Interpolation, extrapolation and estimation procedures

Values of rate constants are often required for modelling at temperatures and pressures well outside the ranges covered by experimental measurements. Therefore, part of the evaluation process is to interpolate between experimental data, or extrapolate from them, to provide information over as wide a range of conditions as possible. To do so evaluators make use of a number of estimation techniques based on their own experience and existing knowledge, as well as theories of chemical kinetics. 

As knowledge increases and theories improve the gap between these two extremes can be expected to diminish, but there is still sufficient uncertainty concerning the mechanisms of many elementary reaction to make estimation an extremely uncertain procedure. On the other hand, estimation can be used with some confidence in many cases where the kinetics of the process are thoroughly understood. 

The interpolation and extrapolation procedures described in this section make use of existing experimental data and are less controversial than estimation techniques but also not without problems. 

So long as there are no indications that the rate constant may be pressure dependent it is usually assumed that it conforms to equation (3.2). When n = 0 equation (3.2) reduces to the Arrhenius equation and will give a linear plot of In k vs. l/T. In practice, n takes on small positive values leading to a degree of curvature of the Arrhenius plot which becomes more pronounced as 1/T becomes smaller. At temperatures less than approximately 1000 K the curvature is usually difficult to detect experimentally at the current precision of measurement. 

There are times when the temperature coefficient has not been measured in this way and, under those circumstances, the evaluator will either use 

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Because data are often very limited, or even non-existent, for a particular reaction the evaluator is also called on to use these sparse data to provide rate constant values for the conditions of interest to the modeller, which may be well outside the range of experimental studies. The interpolation, extrapolation, and estimation procedures involved are touched on in this section but treated in more detail in Section 3.4. 

The former is the classical unimolecular reaction mechanism which has been the subject of continuous theoretical scrutiny since it was first proposed in 1922. The way in which evaluators handle these reactions has been described in Sections 3.2.2 and 3.3.4. The interpolation or extrapolation or, indeed, estimation procedures that may be used for and are analogous to those described earlier in this section. Fc is usually based on fitting the theory to existing experimental data, but theoretical estimation methods are also available [18] and, in some cases, for small molecules at temperatures close to 300 K, a standard value of 0.6 has been used by some evaluators [52] where no experimental data are available. 

Figure 8.13 Idealized plots according to the Method of Standard Additions. Each point plotted is assumed to be the mean of several replicate determinations. The traditional method (left panel) simply plots the observed analytical signal Y vs the amount of calibration standard added x (the black square corresponds to the nonspiked sample, Y = Yq), and estimates the value of X by extrapolation of a least-squares regression line to Y = 0 (see text) however, this procedure implies that the confidence interval at this point (not shown, compare Figure 8.12) has widened considerably. By using a simple transformation from Y to (Y-Yq) the extrapolation procedure is replaced by one of interpolation, thus improving the precision (more narrow confidence interval). Reproduced from Meier and Ziind, Statistical Methods in Analytical Chemistry, 2nd Edition (2000), with permission of John Wiley Sons Inc.

A word of caution should be added with regard to the calculation of the burn-out flux for a pressure intermediate to the main pressure groups that have been correlated this calculation must not be done by taking intermediate y values from Table II. The recommended procedure is to estimate the burn-out flux for the required conditions for the main pressure groups above and below the required pressure, and then to interpolate linearly. It must be also emphasized that while the above correlations can be used with confidence within the experimental ranges of the data, extrapolation outside these ranges should not be taken very far without allowing for a possible reduction in the accuracy obtained. 

To put it more clearly, we select a point and a cluster of points around it in the factor space, that is, we choose a subfactor space in it. The experiment is done in that region and based on it the first model is defined. Such a model is used to estimate the response outside the experimental region. Such an estimation of a response is called extrapolation. If the same is done for points within the experimental region it is then called interpolation. Since extrapolation confidence diminishes with distance from this region, it is done in its vicinity. Extrapolation outcomes are used to choose conditions for performing the next experiment. A further procedure in finding the optimum is repeated. To choose a model for doing the first so-called basic experiment, it is necessary for the model to fulfill certain requirements. 

In order to improve the procedure described above it was adapted to flow mode with the use of an original flow-injection manifold [11], Once a sample and a single standard solution are introduced to the system they can be gradually diluted, and interpolative and extrapolative estimations of analyte concentration in the sample can be obtained for every dilution degree. This procedure allows examination of possible interferences, and is, in effect, a sterling, integrated calibration method that combines in one calibration procedure the set of standards method and the standard addition method with the dilution method. Moreover, it has been demonstrated that any of the calibration methods described above can be integrated with the use of a constructed manifold. 

Given a postulated functional form of the dose-response relationship, the frequency of occurrence of toxic effects may be used to estimate the unknown parameters. tn addition, this estimated dose-response can be extrapolated to provide either (1) estimates of risk probabilities at lower dose levels, or (2) an estimate of the dose level associated with any particular probability of risk. Implicitly, this approach presumes that the true dose-response can be realized within the postulated functional form used in the estimation and extrapolation procedure. Although this presumption is often not critical for interpolation within the range of observed response rates, it may be extremely critical for extrapolation outside this observable range. 

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