Models inadequacy

As in algebraic models, the error term accounts for the measurement error as well as for all model inadequacies. In dynamic systems we have the additional complexity that the error terms may be autocorrelated and in such cases several modifications to the objective function should be performed. Details are provided in Chapter 8. 

The model inadequacy was particularly evident for the systems having different Ba loadings, which showed an increase in the breakthrough time associated with similar NO oxidizing capacity. Work is currently in progress in order to gain a better adequacy of the model to our data in particular the nitrite route has also been included to provide an additional NO adsorption pathway, which is in line with obtained data, and preliminary results obtained in this direction seem to be promising. 

The utility of the overall dependence of the reaction rate upon temperature appears to be slight, perhaps in some cases allowing a discrimination between adsorption and surface reaction classes of models. The study of the residuals of various estimated parameters as a function of temperature can clearly indicate model inadequacies (Section IV) and, in some cases, can lead to model modifications correcting these model inadequacies (Section V). 

The analysis of variance techniques of Section IV,A have been seen to provide information about the overall goodness of fit or about testing the importance of the contribution of certain terms in the model toward providing this overall fit of the data. Although these procedures are quite useful, more subtle model inadequacies can exist, even though the overall goodness of fit is quite acceptable. These inadequacies can often be detected through an analysis of the residuals of the model. 

A plot of the residual versus the predicted value, r, of a model can indicate whether the model truly represents the rate data. For example, residuals that are generally positive at low predicted rates and negative at high predicted rates can indicate a model inadequacy, even though the overall test of an analysis of variance indicates that the model is acceptable. 

Measurement Residual Plot (Model, Sample and Variable Diagnostic) The spectral residuals for the validation data shown in Figure 5-58 are an order of magnitude smaller and less structured than the residuals obtained when the pure spectra were estimated (Figure 5-33). This can be explained as follows Equation 5.18 shows that the reported concentrations and temperatures (C is used in Equation 3.18) are used in the computation of the calibration residuals. Therefore, errors in the reported concentrations and temperatures contribute to the calibration residuals in addition to model inadequacy. In contrast. estimated concentrations and temperatures (C is u.sed in Equation 5-13)... 

The CMB-dlspersion model comparisons had Identified data base deficiencies that would not have been apparent had the model predictions been compared only to measured particulate mass. Correction of the emission inventory and model inadequacies was, however, critical to improving the model s source Impact predictions and to the strategy s success. 

If a linear model is inadequate it means that the response surface is not approximated to the plane. Apart from Fisher s criterion, which is there to judge the lack of fit of a regression model, inadequacy may also be recognized in this way ... 

Removing the model inadequacy by introducing rejected factors (in the phase of screening experiments and linear model analysis) and by an increased number of trial replications, is much more acceptable. 

The contribution of the model inadequacy, i.e., the capability of the assumed mathematical model to fit the instrument response. 

Each point on the curves in Fig. 4.19 corresponds to the mean value of various experimental results. We can notice that, even if we have good trends, the experimental and calculated values do not match well. This can be ascribed to model inadequacies, especially with respect to the liquid exit conditions in that case, we considered that the MWPB output had occurred at x = 0 and at x = H,j when it was experimentally observed that the liquid exit dominantly occurs at x = H,j. This results in a decrease in the mean residence time computed values. If we look at Figs. 4.20 to 4.22, which have been obtained at different operating conditions, we can conclude that we do not have major differences between the computed and experimental values of liquid MWPB hold up then we can consider the equality of the transition probabilities between the individual states of the stochastic model to be realistic. 

Many of the plots just suggested are not limited to ordinary residuals. Weighted residuals, partial residuals, studentized residuals, and others can all be used to aid in model diagnostics. Beyond residual plots, other plots are also informative and can help in detecting model inadequacies. One notable plot is a scatter plot of observed versus predicted values usually with the line of unity overlaid on the plot (Fig. 1.8). The model should show random variation around the line of unity. Systematic deviations from the line indicate model misspecification whereas if the variance of the predicted values increases as the observed values increase then the variance model may be inappropriate. 

The residuals may be distributed in various different ways. First of all they may be scattered more or less symmetrically about zero. This dispersion can be described by a standard deviation of random experimental error. If this is (approximately) constant over the experimental region the system is homoscedastic, as has been assumed up to now. However the analysis of residuals may show that the standard deviation varies within the domain, and the system is heteroscedastic. On the other hand it may reveal systematic errors where the residuals are not distributed symmetrically about zero, but show trends which indicate model inadequacy. 

In the forward selection method, was significant. Now, its contribution is diluted by the x, x, and Xf, variables, with a very different regression equation, and having lost 3 degrees of freedom. Obviously, the two methods may not produce equivalent results. This is often due to model inadequacies, such as XiS, themselves that are correlated, a problem we address in later chapters. The new full model is... 

Since no model can reproduce the pure error sum of squares, the maximum explainable variation is the total sum of squares minus SSpe- In our case, SSt - SSpe = 8930.00 - 45.00 = 8885.00, which corresponds to 8885.00/8930.00 = 99.50% of the total sum of squares. This percentage is close to 100%, because the pure error contribution is relatively small, but it is with this new value that we should compare the variation explained by the regression, 77.79%. The model inadequacy appears clearly in the two first graphs of Fig. 5.8. Once again the residuals are distributed in a curved pattern. 

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