Parameter estimation results

Using the concentration and obscuration measurements allow all of the kinetic parameters of interest to be identified. The simulated data is shown in Figures 1-2. The parameter estimation results corresponding to these measurements are given in Table 1. These results indicate that these measurements may provide enough process information to allow identification, even in the presence of measurement noise. This hypothesis is investigated experimentally in the next section. 

Table 1 Parameter estimation results utilizing simulated process measurements...

Table 3 Parameter estimation results utilizing only concentration data...

Table 5 Parameter estimation results utilizing both concentration and obscuration data assuming spherical particles...

When the proportion of missing data is small and the covariates are static, the dummy variable method seems to work reasonably well, but biased parameter estimates result when the percent of missing data becomes large (Jones, 1996). 

Parameter estimate results from best fit kinetic expressions are listed in Table 4 for each of the hydrolysis reaction rates and yield coefficients. Using these parameters, Fig. 7 compares measured to simulated data for the hydrolysis experiments using the MRE mixture at 100, 400 g/1, and the validation dataset. The values, coefficient of determination, for glucose are 0.99, 1.00, 0.97, respectively, indicating a good fit to all three datasets. 

The parameter estimation results revealed that the governing kinetic parameters are quite well identified typically the standard deviations of the parameters are <22%. and the correlation... 

The Figure 1 shows that the values of t-statistic are less than the critical values of significance level, it means sequence is smooth. So do not use the difference, but ARMA sequence fitting. Consider to use ARMA model. Select the ARMA (2, 2) model as a predictive model. Parameter estimation results are shown in Table 4. 

Figure 15.14 The dependence of parameter estimation results on the Pe number.

FIG U RE A10.11 Parameter estimation results of a wrong model with systematic deviations from experimental data. 

Then, the drift Brownian motion is utilized to establish the reliability model for the first principal component data. The parameter estimation results... 

Table 11.6 Parameter estimation results for the heterogeneous catalysts (Smopex 101) Parameter Value Std. Error Std. Error %)...

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. 

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. 

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 vertices are connected with hues indicating information flow. Measurements from the plant flow to plant data, where raw measurements are converted to typical engineering units. The plant data information flows via reconciliation, rec tification, and interpretation to the plant model. The results of the model (i.e., troubleshooting, model building, or parameter estimation) are then used to improve plant operation through remedial action, control, and design. 

The above assumes that the measurement statistics are known. This is rarely the case. Typically a normal distribution is assumed for the plant and the measurements. Since these distributions are used in the analysis of the data, an incorrect assumption will lead to further bias in the resultant troubleshooting, model, and parameter estimation conclusions. 

Model Development PreHminary modeling of the unit should be done during the familiarization stage. Interactions between database uncertainties and parameter estimates and between measurement errors and parameter estimates coiJd lead to erroneous parameter estimates. Attempting to develop parameter estimates when the model is systematically in error will lead to systematic error in the parameter estimates. Systematic errors in models arise from not properly accounting for the fundamentals and for the equipment boundaries. Consequently, the resultant model does not properly represent the unit and is unusable for design, control, and optimization. Cropley (1987) describes the erroneous parameter estimates obtained from a reactor study when the fundamental mechanism was not properly described within the model. 

Overview Reconciliation adjusts the measurements to close constraints subject to their uncertainty. The numerical methods for reconciliation are based on the restriction that the measurements are only subject to random errors. Since all measurements have some unknown bias, this restriction is violated. The resultant adjusted measurements propagate these biases. Since troubleshooting, model development, ana parameter estimation will ultimately be based on these adjusted measurements, the biases will be incorporated into the conclusions, models, and parameter estimates. This potentially leads to errors in operation, control, and design. 

The presence of errors within the underlying database fudher degrades the accuracy and precision of the parameter e.stimate. If the database contains bias, this will translate into bias in the parameter estimates. In the flash example referenced above, including reasonable database uncertainty in the phase equilibria increases me 95 percent confidence interval to 14. As the database uncertainty increases, the uncertainty in the resultant parameter estimate increases as shown by the trend line represented in Fig. 30-24. Failure to account for the database uncertainty results in poor extrapolations to other operating conditions. 

Increa.se the number of mea.surements included in the mea.sure-ment. set by using mea.surements from repeated. sampling. Including repeated measurements at the same operating conditions reduces the impact of the measurement error on the parameter estimates. The result is a tighter confidence interval on the estimates. 

As with troubleshooting, parameter estimation is not an exact science. The facade of statistical and mathematical routines coupled with sophisticated simulation models masks the underlying uncertainties in the measurements and the models. It must be understood that the resultant parameter values embody all of the uncertainties in the measurements, underlying database, and the model. The impact of these uncertainties can be minimized by exercising sound engineering judgment founded upon a famiharity with unit operation and engineering fundamentals. 

The resultant distribution of the parameter estimates are also unknown. 

Parameter estimation was performed for different agglomeration kernels and the results for one run are compared in Figure 6.24. 

The procedure is to use Eqs. (2-102) and the nonlinear model function. Preliminary parameter estimates Go, bo, are needed. The resulting parameter values... 

Parameter estimation to fit the data is carried out with VARY YM Y1 Y2, FIT M, and OPTIMIZE. The result is optimized values for Ym (0.7835), Y1 (0.6346), and Y2 (1.1770). The statistical summary shows that the residual sum of squares decreases from 0.494 to 0.294 with the parameter optimization compared to that with starting values (Ym=Yl=Y2=l. 0. ) The values of after optimization of Ym, Yl, and Y2 are shown in Figure 2, which illustrates the anchor-pivot method and forced linearization with optimization of the initiator parameters through Yl and Y2. 

Computer tools can contribute significantly to the optimization of processes. Computer data acquisition allows data to be more readily collected, and easy-to-implement control systems can also be achieved. Mathematical modeling can save personnel time, laboratory time and materials, and the tools for solving differential equations, parameter estimation, and optimization problems can be easy to use and result in great productivity gains. Optimizing the control system resulted in faster startup and consequent productivity gains in the extruder laboratory. 

One further question that has a substantial impact on the application of modeling techniques to biomedical problems is the choice of the design. Suppose that in our Gompertz tumor growth example we wanted to decide, given the results of some pilot experiments, when it is most useful to observe the tumor volume. In other words, we wish to choose the time points at which we obtain tumor volume observations in order to maximize the precision of the resulting parameter estimates. 

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