Fitting technique

A straightforward extension of the three-point technique is to utilize a larger number of measured AE-i data pairs, and to analyze the data by using some kind of numerical curve-fitting procedure, usually a nonlinear least-squares method. This increases 

While the above four techniques are widely used in research laboratories and in field testing, other techniques have also been proposed for determining corrosion rate and investigating the mechanism of corrosion reactions from electrochemical measurements. A brief description of some of them is given here for the sake of completeness. These techniques are not widely used, and the errors involved in these techniques have not been examined. Therefore, they will not be included in the critical comparison of techniques. 

Several techniques have been suggested that are intermediate between the three-point technique and the curve-fitting technique. They allow the determination of the corrosion current density and the Tafel slopes from measurements taken near the corrosion potential by using, usually, a graphical approach. Performance that is superior to the three-point technique is generally claimed. 

The use of the inflection point of the polarization curve has also been suggested for corrosion-rate determination. As the inflection point usually occurs near the corrosion potential, measurements at small current densities are sufficient, and it has been claimed that independent measurement of the Tafel slopes is not needed to obtain an approximate value of the corrosion rate. 

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When an element is present on the surface of a sample in several different oxidation states, the peak characteristic of that element will usually consist of a number of components spaced close together. In such cases, it is desirable to separate the peak into its components so that the various oxidation states can be identified. Curve-fitting techniques can be used to synthesize a spectrum and to determine the number of components under a peak, their positions, and their relative intensities. Each component can be characterized by a number of parameters, including position, shape (Gaussian, Lorentzian, or a combination), height, and width. The various components can be summed up and the synthesized spectrum compared to the experimental spectrum to determine the quality of the fit. Obviously, the synthesized spectrum should closely reproduce the experimental spectrum. Mathematically, the quality of the fit will improve as the number of components in a peak is increased. Therefore, it is important to include in a curve fit only those components whose existence can be supported by additional information. 

Range 1 of the mud pump performance characteristic is defined by the performance of the smallest liner, and range 2 is defined by the remaining liners. The pressure loss in a circulating system, except for bit (p ), can be estimated from numerous theoretical formulas or from a flowrate test. Data obtained from a flowrate test can be approximated using a curve-fitting technique by the following function ... 

The method of Lew and Angus allows the advantage of nonlinear fitting techniques to yield competitive antagonist pKB values. 

FIGURE 11.12 Overextrapolation of data, (a) Nonlinear curve fitting techniques estimate an ordinate maximal asymptote that is nearly 100% beyond the last available data point, (b) The curve fitting procedure estimates a maximal asymptote much closer to the highest available data point. A useful rule is to reject fits that cause an estimated maximal asymptote that is >25% the value of the highest available data point. 

Nonlinear fitting techniques (for example, to Equation 12.3) are used to fit the data points to curves. The ICSo values form the fit curves are shown in Table 12.2b. 

The algebraic solution is the classical fitting technique, as exemplified by the linear regression (Chapter 2). The advantage lies in the clear formulation of the numerical algorithm to be used and in the uniqueness of the solution. If one is free to choose the calibration concentrations and the number of... 

The analysis of absorption data in humans has moved away from the more traditional modeling and data fitting techniques [35]. Absorption processes are now more often characterized by a mean absorption (or input) time (i.e., the average amount of time that the drug molecules spend at the absorption site) or by a process called deconvolution. The former analysis results in a single value (similar to absorption half-life) and the latter results in a profile of the absorption process as a function of time (e.g., absorption rate or cumulative amount absorbed vs. time). These approaches offer additional ways of interpreting the absorption process. 

The values of a and P in Eq. (125) are obtained by nonlinear curve-fitting techniques from which the uptake Pe and Ke values as a function of BSA concen-... 

Statistics available in the system include a large set of commonly used analysis techniques, as well as advanced nonlinear curve fitting techniques. Statistical results can be displayed numerically or graphically. 

Also global fitting techniques, where the space invariance (or any other invariance property) of one or more fitting parameters is exploited, have been successfully used to analyze fluorescence lifetime images [45, 46]. When applicable, global analysis techniques provide more homogeneous SNRs and reduce the number of fitted parameters. 

Using modern computer curve-fitting techniques,102 equations (17) and (27) can be fitted simultaneously with variables X, log CH+ and D, obtaining pATBH+. m, DB and DA as coefficients, which is a more precise technique than calculating log/ values first and using equation (17) and the error function61 discussed above. 

The probit relationship of Equation 2-4 transforms the sigmoid shape of the normal response versus dose curve into a straight line when plotted using a linear probit scale, as shown in Figure 2-10. Standard curve-fitting techniques are used to determine the best-fitting straight line. 

Muj taba, I. M. and S. Macchietto. Efficient Optimization of Batch Distillation with Chemical Reaction Using Polynomial Curve Fitting Techniques. Ind Eng Chem Res 36 2287-2295 (1997). 

The surface complexation models used are only qualitatively correct at the molecular level, even though good quantitative description of titration data and adsorption isotherms and surface charge can be obtained by curve fitting techniques. Titration and adsorption experiments are not sensitive to the detailed structure of the interfacial region (Sposito, 1984) but the equilibrium constants given reflect - in a mean field statistical sense - quantitatively the extent of interaction. 

Figure 11.8. Illustration of linearized digital curve fit technique.

A third problem with simulation of high resolution diffraction data is that there is no unique instrament function. In the analysis of powder diffraction data, the instalment function can be defined, giving a characteristic shape to all diffraction peaks. Deconvolution of these peaks is therefore possible and fitting techniques such as that of Rietveld can be used to fit overlapping diffraction peaks. No such procedure is possible in high resolution diffraction as the shape of the rocking curve profile depends dramatically on specimen thickness and perfection. Unless you know the answer first, you cannot know the peak shape. 

The parameter values on the right-hand side of the equation, are the P values, and were determined by a curve fitting technique (section 4.14). Introducing these values in Eq. (4.71) gives the solid line through the points in Fig. 4.18. 

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