Minimizing difference between theoretical

The fitting procedure results in a set of model parameters, minimizing the difference between theoretical and measured concentration profile. A discussion of statistical based objective functions and optimization procedures is beyond the scope of this book. For further information see, for example, Lapidus (1962), Barns (1994), Korns (2000) and Press et al. (2002, http //www.nr.com/). 

K is a correcting flmction accounting for residual deviations and minimizes the differences between theoretical and values approximated with the help of the empirical correlation. 

Machine tool geometric error identification strategy depends on the type and configuration of the machine as well as the purpose of verification [5, 6].The difference between theoretical and real point represents the influence combined of machine errors for each point, volumetric error (ve) Eq, 1. Minimizing the machine tool volumetric error using non-lineal optimization techniques, the approximation functions of each error can be obtained, if there are enough measured points. 

There is little difference between the two methods in the current example since the data are of high quality. However, the sequential approach of first minimizing and then minimizing is somewhat better for this example and is preferred in general. Figure 7.4 shows the correlation. It is theoretically possible to fit both kj and kn by minimizing S, but this is prone to great error. 

In this case, Sy is the best, theoretical, C chemical-shift value, 5b is the chemical shift of the di[ C]methylamino group in the nonprotonated form, and A is the chemical-shift difference between the di[ C]methylamino group in the protonated and nonprotonated forms. The best fit was obtained when E, [ST(i) 5obs(i)] was minimized Sobs is the chemical-shift value observed at that given pH value. 

Note added in proof. It should be emphasized that the parametric fit used by Scheraga and co-workers is not achieved by minimizing the theoretical free energy function with respect to said parameters. Rather, the difference between the theoretical free energy function and experimental data is minimized with respect to these parameters. The method is, therefore, related to curve fitting. 

Once equilibrium has been reached, the height difference between the two liquid surfaces is all that remains to be measured. The primary factor to note here is that capillaries are used to minimize the dilution effects. This means that corrections for capillary rise must be taken into account unless the apparatus allows the difference between two carefully matched capillaries to be measured. We discuss capillary rise in Chapter 6, Sections 6.2 and 6.4. Finally, there is an extremely important practical reason, in addition to the theoretical requirement of isothermal conditions, for good thermostating in osmometry experiments. The apparatus consists of a large liquid volume attached to a capillary and therefore has the characteristics of a liquid thermometer The location of the meniscus is quite sensitive to temperature fluctuations. 

A back Fourier transform on individual peak can be then applied to isolate individual contributions. The so-obtained filtered EXAFS signal is analyzed in order to determine the related structural parameters N, R and a. A common method consists to build a theoretical model from the relation (2) using known electronic parameters for X(k), So2, Aj(ir,k) and Oy (k), and, by treating N, R, and a as free parameters, to minimize the difference between the theoretical and experimental curves. The electronic parameters can be evaluated from EXAFS signals recorded for references of known structures or from theoretical calculations using the efficient ab initio FeFF codes [7, 8],... 

A numerical method is said to be convergent if the global discretization error tends to zero when the step size tends to zero. The global discretization error is the difference between the computed solution (neglecting round-off errors) and the theoretical solution. Convergence is a minimal property of a numerical method and there is no use of a divergent method. Most numerical methods are convergent if they are consistent and stable. 

Since Eq. (16) is nonlinear, one must use a nonlinear least-squares fitting program or, as described here, make use of the Solver option available as an add-in tool in Excel. An example of the use of Solver is given in Chapter HI. In the present apphcation, initial estimated values for the four fitting parameters (Q, Cj, P, and j8) are entered into four worksheet cells. For each of the Ndata points, these cells are used to calculate r and then to obtain a theoretical < >obs value from Eq. (16). The difference between the experimental and theoretical < >obs value (residual) is squared and the sum of these squares (essentially proportional to is placed in a test location. Solver is then run iteratively to adjust the fitting parameters so as to minimize this sum of residuals squared. 

We can state categorically that no data have come to our attention that are of sufficient accuracy to test this difference between Bueche s theoretical result and the suggested empirical expression. This is due to the extreme dependence of f on Data of sufficient accuracy would have to be on a polymer with a very low Tg so that variation of f with 95 would be minimized simply because the change in (T — Tg) with would be necessarily small. 

Fourier domain fitting. The Fourier transform of the experimental elution curve is calculated. The parameters a and 3 are then determined using a fitting procedure in the Fourier domain that is equivalent to a least-squares criterion in the time domain. With Fourier domain estimation, model parameters are chosen to minimize the difference between the Fourier transforms of experimental and theoretical elution curves. The Fourier transform of a bounded, time varying response curve, f(t), is defined as... 

The best criterion for minimizing the difference between the theoretical and experimental transform is not immediately obvious. In the time domain, a least-squares criterion is usually preferred (221 that is, parameters are selected that minimize the least-squares objective function... 

Fortunately, there is such a method, which is both simple and generally applicable, even to mixtures of polyprotic acids and bases. It is based on the fact that we have available a closed-form mathematical expression for the progress of the titration. We can simply compare the experimental data with an appropriate theoretical curve in which the unknown parameters (the sample concentration, and perhaps also the dissociation constant) are treated as variables. By trial and error we can then find values for those variables that will minimize the sum of the squares of the differences between the theoretical and the experimental curve. In other words, we use a least-squares criterion to fit a theoretical curve to the experimental data, using the entire data set. Here we will demonstrate this method for the same system that we have used so far the titration of a single monoprotic acid with a single, strong monoprotic base. 

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