Estimation of Parameter

Parameter estimation for a given model deals vith optimising some parameters or their evaluation from experimental data. It is based on setting the best values for the parameters using experimental data. Parameter estimation is the calculation of the non-process parameters, i.e. the parameters that are not specific to the process. Physical and chemical properties are examples of such non-process parameters. Typical stages of the parameter estimation procedure are (i) the choice of the experimental points, (ii) the experimental ivork, i.e. the measurement of the values, (hi) the estimation of the parameters and analysis of the accuracy of the results, (iv) if the results are not accurate enough, additional experiments are carried out and the procedure is restarted from stage (i). 

In parameter estimation, the parameters are optimised, and the variables are given fixed values. Optimality in parameter estimation consists in establishing the best match between the experimental data and the values calculated by the model. All the procedures for the identification of parameters comply with the optimality requirements [2.12], 

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FORMAT (IH1,57HMAXlMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM VL IE DATA//1X,40HCONTROL PARAMETERS WERE SET AS FOLLOWS -/)... 

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM VLE DATA... 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

The change in concentration with time of a given molecular weight can be estimated and used to provide a guide as to choice of model or an approximate estimate of parameters. 

Estimation of parameters. Model parameters in the selected model are then estimated. If available, some model parameters (e.g. thermodynamic properties, heat- and mass-transfer coefficient, etc.) are taken from literature. This is usually not possible for kinetic parameters. These should be estimated based on data obtained from laboratory expieriments, if possible carried out isothermal ly and not falsified by heat- and mass-transport phenomena. The methods for parameter estimation, also the kinetic parameters in complex organic systems, and for discrimination between models are discussed in more detail in Section 5.4.4. More information on parameter estimation the reader will find in review papers by Kittrell (1970), or Froment and Hosten (1981) or in the book by Froment and Bischoff (1990). 

The category of algebraic equation models is quite general and it encompasses many types of engineering models. For example, any discrete dsmamic model described by a set of difference equations falls in this category for parameter estimation purposes. These models could be either deterministic or stochastic in nature or even combinations of the two. Although on-line techniques are available for the estimation of parameters in sampled data systems, off-line techniques... 

Estimation of parameters present in partial differential equations is a very complex issue. Quite often by proper discretization of the spatial derivatives we transform the governing PDEs into a large number of ODEs. Hence, the problem can be transformed into one described by ODEs and be tackled with similar techniques. However, the fact that in such cases we have a system of high dimensionality requires particular attention. Parameter estimation for systems described by PDEs is examined in Chapter 11. 

Even if we make the stringent assumption that errors in the measurement of each variable ( >,. , M.2,...,N, j=l,2,...,R) are independently and identically distributed (i.i.d.) normally with zero mean and constant variance, it is rather difficult to establish the exact distribution of the error term e, in Equation 2.35. This is particularly true when the expression is highly nonlinear. For example, this situation arises in the estimation of parameters for nonlinear thermodynamic models and in the treatment of potentiometric titration data (Sutton and MacGregor. 1977 Sachs. 1976 Englezos et al., 1990a, 1990b). 

The only drawback in using this method is that any numerical errors introduced in the estimation of the time derivatives of the state variables have a direct effect on the estimated parameter values. Furthermore, by this approach we can not readily calculate confidence intervals for the unknown parameters. This method is the standard procedure used by the General Algebraic Modeling System (GAMS) for the estimation of parameters in ODE models when all state variables are observed. 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

Table 16.24 HP A Hydrogenation Systematic Estimation of Parameter Values Using the Data Collected at 318 K... 

Patino-Leal, H., and P.M. Reilly, "Statistical Estimation of Parameters in Vapor-Liquid Equilibrium", AIChEJ., 28(4), 580-587 (1982). 

Chapter 9 deals with estimation of parameters subject to equality and inequality constraints whereas Chapter 10 examines systems described by partial differential equations (PDE). Examples are provided in Chapters 14 and 18. 

Table I. Estimates of Parameters for Start of Iterative Fitting...

There are a number of different methods for finding point estimators of parameters. Following Mikhail (1976) we mention here ... 

Many of the models encountered in reaction modeling are not linear in the parameters, as was assumed previously through Eq. (20). Although the principles involved are very similar to those of the previous subsections, the parameter-estimation procedure must now be iteratively applied to a nonlinear surface. This brings up numerous complications, such as initial estimates of parameters, efficiency and effectiveness of convergence algorithms, multiple minima in the least-squares surface, and poor surface conditioning. 

In summary, GC adjusts for population stratification without the assumption or estimation of parameters such as the number of subpopulations involved in the study. It provides control of false-positive results caused by population structure as well as by multiple testing. One possible drawback of this method is that the correction of the test statistic is constant across the genome. As a result, GC may have less power in certain situations. 

Rouen D, Scher H, Blunt M (1997) On the structure and flow processes in the capillary fringe of phreatic aquifers. Transp Porous Media 28 159-180 Rose CW (1993) The transport of adsorbed chemicals in eroded sediments. In Russo D, Dagan G (eds) Water flow and solute transport in soils. Springer, Heidelberg, pp 180-199 Rosenberry DO, Winter TC (1997) Dynamics of water-table fluctuations in an upland between two prairie-pothole wetlands in North Dakota. J Hydrol 191 266-289 Russo D (1997) On the estimation of parameters of log-unsaturated conductivity covariance from solute transport data. Adv Water Resour 20 191-205 Russo D, Toiber-Yasur 1, Laufer A, Yaron B (1998) Numerical analysis of field scale transport of bromacU. Adv Water Resour 21 637-647... 

Doctor, P. G. Nelson, R. W. "Geostatistical Estimation of Parameters for Transport Modeling" Pacific Northwest Laboratory Richland, Washington, 1980 PNL-SA-8482. 

In the problem of selecting a distribution for a ID model of variation, there are 2 kinds of variables, namely, 1) the data, which we know and 2) distribution parameters, which will be assigned values based on the data. Here we will often follow statistical terminology by using the term estimation (of parameters) instead of fitting. In statistical terminology, the values assigned to distribution parameters are termed estimates the expressions used to compute estimates are estimators. ... 

The effect of the 7V-substituents on the geometry of the 1,2,3-triazole ring is studied by analyzing 35 x-ray crystal structures of substituted triazoles and benzotriazoles. The analyses employ least-squares estimation of parameters. Two models for the substitution effects on the endocyclic angles... 

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