Model confidence

By using normal distribution, we can introduce other random variables, which are very important for testing the significance of Poi Pn P2) Pi2--- coefHcients as well as for testing the model confidence (see Fig. 5.3). 

At this point, we have to think about the problem of the model confidence. For this purpose we have to consider that ... 

The same procedure is used if we increase the polynomial degree given by the regression equation. In this case, the tests of the coefficient significance and model confidence are implemented as shown in the example developed in Section 5.4.1.1. It is important to note that we must use relation (5.59) for the calculation of the variances around the mean value of Pj. 

The regression analysis, when the relationship between the process variables is given by a matrix, is frequently used to solve the problems of identification and confidence of the coefficients as well as the problem of a model confidence. The matrix expression is used frequently in processes with more than two independent variables which present simultaneous interactive effects with a dependent variable. In this case, the formulation of the problem is similar to the formulation described in the previous section. Thus, we will use the statistical data from Table 5.11 again in order to identify the coefficients with the following relation ... 

Finally, we conclude this analysis with the estimation of the model confidence. 

The Lewis model has become so famous and has for years been used so mechanically that some of the considerations underlying it are here presented. They demonstrate how deeply Lewis was concerned with the underlying physical cause of valence rather than with a simple rule of thumb. He further points up an operational difficulty in Bohr s atomic model if it gives no information as to the electron s movement within an orbit, we have no business postulating its movement. Furthermore, electrons orbiting around a nucleus could not possibly explain directed valences. Lewis, therefore, proposed a static model, confident that theoretical chemistry would somehow, some day, confirm it ... 

If the terminal model adequately explains the copolymer composition, as is often the case, the terminal model is usually assumed to apply. Even where statistical tests show that the penultimate model does not provide a significantly better fit to experimental data than the tenninal model, this should not be construed as evidence that penultimate unit effects are unimportant. It is necessary to test for model discrimination, rather than merely for fit to a given model. In this context, it is important to remember that composition data are of very low power when it comes to model discrimination. For MMA-S copolymerization, even though experimental precision is high, the penultimate model confidence intervals are quite large 0.4[Pg.348]

The general idea is to incorporate the model confidence bounds into the analysis. Classical Design of Experiments statistics incorporates the confidence boimds into answering questions tike ... is the effect of altering a certain process or product variable onto a certain metric significant ... 

Process modeling and simulation ate nevertheless extremely important tools in the design and evaluation of process control strategies for separation processes. There is a strong need, however, for better process mo ls for a variety of separations as well as process data with which to confirm tiiese models. Confidence in complex process models, especially those that can be used to study process dynamics, can come only from experimental verification of these models. This will require more sophisticated process sensors than those commonly available for temperature, pressure, pH, and differential pressure. Direct, reliable measurement of stream composition, viscosity, turbidity, conductivity, and so on is important not only for process model verification but also for actual process control applications. Other probes, which could be used to provide a better estimate of the state of the system, are needed to contribute to the understanding of the process in the same time frame as that of changes occurring in the process. 

Dynamic range of the data Experimental error Applicability domain of the model Confidence intervals on correlation coefficients... 

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. 

Large confidence regions are obtained for the parameters because of the random error in the data. For a "correct" model, the regions become vanishingly small as the random error becomes very small or as the number of experimental measurements becomes very large. 

Four replicate measurements were made at the center of the factorial design, giving responses of 0.334, 0.336, 0.346, and 0.323. Determine if a first-order empirical model is appropriate for this system. Use a 90% confidence interval when accounting for the effect of random error. 

Because exceeds the confidence interval s upper limit of 0.346, there is reason to believe that a 2 factorial design and a first-order empirical model are inappropriate for this system. A complete empirical model for this system is presented in problem 10 in the end-of-chapter problem set. 

HETP values obtained in this way have been compared to measured values in data banks (69) and statistical analysis reveals that the agreement is better when equations 79 and 80 are used to predict and than with the other models tested. Even so, a design at 95% confidence level would require a safety factor of 1.7 to account for scatter. 

Often the goal of a data analysis problem requites more than simple classification of samples into known categories. It is very often desirable to have a means to detect oudiers and to derive an estimate of the level of confidence in a classification result. These ate things that go beyond sttictiy nonparametric pattern recognition procedures. Also of interest is the abiUty to empirically model each category so that it is possible to make quantitative correlations and predictions with external continuous properties. As a result, a modeling and classification method called SIMCA has been developed to provide these capabihties (29—31). 

The usual practice in these appHcations is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be appHed only to problems in which there is a weU-defined, clear association between the independent and dependent variables. The generalization of statistics to the associated confidence intervals for nonlinear coefficients is not well developed. 

Instead of radical reactions, models based on molecular reactions have been proposed for the cracking of simple alkanes and Hquid feeds like naphtha and gas oil (40—42). However, the vaUdity of these models is limited, and caimot be extrapolated outside the range with confidence. With sophisticated algorithms and high speed computers available, this molecular reaction approach is not recommended. 

Over 25 years ago the coking factor of the radiant coil was empirically correlated to operating conditions (48). It has been assumed that the mass transfer of coke precursors from the bulk of the gas to the walls was controlling the rate of deposition (39). Kinetic models (24,49,50) were developed based on the chemical reaction at the wall as a controlling step. Bench-scale data (51—53) appear to indicate that a chemical reaction controls. However, flow regimes of bench-scale reactors are so different from the commercial furnaces that scale-up of bench-scale results caimot be confidently appHed to commercial furnaces. For example. Figure 3 shows the coke deposited on a controlled cylindrical specimen in a continuous stirred tank reactor (CSTR) and the rate of coke deposition. The deposition rate decreases with time and attains a pseudo steady value. Though this is achieved in a matter of rninutes in bench-scale reactors, it takes a few days in a commercial furnace. 

Extended Plant-Performance Triangle The historical representation of plant-performance analysis in Fig. 30-1 misses one of the principal a ects identification. Identification establishes troubleshooting hypotheses and measurements that will support the level of confidence required in the resultant model (i.e., which measurements will be most beneficial). Unfortunately, the relative impact of the measurements on the desired end use of the analysis is frequently overlooked. The most important technical step in the analysis procedures is to identify which measurements should be made. This is one of the roles of the plant-performance engineer. Figure 30-3 includes identification in the plant-performance triangle. 

Systematic Measurement Error Fourth, measurements are subject to unknown systematic errors. These result from worn instruments (e.g., eroded orifice plates, improper sampling, and other causes). While many of these might be identifiable, others require confidence in all other measurements and, occasionally, the model in order to identify and evaluate. Therefore, many systematic errors go unnoticed. 

Required Sensitivity This is difficult to establish a priori. It is important to recognize that no matter the sophistication, the model will not be an absolute representation of the unit. The confidence in the model is compromised by the parameter estimates that, in theoiy, represent a limitation in the equipment performance but actually embody a host of limitations. Three principal limitations affecting the accuracy of model parameters are ... 

---