Errors and models

Thus we wish to determine systematic variations in the presence of noise  

To be of any practical value, a response surface model should give a satisfactory description of the variation of y in the experimental domain. This means that the model error R(x) should be negligible, compared to the experimental error. By multiple linear regression, least squares estimates of the model parameters would minimize the model error. Model fitting by least squares multiple linear regression is described in the next section. 

From the response surface model it is possible to calculate (predict) the response for any settings of the experimental variables. This makes it possible to compute the residuals, e, as the difference between the observed response,y , in an experiment i , and the response predicted from the model, y, for the corresponding experimental conditions. 

If the model were absolutely perfect, the residuals would be a measure of the experimental error onfy. If the model is adequate, the variance of the residuals should not be significantly greater than the variance of experimental error. This can be checked by an F-test, provided that an independent estimate of the experimental error variance is available. Such an estimate can be obtained through replication of one or more experiment. 

In the sections below a brief outline is given of how experimental data can be used to fit a response surface model and how statistical methods can be used to evaluate the results. The author s intensions in this introductory chapter is to give the reader a feeling for how statistical tools can be used in an experimental context. Detailed descriptions follow in subsequent chapters. 

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Interaction between measurement error and model parameters... 

Let ai..as obs denote the average pure component spectra of the observable species over the entire set of measurements. Then a model for the spectroscopic data can be constructed (Eq. (5)) where c is the concentration and e represents both experimental error and model error (non-linearities) [47]. ... 

Step 8 Measuring Results and Monitoring Performance The evaluation of MPC system performance is not easy, and widely accepted metrics and monitoring strategies are not available, ffow-ever, useful diagnostic information is provided by basic statistics such as the means and standard deviations for both measured variables and calculated quantities, such as control errors and model residuals. Another useful statistic is the relative amount of time that an input is saturated or a constraint is violated, expressed as a percentage of the total time the MPC system is in service. 

In the framework of the TDM model, the transport coefficient is the last parameter to be determined according to Fig. 6.9. All prior experimental errors and model inaccuracies are lumped into this parameter. In addition it cannot be excluded that the mass transfer depends on concentration because of surface diffusion or adsorption kinetics. However, in many cases, e.g. for the target solutes discussed in this book, the transfer coefficient can be assumed to be independent of operating conditions (especially flow rate) with reasonable accuracy. 

After building the model, it is necessary to test its performance and validate it with independent measurement data. Basically, there are two reasons why measured data might be out of line with model predictions measurement or sampling errors and model failure. If sampling and measurement errors can be excluded, the model needs adjustment and modification. An adequate model is seldom obtained at the first attempt. In general, an iterative procedure is needed, where improvements are continuously made until an adequate model is achieved and measurement results are within the confidence limits of the model. 

According to the approach summarized in Figure 6.9 for the transport-dispersive model, the transport coefficient is the last parameter to be determined. All prior experimental errors and model inaccuracies are lumped no v into this final parameter. 

One critical factor to keep in mind when building and evaluating predictive models is that every experimental data point has an error associated with it. For example, if we measure the Log 5 of a compound as -6 and that data point has an error of 0.3 log units, the acmal value could be anywhere between -6.3 and -5.7. In a 2009 paper, Brown and coworkers [41] examined the relationship between experimental error and model performance. They carried out a series of theoretical experiments where Gaussian distributed random values were added to data to simulate experimental errors. The authors then calculated the correlation between the measured values and the same values with this simulated error. This correlation can be thought of as the maximum correlation possible given the error in the measurement. As we saw earlier. 

The last assumption in the list above involves a model G of gas dispersion, and in effect states that the model error AG u,v) = G u,v) - T is 0. The implication is that there is no need to consider model output uncertainty, i.e. uncertainty on the value of the model error. The terms model error and model output uncertainty is here used as defined by Aven Zio (2013). 

Subroutine REGRES. REGRES is the main subroutine responsible for performing the regression. It solves for the parameters in nonlinear models where all the measured variables are subject to error and are related by one or two constraints. It uses subroutines FUNG, FUNDR, SUMSQ, and SYMINV. 

Nonetheless, these methods only estimate organ-averaged radiation dose. Any process which results in high concentrations of radioactivity in organs outside the MIRD tables or in very small volumes within an organ can result in significant error. In addition, the kinetic behavior of materials in the body can have a dramatic effect on radiation dose and models of material transport are constandy refined. Thus radiation dosimetry remains an area of significant research activity. 

In concentrated wstems the change in gas aud liquid flow rates within the tower and the heat effects accompanying the absorption of all the components must be considered. A trial-aud-error calculation from one theoretical stage to the next usually is required if accurate results are to be obtained, aud in such cases calculation procedures similar to those described in Sec. 13 normally are employed. A computer procedure for multicomponent adiabatic absorber design has been described by Feiutnch aud Treybal [Jnd. Eng. Chem. Process Des. Dev., 17, 505 (1978)]. Also see Holland, Fundamentals and Modeling of Separation Processes, Prentice Hall, Englewood Cliffs, N.J., 1975. 

Measurement Error Uncertainty in the interpretation of unit performance results from statistical errors in the measurements, low levels of process understanding, and differences in unit and modeled performance (Frey, H.C., and E. Rubin, Evaluate Uncertainties in Advanced Process Technologies, Chemical Engineering Progress, May 1992, 63-70). It is difficult to determine which measurements will provide the most insight into unit performance. A necessary first step is the understanding of the measurement errors hkely to be encountered. 

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. 

However, given that reconciliation will not always adjust measurements, even when they contain large random and gross error, the adjustments will not necessarily indicate that gross error is present. Further, the constraints may also be incorrect due to simphfications, leaks, and so on. Therefore, for specific model development, scrutiny of the individual measurement adjustments coupled with reconciliation and model building should be used to isolate gross errors. 

Applying the information-processing model to each of the operator tasks can provide insights into the potential for human error and also suggest solutions for preventing errors. 

You should consider obtaining internal and external quality assurance reviews of the study (to ferret out errors in modeling, data, etc.). Independent peer reviews of the QRA results can be helpful by presenting alternate viewpoints, and you should include outside experts (either consultants or personnel from another plant) on the QRA review panel. You should also set up a mechanism wherein disputes between QRA team members (e.g., technical arguments about safety issues) can be voiced and reconciled. All of these factors play an essential role in producing a defendable, high-quality QRA. Once the QRA is complete, you must formally document your response to the project team s final report and any recommendations it contains. 

In general, the R factor is between 0.15 and 0.20 for a well-determined protein structure. The residual difference rarely is due to large errors in the model of the protein molecule, but rather it is an inevitable consequence of errors and imperfections in the data. These derive from various sources, including slight variations in conformation of the protein molecules and inaccurate corrections both for the presence of solvent and for differences in the orientation of the microcrystals from which the crystal is built. This means that the final model represents an average of molecules that are slightly different both in conformation and orientation, and not surprisingly the model never corresponds precisely to the actual crystal. 

HWR v.trijbilny inosily driven by modeling of human errors and availability of alternative boron injection system... 

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