Functions of model parameters

Standard errors and confidence intervals for functions of model parameters can be found using expectation theory, in the case of a linear function, or using the delta method (which is also sometimes called propagation of errors), in the case of a nonlinear function (Rice, 1988). Begin by assuming that 0 is the estimator for 0 and X is the variance-covariance matrix for 0. For a linear combination of observed model parameters 

Up to this point normality has not been assumed. Equations (3.55) and (3.56) do not depend on normality for their validity. But, if 0 was normally distributed then g(0) will also be normally distributed. So returning to the example, since Vc and Vp were normally distributed, then Vss will be normally distributed. 

If g(0), be it univariate or multivariate, is a nonlinear function then an approach repeatedly seen throughout this book will be used—the function will first be linearized using a first-order Taylor series and then the expected value and variance will be found using Eqs. (3.55) and (3.56), respectively. This is the so-called delta method. If g(0) is a univariate, nonlinear function then to a first-order Taylor series approximation about 0 would be 

The expected value can now be found through a linear combination of random variables 

For example, suppose the terminal elimination rate constant ( z,) was estimated using four observations. If the mean and variance of z was 0.154 per hour and 6.35E — 4 (per hour)2, respectively. Then the mean and variance for half life O1/2) defined as Ln(2)/ z, would be 4.5 h and 

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The sum of the squared differences between calculated and measures pressures is minimized as a function of model parameters. This method, often called Barker s method (Barker, 1953), ignores information contained in vapor-phase mole fraction measurements such information is normally only used for consistency tests, as discussed by Van Ness et al. (1973). Nevertheless, when high-quality experimental data are available. Barker s method often gives excellent results (Abbott and Van Ness, 1975). 

Figure 5-2 The plot of the misfit functional value as a function of model parameters tn. The vector of the steepest ascent, l(m ), shows the direction of "climbing on the hill" along the misfit functional surface. The intersection between the vertical plane P drawn through the direction of the steepest descent at point m and the misfit functional surface is shown by a solid parabola-type curve. The steepest descent step begins at a point 0(m ) and ends at a point 0(m +i) at the minimum of this curve. The second parabola-type curve (on the left) is drawn for one of the subsequent iteration points. Repeating the steepest descent iteration, we move along the set of mutually orthogonal segments, as shown by the solid arrows in the space M of the model parameters.

Figure 5-4 The plot of the misfit functional value as a function of model parameters m. In the framework of the Newton method one tries to solve the problem of minimization in one step. The direction of this step is shown by the arrows in the space M of model parameters and at the misfit surface.

Catalyst deactivation in large-pore slab catalysts, where intrapaiticle convection, diffusion and first order reaction are the competing processes, is analyzed by uniform and shell-progressive models. Analytical solutions arc provid as well as plots of effectiveness factors as a function of model parameters as a basis for steady-state reactor design. 

The response produced by Eq. (8-26), c t), can be found by inverting the transfer function, and it is also shown in Fig. 8-21 for a set of model parameters, K, T, and 0, fitted to the data. These parameters are calculated using optimization to minimize the squarea difference between the model predictions and the data, i.e., a least squares approach. Let each measured data point be represented by Cj (measured response), tj (time of measured response),j = 1 to n. Then the least squares problem can be formulated as ... 

Fault detection is a monitoring procedure intended to identify deteriorating unit performance. The unit can be monitored by focusing on values of important unit measurements or on values of model parameters. Step changes or drift in these values are used to identify that a fault (deteriorated performance in unit functioning or effectiveness) has occurred in the unit. Fault detection should be an ongoing procedure for unit monitoring. However, it is also used to compare performance from one formal unit test to another. 

This problem can be cast in linear programming form in which the coefficients are functions of time. In fact, many linear programming problems occurring in applications may be cast in this parametric form. For example, in the petroleum industry it has been found useful to parameterize the outputs as functions of time. In Leontieff models, this dependence of the coefficients on time is an essential part of the problem. Of special interest is the general case where the inputs, the outputs, and the costs all vary with time. When the variation of the coefficients with time is known, it is then desirable to obtain the solution as a function of time, avoiding repetitions for specific values. Here, we give by means of an example, a method of evaluating the extreme value of the parameterized problem based on the simplex process. We show how to set up a correspondence between intervals of parameter values and solutions. In that case the solution, which is a function of time, would apply to the values of the parameter in an interval. For each value in an interval, the solution vector and the extreme value may be evaluated as functions of the parameter. 

