Extrapolation problems

Furthermore, there are two other aspects to the extrapolation problem one structural and one statistical. An illustrative example of these various cases can be found in a dataset of benzamides (S16.1). that one of the present authors (U.N.) published some time ago [44]. If one develops a PLS model based on the same descriptors and the same, experimental design-based, training set (compounds 1-16) augmented by compound 17 (Table 16.8) in order to prove the points raised above [the prediction limit (1.502) set to two times the overall RSD of the model (0.751) which roughly gives 95% confidence interval], one can observe the following with respect to predictions on the remaining test set compounds ... 

These types of extrapolation problems are not the only ones that have to be tackled. Here are some more ... 

The diversity of existing extrapolation techniques also relates to the types of extrapolation problems. Extrapolation can consist of range extrapolation, implying intra- or extrapolation using an available data set. It can also be a specific extrapolation from 1 data set to parameters in another realm (e.g., from total concentrations to bioavailable concentrations, or from species sensitivities to community-level responses). 

Finally, the diversity of extrapolation techniques relates to the diversity of technical solutions that have been defined in the face of the various extrapolation problems. Methods may range from simple to complex, or from empirical-statistical methods that describe sets of observations (but do not aim to explain them) to mechanism-based approaches (in which a hypothesized mechanism was guiding in the derivation of the extrapolation method). In addition, they may range from those routinely accepted in formal risk assessment frameworks to unique problem-specific approaches, and from laboratory-based extrapolations consisting of 1 or various kinds of modeling to physical experiments that are set up to mimic the situation of concern (with the aim to reduce the need for extrapolation modeling). 

An axis for problem type, that is, the scientific problem to be solved by the extrapolation problem (e.g., from simple and generic assessment questions to more complex and specific ones)... 

Type of extrapolation Problem definition Problem subdefinition... 

In Table 10.4, the same overview is presented however, it is now organized from the perspective of the subjects of extrapolation (e.g., matrix and media extrapolations), so that one can inspect the presence of extrapolation methods, at different tiers, for a given extrapolation problem. 

To facilitate working with extrapolation problems, a set of questions is presented. Given the assessment problem, these questions can be asked by the assessor. These questions have been ordered according to the steps identified earlier. Together with Table 10.3 and Table 10.4, these questions show how a risk assessment proceeds. Note that 1 question can occur various times, because of the fact that 1 issue is encountered, for example, from the perspectives of exposure extrapolation (time-varying exposure) and of effects extrapolation (time-varying, age-specific sensitivity). 

What is the order of handling extrapolation problems Following the cause-to-effect chain of events, for example, (Q)SARs first, then matrix and media, and then mixture assessment. 

Forbes V.E., Calow P., Sibly R. (2008) The extrapolation problem and how population modelling can help. Environmental Toxicology and Chemistry 27(10) 1987-1994. 

Empirical QSPR Correlations In quantitative structure property relationship (QSPR) methods, physical properties are correlated with molecular descriptors that characterize the molecular and electronic structure of the molecule. Large amounts of experimental data are used to statistically determine the most significant descriptors to be used in the correlation and their contributions. The resultant correlations are simple to apply if the descriptors are available. Descriptors must generally be generated by the user with computational chemistry software, although the DIPPR 801 database now contains a table of molecular descriptors for most of the compounds in it. QSPR methods are often very accurate for specific families of compounds for which the correlation was developed, but extrapolation problems are even more of an issue than with GC methods. 

For threshold effects the high-to-low dose extrapolation problem is solved if the threshold dose for the human population can be identified. If the threshold dose is known for a particular chemical, and the dose of that chemical received by individuals from whatever environmental sources create their exposure is also known, then it will be possible to understand whether those individuals are at risk (whether the environmental dose they receive exceeds the threshold dose) and the extent of that risk (the fraction of the population experiencing a dose exceeding the threshold). The problem for threshold toxicants, to be dealt with in the chapter on risk assessment, is somewhat more complex than this, but not greatly so. 

However, many statistical modeling techniques do not, in an easy and straightforward way, by default, enable the estimation of whether a prediction is an interpolation to the model, thus rendering the prediction more credibility or an extrapolation to the model in which case the prediction must be evaluated with greater care. Furthermore, there are two aspects to the extrapolation problem one structural and the other statistical. Considerable research has been devoted to the problem of ADs. For a recent compilation on this issue, see Ref. [120]. 

Knasmuller, S., Steinkellner, H., Majer, B. J., Nobis, E. C., Scharf, G., and Kassie, F. (2002). Search for dietary antimutagens and anti carcinogens Methodological aspects and extrapolation problems. Food Chem Toxicol 40, 1051-1062. 

The high to low dose extrapolation problem is conceptually straight-forward. The probability of a toxic response is modeled by a dose-response function P(D) which represents the probability of a toxic response when exposed to D units of the toxic agent. A general mathematical model is chosen to describe this functional relationship, its unknown parameters are estimated from the available data, and this estimated dose-response function P(D) is then used to either (1) estimate the response measure at a particular low dose level of interest or (2) estimate that dose level corresponding to a desired low level of response (this dose estimate is commonly known as the virtually safe dose, VSD). 

Model based control schemes such as model predictive control are highly related to the accuracy of the process model. A regional-knowledge index is proposed in this study and applied in the analysis of dynamic artificial neural network models in process control. To tackle the extrapolation problem and assure stability of the control system, we propose to run a neural adaptive controller in parallel with a model predictive control. A coordinator weights the outputs of these two controllers to make the final control decision. The proposed analysis method and the modified model predictive control architecture have been applied to a neutralization process and excellent control performance is observed in this highly nonlinear system. 

Figure 3.56 illustrates how ([r (0P) changes with time t. The boundaries of the three time regimes can also be obtained as an interaction between two lines that correspond to the relevant sections and their extrapolates (Problem 3.27). The diffusion characteristics show a crossover from the single-bead diffusion to the A -bead... 

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