Fitting methods

Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte s concentration is determined by solving the appropriate rate equation at one or two discrete times. The relationship between the analyte s concentration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Miscellaneous Methods At the beginning of this section we noted that kinetic methods are susceptible to significant errors when experimental variables affecting the reaction s rate are difficult to control. Many variables, such as temperature, can be controlled with proper instrumentation. Other variables, such as interferents in the sample matrix, are more difficult to control and may lead to significant errors. Although not discussed in this text, direct-computation and curve-fitting methods have been developed that compensate for these sources of error. ... 

Equation 13.23 can also be used as the basis for a curve-fitting method. As shown in Figure 13.14, a plot of In(Ct) as a function of time consists of two regions. At short times the plot is curved since A and B are reacting simultaneously. At later times, however, the concentration of the faster-reacting component. A, decreases to 0, and equation 13.23 simplifies to... 

Among these models, the Fukui-Kaneko model is regarded as the most accurate one because it is derived from the linearized Boltzmaim equation. However, the flow rate coefficient, Qp, in the Fukui-Kaneko model, is not a unified expression, as defined in Eqs (6)-(8). This would cause some inconveniences in a practical numerical solution of the modified Reynolds equation. Recently, Huang and Hu [17] proposed a representation of Qp, as shown in Eq (10), for the whole range of D from 0.01 to 100, by a data-fitting method, to replace the segmented expressions of the Fukui-Kaneko model... 

Most of the well-developed methods available for solving such a system falls within the categories of direct or exact-fitted methods and iterative or successive-approximate methods which are gaining increasing popularity. 

The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. 

The IC50 can thus be accurately determined by fitting the concentration-response data to Equation (5.1) through nonlinear curve-fitting methods. Some investigators prefer to plot data in terms of % inhibition rather than fractional activity. Using the mass-balance relationships discussed above, we can easily recast Equation (5.1) as follows ... 

The model gives a unique answer in the limit of an infinite basis set, whereas density matrix fitting methods do not [10, 11]. 

Do so, we use the formalism of the variational density fitting method [55, 56] where the Coulomb self-interaction energy of the error is minimized ... 

Sine fitting methods for estimating phase and modulation... 

Stability constant too large to be reliably calculated using curve-fitting method. No evidence of binding was seen. 

One could argue whether PCR and PLS should be part of the chapter Model-Based Analyses or Model-Free Analyses. Both, PCR and PLS, are clearly not hard-model fitting methods in the way presented in Chapter 4, nor are they pure model-free analyses. They are somewhere in between, maybe closer to model-free analyses and that is the reason for discussing them here. 

These problems can be fully or partially avoided by (i) an improved instrumental resolution, (ii) the introduction of pattern indexing methods and (iii) the introduction of fitting methods. 

Residual Standard Deviation Another important approach that can be used to evaluate the applicability domain is the degree-of-fit method developed originally by Undberg et al. [40] and modified recently by Cho et al. [6]. According to the original method, the predicted y values are considered to be reliable if the following condihon is met ... 

Results of these studies are very encouraging and Indicate that a reasonably fast, accurate, and practical method for the quantitative determination of minerals In complex solids can be achieved with this approach, particularly If multivariate least squares curve fitting methods can be automated. 

After rearranging Eq. (2), the values of ixDCD and KB can be estimated by nonlinear least squares curve-fitting methods (similar to the Michaelis-Menten equation) or the expression can be rearranged under different linear forms (y = rnx I n), where y = (jueff yiD) and x = [CD], Well known are... 

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