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** Estimation of model parameters **

Unfortunately, models are rarely exac4. The semblance of sophistication inherent in the model and used to develop parameter estimates frequentlv masks their deficiencies. Models are only approximate, and their predictions when the parameter estimates are based on analysis of plant performance must be considered as approximate. Vahdation of the model and the parameter estimates using other operating conditions will reduce the likelihood that the conclusions have significant [Pg.2578]

The purpose is to develop estimates of significant model parameters that provide the best estimate of unit operation. The unit operation is embodied in the measurements. [Pg.2573]

Using a normal probability plot of the estimated model parameters, determine the significant parameters. Analyse this reduced model. [Pg.171]

The ANOVA table shown in Table 2.14 indicates that there was no significant lack-of-fit of the model. Parameter estimates and t-statistics for this model are shown in Table 2.15. [Pg.53]

As just mentioned, the T-test tests the significance of a particular parameter estimate. What is really needed is also a test of the overall significance of a model. To start, the total sum of squares of the observed data, SStotai, is partitioned into a component due to regression, SSregression, and a component due to residual, unexplained error, SSE, [Pg.61]

One of the main differences between the polymerization kinetics with coordination catalysts and free-radical initiators is that the former depends on the characteristics of the active site as well as on monomer type, while the latter is almost exclusively regulated by monomer type. As we will see, even though this may not constitute a problem for establishing an operative mechanism for coordination polymerization, it creates a significant challenge for model parameter estimation. [Pg.383]

Linear models with respect to the parameters represent the simplest case of parameter estimation from a computational point of view because there is no need for iterative computations. Unfortunately, the majority of process models encountered in chemical engineering practice are nonlinear. Linear regression has received considerable attention due to its significance as a tool in a variety of disciplines. Hence, there is a plethora of books on the subject (e.g., Draper and Smith, 1998 Freund and Minton, 1979 Hocking, 1996 Montgomery and Peck, 1992 Seber, 1977). The majority of these books has been written by statisticians. [Pg.23]

Let be an estimate of the experimental error variance with r degrees of freedom let B be the vector of estimated model parameters let p be the number of estimated model parameters and let be the critical F value with (p,r) degrees of freedom and the significance level a. The boundaries of the joint confidence region is defined by the equation [Pg.117]

The spectral density estimators at different frequencies possess tractable properties so that they follow independent Wishart distributions in a certain frequency band, regardless of the distribution of the signal in the time domain. The method is efficient in the sense that most of the information from the data for identification of the model parameters, especially those related to the frequency stmcture, concentrates in a very limited bandwidth around the peaks in the spectrum. Therefore, the number of frequencies involved in the computation of the posterior PDF is significantly smaller than the total number of frequencies in the spectmm, i.e., INT(A /2)+l. However, computation of the inverse and determinant of the matrices [Sy,iv(r A)] 6 kefC, is required for each frequency included in establishing the [Pg.188]

If it is assumed that interaction effects between three or more variables can be neglected, it is seen that the estimated effect can be used as estimators of the model parameters. The 16 model parameters have been estimated from 16 runs. There are no degrees of freedom left to give an estimate of the residual variance which might have been used as an estimate of the error variance. Neither was any previous error estimate available. The significance of the estimated parameters must be assessed from a Normal probability plot. [Pg.157]

The only approximation made in the Bayesian time-domain approach is that the system response at a particular time step estimated by its entire history is essentially the same as conditioning on a significantly smaller number of previous time steps. In practice, the time-domain approach provides virtually an exact solution in the sense that the Bayesian approach utilizes the complete information inherited in the measurement. Therefore, the Bayesian time-domain approach provides more accurate statistical inference of the model parameters with the information in the data. [Pg.189]

This means that three- and four-factor interaction effects were assumed to be small compared to the experimental error. The model contains 11 parameters and the experimental design contains 16 runs. The excluded higher order interaction effects allow an estimate of the residual sum of squares, SSE, which will give an estimate of the experimental error variance, s with 16 - 11 = 5 degrees of freedom. This estimate can then be used to compute confidence limits for the estimated model parameters so that their significance can be evaluated. [Pg.113]

** Estimation of model parameters **

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