Parameter estimation procedures

To offer more flexibility we adopt an approach, based on the transient simulation model TRNSYS (Klein et al., 1976), making use of the Lund DST borehole model (Hellstrom, 1989). The parameter estimation procedure is carried out using the GenOPT (Wetter, 2004) package with the Nelder and Mead Simplex minimization algorithm (Nelder and Mead, 1965) or Hooke and Jeeves minimization algorithm (Hooke and Jeeves, 1961). 

In principle all parameters of the model can be entered in the parameter estimation procedure. For the time being we limit the parameters to be calibrated to the ground thermal conductivity, ground heat capacity and borehole filling conductivity. 

At this point, no universal parameter estimation protocol has been adopted, although standard methods have been applied to individual models. The great flexibility that PB-PK models offer in terms of characterizing all types of drug transport makes routine parameter estimation methods unnecessary and unattractive. As with any modeling endeavor, the modeler should have a clear idea of which parameter estimation procedures will be employed so that experiments can be designed properly. 

To determine the form of the rate law, values of (-rA) as a function of cA may be obtained from a series of such experiments operated at various conditions. For a given reactor (V) operated at a given % conditions are changed by varying either cAo or q. For a rate law given by (—rA) = kAcA, the parameter-estimation procedure is die same as that in the differential method for a BR in the use of equation 3.4-2 (linearized form of the rate law) to determine kA and n. The use of a CSTR generates point ( -rA) data directly without the need to differentiate cA data (unlike the differential method with aBR). 

ESE envelope modulation. In the context of the present paper the nuclear modulation effect in ESE is of particular interest110, mi. Rowan et al.1 1) have shown that the amplitude of the two- and three-pulse echoes1081 does not always decay smoothly as a function of the pulse time interval r. Instead, an oscillation in the envelope of the echo associated with the hf frequencies of nuclei near the unpaired electron is observed. In systems with a large number of interacting nuclei the analysis of this modulated envelope by computer simulation has proved to be difficult in the time domain. However, it has been shown by Mims1121 that the Fourier transform of the modulation data of a three-pulse echo into the frequency domain yields a spectrum similar to that of an ENDOR spectrum. Merks and de Beer1131 have demonstrated that the display in the frequency domain has many advantages over the parameter estimation procedure in the time domain. 

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. 

The iterative parameter-estimation procedure can also be represented as in Fig. 29. An experimenter performs an experiment, obtains the data and estimates the parameters, for example, by nonlinear least squares. Then he checks the confidence region of the parameter estimates, perhaps through A as shown in Fig. 27. If it is too large, the experimenter finds those experimental settings that maximize the parameter estimation criterion of Eq. (144), that is, experimental settings that will reduce the size of confidence region as... 

The statistics parameters estimation procedure is based on two experimental observations ... 

Signal processing may also involve parameter estimation methods (e.g., resolution of a spectral curve into the sum of Gaussian functions). Even in such cases, however, we may need non-parametric methods to approximate the position, height and half-width of the peaks, used as initial estimates in the parameter estimation procedure. 

These partial derivatives provide a lot of information (ref. 10). They show how parameter perturbations (e.g., uncertainties in parameter values) affect the solution. Identifying the unimportant parameters the analysis may help to simplify the model. Sensitivities are also needed by efficient parameter estimation procedures of the Gauss - Newton type. Since the solution y(t,p) is rarely available in analytic form, calculation of the coefficients Sj(t,p) is not easy. The simplest method is to perturb the parameter pj, solve the differential equation with the modified parameter set and estimate the partial derivatives by divided differences. This "brute force" approach is not only time consuming (i.e., one has to solve np+1 sets of ny differential equations), but may be rather unreliable due to the roundoff errors. A much better approach is solving the sensitivity equations... 

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

After each new experiment the parameter estimation procedure is repeated, the new information extracted. 

After the formulation stage, we have all the equations of the model, but they are not useful yet, because parameters in the equations do not have a particular value. Consequently, the model cannot be used to reproduce the behavior of a physical entity. The parameter estimation procedure consists of obtaining a set of parameters that allows simulation with the model. In many cases, parameters can be found in literature, but in other cases it is required to fit the model to the experimental behavior by using mathematical procedures. The easier and more used types of procedures are those based on the use of optimization algorithms to make minimum the differences between the experimental observations and the model outputs. The more frequently used criterion to optimize the values of the parameters is the least square regression coefficient. In this procedure, a set of values is proposed for all model parameters (one for every parameter) and the model is run. After that, the error criterion is calculated as the sum of the squares of the residues (differences between the values of every experimental and modeled value). Then, an optimization procedure is used to change the values of the model parameters in order to get the minimum value of this criterion. 

To further increase accuracy without an increase of computer time the mesh density of spatial pivots may vary non-uniformly. This allows a higher mesh density at locations where the spatial gradients are larger. When a parameter estimation procedure is superposed, the mesh points can be chosen at exactly those positions where experimental data (e.g. wall temperatures) are taken. 

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). 

The development of new polymeric structures for different technological applications usually requires knowledge about properties of this material. The prediction of properties using additive group contribution method is a valuable procedure adopted during the developments presented here. The group contribution method concept was applied to obtain viscosity data versus temperature, an intermediate step of the free-volume parameters estimation procedure (equation (2) inputs). Detailed concepts about prediction of polymer properties were studied and applied as presented in specific literature (Van Krevelen, 1992 Bicerano, 2002). Equations (4) and (5) are the key equations of the procedure to obtain zero shear viscosity predicted data. The references adopted in this section also allows to predict many others polymer properties. 

The one-dimensional column experiments and parameter-estimation procedure described in this work provides rate constants, or time scales, for sorption/desorption that are independent of the flow and the large-scale media geometry. By using model sorbents the rate constants can be related to the dominant binding interaction, which helps define their applicability. 

The equation in this form is a useful tool to estimate parameters of reaction kinetics. Instead of performing nonlinear parameter estimation procedures, the functional dependence of XSf on X/(1 - X) can be plotted. The plot should look like a straight line whose slope is (-K m) and whose intersection with the XSf axis is given by the point of coordinate (Vmaxr). Hence, it furnishes a rapid graphic procedure to obtain rough estimates of kinetic parameters. 

A way to improve the parameter estimation procedure is to orthogonalize (centralize) the experiments according to the following procedure. The abscissa axis is shifted to the average value of the experiments (x), x =x-x. After applying this to the Arrhenius law we obtain... 

A more detailed description of the mathematical model presented herein, the model solution, parameter estimation procedure and the confidence limits equations are described in detail in references 5,8 and 11. 

A Fault Parameter Estimation Procedure Based on User Defined Scilab Functions... 

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