The following diagrams show typical pressure curves in the flow region of the calender nip as a function of various parameters. The following parameters are used for the presented model calculations ... 

For large values of z a fully developed case is reached in which the velocities are only functions of r and 0. In the fully developed case the weight fraction polymer increases linearly in z with the same slope for all r and 0. An implicit finite difference scheme was used to solve the model equations, and for the fully developed case the finite difference method was combined with a continuation method in order to efficiently obtain solutions as a function of the parameters (see Reference 14). It was determined that except for very large Grashof... 

When the model equations are linear functions of the parameters the problem is called linear estimation. Nonlinear estimation refers to the more general and most frequently encountered situation where the model equations are nonlinear functions of the parameters. 

The order parameter is directly available from the calculations and the SCF results are given in Figure 17. The absolute values of the order parameter are a strong function of head-group area. Unlike in most SCF models, we do not use this as an input value it comes out as a result of the calculations. As such, it is somewhat of a function of the parameter choice. The qualitative trends of how the order distributes along the contour of the tails are rather more generic, i.e. independent of the exact values of the interaction parameters. The result in Figure 17 is consistent with the simulation results, as well as with the available experimental data. The order drops off to a low value at the very end of the tails. There is a semi-plateau in the order parameter for positions t = 6 — 14,... 

Figure 31 shows the largest eigenvalue of the Jacobian at the experimentally observed metabolic state as a function of the parameter 0 TP. Similar to Fig. 28 obtained for the minimal model, several dynamic regimes can be distinguished. In particular, for sufficient strength of the inhibition parameter, the system undergoes a Hopf bifurcation and the pathway indeed facilitates sustained oscillations at the observed state. 

The approach to the mathematical definition of the interface model is very simple. For every layer in the interface, the charge is defined once as a function of chemical parameters and once as a function of electrostatic parameters. The functions for charge are set equal to each other and solved for the unknown electrochemical potentials. Mathematical techniques for solving the equations have been worked out and described in detail (9). 

There are several reasons why the sum of squares, i.e. the sum of squared differences between the measured and modelled data, is used to define the quality of a fit and thus is minimised as a function of the parameters. It is instructive to consider alternatives to the sum of squares, (a) Minimal sum of differences - is not an option, as positive and negative differences cancel each other out. Huge deviations in both directions can result in zero sums. 

The residuals are a function of the parameters. Note that they are also a function of the model and the data, but we take these as given and ignore this for the time being. The sum of squares, ssq, is the sum of all the squares of the individual residuals and thus is also a function of the parameters ... 

Figure 5-31 clearly features three minima at the correct positions, Aiesi=0, and the two rate constants used to generate the data ltest=0.03 and Xtest=0.1. A very interesting feature of the whole method is that the rate constants are completely independent. Each minimum, or rate constant, is defined on its own, completely independent of all the others. This is in clear contrast to normal, hard-modelling data fitting where the residuals are a function of all parameters together. 

For this reason considerable effort goes into the development of the parameters which appear in the energy function (2, ). This parameterization is generally accomplished by the matching of calculated properties to experimental measurements, as a function of the parameter set for selected small model compounds. 

As discussed above, the potential octane boost which can be achieved from ZSM-5 addition is a function of five parameters the regenerator temperature and steam partial pressure (which determine the activity maintenance) the base and ZSM-5 catalyst makeup rates (which determine the catalyst age) and the base gasoline octane. The sensitivity of the model to these parameters is discussed below. 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

Abstract To design an adsorption cartridge, it is necessary to be able to predict the service life as a function of several parameters. This prediction needs a model of the breakthrough curve of the toxic from the activated carbon bed. The most popular equation is the Wheeler-Jonas equation. We study the properties of this equation and show that it satisfies the constant pattern behaviour of travelling adsorption fronts. We compare this equation with other models of chemical engineering, mainly the linear driving force (LDF) approximation. It is shown that the different models lead to a different service life. And thus it is very important to choose the proper model. The LDF model has more physical significance and is recommended in combination with Dubinin-Radushkevitch (DR) isotherm even if no analytical solution exists. A numerical solution of the system equation must be used. 

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