Error terms, analytical model

Before stepping through the several dimensions, it is worthwhile to examine the general analytical model which applies and, through that, consider the implications of the necessary assumptions in practical applications. To begin, let us express the observed response (y) and its error (e) in terms of the blank (B) and concentrations of all contributing analytes (xj). 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

In analytical chemistry we expect the calibration (standardization) not to depart much from a straight line. Should a slight deviation occur, a quadratic trend will usually describe the trend acceptably well. This means that we can make a decision by seeing whether a linear or a quadratic curve fits the data better. Thus, Mandel s test" compares the residual standard error of both models by an test [see eqn (2.18)] and helps us in deciding whether additional terms should be added to the straight line fit (see more details in Appendix 1). The practical equation to be used is ... 

The book has been written around a framework of linear models and matrix least squares. Because we authors are so often involved in the measurement aspects of investigations, we have a special fondness for the estimation of purely experimental uncertainty. The text reflects this prejudice. We also prefer the term purely experimental uncertainty rather than the traditional pure error , for reasons we as analytical chemists believe should be obvious. 

NIR models are validated in order to ensure quality in the analytical results obtained in applying the method developed to samples independent of those used in the calibration process. Although constructing the model involves the use of validation techniques that allow some basic characteristics of the model to be established, a set of samples not employed in the calibration process is required for prediction in order to conhrm the goodness of the model. Such samples can be selected from the initial set, and should possess the same properties as those in the calibration set. The quality of the results is assessed in terms of parameters such as the relative standard error of prediction (RSEP) or the root mean square error of prediction (RMSEP). 

The real power in the multi-coefficient models, however, derives from the potential for the coefficients to make up for more severe approximations in the quantities used for (/) in Eq. (7.62). At present, Truhlar and co-workers have codified some 20 different multicoefficient models, some of which they term minimal , meaning that relatively few terms enter into analogs of Eq. (7.62), and in particular the optimized coefficients absorb the spin-orbit and core-correlation terms, so they are not separately estimated. Different models can thus be chosen for an individual problem based on error tolerance, resource constraints, need to optimize TS geometries at levels beyond MP2, etc. Moreover, for some of the minimal models, analytic derivatives are available on a term-by-term basis, meaning that analytic derivatives for the composite energy can be computed simply as the sum over tenns. 

The algorithm used is attributed to J. B. J. Read. For many manipulations on large matrices it is only practical for use with a fairly large computer. The data are arranged in two matrices by sample i and nuclide j one matrix, V, contains the amount of each nuclide in each sample the other matrix, E, contains the variances of these numbers, as estimated from counting statistics, agreement between replicate analyses, and known analytical errors. It is also possible to add an arbitrary term Fik to each variance to account for random effects between samples not considered in the model this is usually done in terms of an additional fractional error. Zeroes are inserted for missing data in cases in which not all nuclides were measured in every sample. 

In the presented framework the results by Klik and Yao (including the analytical formulas for them missing in Ref. 93) can be obtained immediately if to take the function cpj in a stepwise form (4.145) and not to allow for the corrections caused by the finiteness of its derivative at x = 0. In our terms this means to stop at set (4.167), namely, zero-derivative solution, and not to go further. The emerging error is, however, uncontrollable and not at all small. As an illustration, in Figure 4.12 we show the result obtained with this model (dashed lines) for the cubic susceptibility y in a textured system where the particle common axis n is tilted under the angle p = ji/3 to the probing field. One can see that deviations are substantial. 

Fitting of model potentials to supermolecular interactions like in Eq. (3-1) has its disadvantages the calculations have to be repeated many times and a predefined analytical expression of the model potentials in terms of atomic parameters is required. These parameters are not easily transferred to other situations. The internal state of, e.g., system A depends on the presence of another system X, and this will be different near system Y. Furthermore, the chosen analytical form of the potential may give rise to problems and errors. For example, Hartree-Fock (HF) calculations fitted to a power series in 1/r suggest that the 1/r6 terms have to do with dispersion, which is not part of the HF energy. Finally, such empirical potentials are best for describing situations close to those to which the parameters are fitted. Whenever the situation is very different from that, the results will be doubtful. 

The validation process determines the amount of error owing to variation among the values in the population. It is used to check for the existence of a relationship between the calibration set and the validation set. Manufacturers of NIRS instrumentation include software packages that allow the operator to predict analytical results on data files that have been stored, thus allowing for validation of the calibration equation and testing for errors in the developed calibration. This enables calibration equation performance testing in terms of precision. The validity of these models depends on the ability of the calibration set to accurately represent the samples in the prediction set. 

A proper model validation procedure consists of a model verification part and a part where the model predictions are compared to experimental data [61]. The model verification may be performed by the method of manufactured solutions[14 7, 163]. The method of manufactured solutions consists in proposing an analytical solution, preferably one that is infinitely differentiable and not trivially reproduced by the numerical approximation, and the produced residuals are simply treated as source terms that produce the desired or prescribed solution. These source terms or residuals are referred to as the consistent forcing functions. This method can be used to confirm that there are no programming errors in the code and to monitor the truncation error behavior during the iteration process. 

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