Parameter fitting 422 Subject

With formulae (3.58), (3.59) and (3.66) Q-branch contours are calculated for CARS spectra of spherical rotators at various pressures and for various magnitudes of parameter y (Fig. 3.14). For comparison with experimental data, obtained in [162], the characteristic parameters of the spectra were extracted from these contours half-widths and shifts of the maximum subject to the density. They are plotted in Fig. 3.15 and Fig. 3.16. The corresponding experimental dependences for methane were plotted by one-parameter fitting. As a result, the cross-section for rotational energy relaxation oe is found ... 

The activation parameters are subjected to errors, especially in the precise determination of temperatures such as Tc, the frequency separation (Av), line widths, coupling constants, variation of the concentration with the temperature etc. Although the experimental determination of the activation parameters could have been performed accurately, it should not be pretended to possess excessive accuracy. Caution is advised especially with entropies of activation, due to the inherent imprecision of the method (ordinates at the origin from least-squares line fitting) ... 

Solver needs reasonably close initial estimates of the parameter values, otherwise it can easily produce non-optimal results. If at all possible, subject a subset of the data to a linear least squares analysis to get an idea of at least some of the parameter values. And wherever possible, use a graph to see how close your initial estimate is, and follow the progress of the iterations visually, even though that slows down Solver. Where this is not feasible, as in multi- parameter fits, at least inspect the orders of magnitude of the values in the two columns Solver compares yexp and yca c. 

The problem of finding a global minimum in parameter fitting is partly due to a disparity in the relative size of the components of the kinetic rate constant and partly to the inherent problems of fitting sums of exponential terms to data. Perhaps there is no solution and we will always be troubled by this issue. However, now that the data is available and the need for better fitting procedures is before us, there is an incentive to improve parameter optimization routines to reduce the subjectivity present in these procedures. Then again, subjectivity in this type of data-fitting may well be inevitable. 

The solid curves Figure 10.9 are best fitted to Eq. (45), with K] and 02 as fitting parameters. As can be seen, Eq. (45) satisfactorily describes the nonstoichiometry, even though the best-fitted values for the fitting parameters are subjected to rather large uncertainties. The latter is again attributed to the still poor precision of the measurement, particularly in the vicinity of the stoichiometric point. 

Subject-Based Retrieval Parameters. There are numerous means by which the subject content of a patent can be expressed, and which a searcher can use in developing a search strategy. Different databases offer differing subsets of these means. Effective strategies should in general not be limited to a single type of retrieval parameter rather, they should be built from different parameters and modified as needed to provide the strategy best fitted to the subject at hand. 

Data points are subjected to nonlinear curve fitting. For these data, Equation 12.5 is used to fit the curve with basal = 0. The fitting parameters for histamine and E-2-P are given in Table 12.4b. The curves are shown in Figure 12.5a. 

The Avrami—Erofe ev equation, eqn. (6), has been successfully used in kinetic analyses of many solid phase decomposition reactions examples are given in Chaps. 4 and 5. For no substance, however, has this expression been more comprehensively applied than in the decomposition of ammonium perchlorate. The value of n for the low temperature reaction of large crystals [268] is reduced at a 0.2 from 4 to 3, corresponding to the completion of nucleation. More recently, the same rate process has been the subject of a particularly detailed and rigorous re-analysis by Jacobs and Ng [452] who used a computer to optimize curve fitting. The main reaction (0.01 < a < 1.0) was well described by the exact Avrami equation, eqn. (4), and kinetic interpretation also included an examination of the rates of development and of multiplication of nuclei during the induction period (a < 0.01). The complete kinetic expressions required to describe quantitatively the overall reaction required a total of ten parameters. 

It is important to note that the fitting according to eq. (1) requires zero intercept behavior i.e., F =. 00 for H (for which Oj = Or =. 00). While we recognize that the data for the unsubstituted (H) member of a set may be as subject to experimental error as any other member, such error is generally relatively small for a set of reliable data. Any constant error from this source will be distributed among all of the substituents in such a manner as to achieve best fit. Any loss in precision of fitting of the set which may result by such a procedure we believe is a small price to pay compared to the violence done by introduction in eq. (I) of a completely variable constant parameter. The latter procedure has been utilized by other authors both in treatments by the simple Hammett equation and by the dual substituent parameter equation. 

Sample fits of observed volumes at different times (open circles) and their OLS-predicted time course (solid line) can be seen for a few of the subjects in Figure 3.4, a-d the OLS estimates of the parameter values for all subjects are reported in Table 3.1. 

It is seen that this process is essentially a least square fit of atp eg and pifirregby (f>i and (fE, subject to a minimum energy condition which allows to determine a and /3. Note that a and fi are related by the norm of so that there is in fact a single parameter in this minimisation. 

The unknown model parameters will be obtained by minimizing a suitable objective function. The objective function is a measure of the discrepancy or the departure of the data from the model i.e., the lack of fit (Bard, 1974 Seinfeld and Lapidus, 1974). Thus, our problem can also be viewed as an optimization problem and one can in principle employ a variety of solution methods available for such problems (Edgar and Himmelblau, 1988 Gill et al. 1981 Reklaitis, 1983 Scales, 1985). Finally it should be noted that engineers use the term parameter estimation whereas statisticians use such terms as nonlinear or linear regression analysis to describe the subject presented in this book. 

Two issues present themselves when the question of PB-PK model validation is raised. The first issue is the accuracy with which the model predicts actual drug concentrations. The actual concentration-time data have most likely been used to estimate certain total parameters. Quantitative assessment, via goodness-of-fit tests, should be done to assess the accuracy of the model predictions. Too often, model acceptance is based on subjective evaluation of graphical comparisons of observed and predicted concentration values. 

Modeling relaxation-influenced processes has been the subject of much theoretical work, which provides valuable insight into the physical process of solvent sorption [119], But these models are too complex to be useful in correlating data. However, in cases where the transport exponent is 0.5, it is simple to apply a diffusion analysis to the data. Such an analysis can usually fit such data well with a single parameter and provides dimensional scaling directly, plus the rate constant—the diffusion coefficient—has more intuitive significance than an empirical parameter like k. 

A phrase such as "seriously contaminate" is known as a linguistic variable it gives an indication of the magnitude of a parameter, but does not provide its exact value. Subjective knowledge is expressed by statements that contain vague terms, qualifications, probabilities, or judgmental data. Objects described by these vague statements are more difficult to fit into crisp sets. 

In equation 3.4-18, the right side is linear with respect to both the parameters and the variables, j/the variables are interpreted as 1/T, In cA, In cB,.. . . However, the transformation of the function from a nonlinear to a linear form may result in a poorer fit. For example, in the Arrhenius equation, it is usually better to estimate A and EA by nonlinear regression applied to k = A exp( —EJRT), equation 3.1-8, than by linear regression applied to Ini = In A — EJRT, equation 3.1-7. This is because the linearization is statistically valid only if the experimental data are subject to constant relative errors (i.e., measurements are subject to fixed percentage errors) if, as is more often the case, constant absolute errors are observed, linearization misrepresents the error distribution, and leads to incorrect parameter estimates. 

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