Best-fit model

Figure Bl.26.17. (a) Observed and calculated ellipsometric [A(1), T(>l)] spectra for the Y2O3 film on vitreous silica. Angle of incidence 75°. (b) Best-fit model of the Y2O3 film on vitreous silica (Chindaudom P and Vedam K 1994 Physics of Thin Films vol 19, ed K Vedam (New York Academic) p 191).

Models are fitted to scattering data by means of nonlinear regression [270] and related computer programs [154,271], The quality of the parameterization (by structural parameters) and of the fit are estimated. The best fitting model is accepted. The found values for the structural parameters are plotted vs. environmental parameters of the experiment and discussed. Environmental parameters that come into question are, for example, the materials composition, its temperature, elongation, or the elapsed time. 

Another interesting comparison is between the coefficients for the functions representing the best-fitting models for the data and the coefficients for the functions that result from performing the linearity test as described in the previous chapter [4], We have not looked at these before since they are not directly involved in the linearity test. Now, however, we consider them for their pedagogic interest. These coefficients, for the case of testing a quadratic nonlinearity of the data from Figure 67-1, are listed in Table 67-2. 

Fig. 3. Beryllium spectra for the two NGC6397 stars observed by Pasquini et al. 2004. The spectrum of a comparison field star is also added. Best fit models (red dots) are overimposed.

Fig. 1. Left panel Metallicity distribution for our 487 red giants in Carina. The dashed line is the best-fit model prediction from [3]. The right plot shows spatially separated MDFs in three different regions of Carina, measured along the major and minor axes.

An especially important aspect of our protocol is the collection and analysis of data over different time spans. 18 This is done using the same number of channels to collect about the same number of photons, but with different channel delays. Typically, data on four time spans, 0 to 16-20 ns, 0 to 35 10 ns, 0 to 65-80 ns, and 0 to 120-130 ns, are collected and analyzed. In our experience, the. constancy of the best-fit model parameters with variation in time span is the most sensitive and reliable test for the shape of the model function. 

Figure 4. Deconvolution of the denaturational endotherm for native CBHI at pH 4.80. Circles represent experimental values for differential heat capacity the solid curves represent the overall best fit model and the two sequential component transitions that contribute to the overall fit (See text).

Figure 7.15. (A) Crystal truncation rods along the (00L), (10L), (11L), and (20L) zones, showing results of modeling [38] (B) Best-fit model for the a-Al203 (0001)/water interface [38] (C) CTR diffraction data for the UHV clean a-Al203 (0001) surface [168] (D) Best-fit model of the UHV-clean a-Al203 (0001) surface showing that the surface is terminated by A1 atoms [168].

A convenient way to illustrate the behavior of the model for the example of ZnS deposition is to plot the measured deposition rate, r(d, ZnS), as a function of the incident-flux rate of one element when the incident-flux rate of the second element is fixed. An example is shown in Figure 13 for the deposition of ZnS as a function of the incident-flux rate of sulfur at a substrate temperature of 200 °C. Experimental data points and curves representing the best-fit model predictions are shown for each of four zinc incident-flux rates. A nonlinear least-square procedure was used to obtain the following values for the model parameters that best fit equation 40 to the experimental data 8(Zn) = 0.6-0.7, 8(S) = 0.5-0.7, and K(ZnS) > 1015 cm2-s/ZnS. 

Table 7.5 Best fitting models feature composition and FI values.

Excel provides some built-in tools for fitting models to data sets. By far the most common routine method for experimental data analysis is linear regression, from which the best-fit model is obtained by minimizing the least-squares error between the y-test data and an array of predicted y data calculated according to a linear... 

Fig. 3.8. (a) Experimental (dotted lines) and best-fit model (solid lines) IRSE spectra of a (0001) ZnO bulk sample. The ZnO phonon modes, as obtained by IRSE, are marked by vertical arrows, (b) Experimental (dotted lines) and best-fit model (solid lines) IRSE spectra ( h only) of a (0001) ZnO thin film on (0001) sapphire (upper panel, thickness d 1970 nm) and of a bare (0001) sapphire substrate (lower panel). The ZnO phonon modes, as obtained by IRSE, are marked by arrows. The IR-active modes of sapphire are indicated by solid (TO) and dotted (LO) vertical markers. Reprinted with permission from [38]... 

The refinement processes were carried out with several variations such as baseline correction, cell parameters, zero correction, scale factor, temperature factor, etc. Li1.1V0.9O2 is assigned to the hexagonal system with an R-3m space group with cell parameter of fl=2.853 A, c=14.698A, and cell volume of V=103.62 A3. The best-fitted model is shown in Figure 3.2(a). 

By constructing the design matrix and then using the pseudo-inverse, calculate die coefficients for die best fit model given by the equation... 

In regression, the equation A B.C is an approximation for example, A may represent a series of spectra diat are approximately equal to the product of two matrices such as scores and loadings matrices, hence this approach is important to obtain the best fit model for C knowing A and B or for B knowing A and C. 

All models, however, have the same deficiency, that is, the concentrations in the solid and solution can be measured experimentally however, the surface activity coefficients, the surface electric work, as well as the energy distribution function can only be estimated. It means that the models are adapted to the experimental data and the best-fitted model is used, and therefore the selected model has no thermodynamically significant meaning (Cemik et al. 1995). In this chapter, these problems will be illustrated and discussed. 

As clearly illustrated in Figures 12.9 and 12.11, good descriptions of the effluent results from BC-I and BC-II were achieved when the first-order kinetic model was implemented. Furthermore, increased sorption was realized for the higher input concentration (C0 = 0.005 M) of the BC-I column than for BC-II. This finding is based on the ratio of the parameters (k, k2) that provided the best-fit model (see Table 12.3). For BC-I, the value of (k, k2) was 2.76 compared to a value of 0.98 for BC-II. Such trends are consistent with the upper Bs layer and indicate sorption dependency on the dominant concentration within the soil column as influenced by the S04 input (C0). 

Fig. 8 Illustration of the sapphire (0001) surface truncated bulk and best fit model obtained from crystal truncation rods measurements. The two top planes are strongly shifted with respect to their bulk positions in a nearly bond-length conservative displacement (from Ref. 124).

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