the data to the appropriate equation with a computer program in order to estimate the kinetic constants. 

Graphical analysis must always precede the statistical analysis. It is imperative to keep short the time elapsed between data acquisition and data analysis, and in most cases, it is advisable to perform the graphical analysis even while the experiment is still in progress. Wien the data clearly define the nature ofthe rate or binding equation, statistical analysis is not needed to do this. Nevertheless, for a definitive work, statistical methods are necessary for parameter estimation as well as for model discrimination (Senear Bolen, 1992). Computer programs are now available for even the most sophisticated problems in enzyme kinetics (see Section 18.2.4). 

In section 4 we indicate how the probit function (lognormal distribution), the normal and WeibuU distribution function parameters are obtained from experimental data. We use the graphical method, point estimator and momentum method within a resambling approach. In section 5 we summarize and conclude. 

We compared several methods (point estimator determined by the hkelihood principle or the momentum method and the graphical method) that determine the parameters of the lognormal, the normal and the Weibull distribution. We foimd that in aU cases resampling must be applied because of the few data that is available. 

Figure 6.45. Graphical method of parameter estimation in case of Equ. 6.116 representing a pseudohomogeneous approach to biofilm processing, Kornegay and Andrews (1969) and Kornegay and Andrews (1968).

Figure 17. Graphical method for the estimation of the E and C parameters of a polymer using two solvents of known E and C values, and a knowledge of the exothermic acid-base interaction between the polymer and solvent 1 (-Z Wabi) and solvent 2 (-A//ab2) ...

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