Model fitting

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

An alternative suggestion, based on a mathematical model fitted to experimental data, is that initiation occurs by thermolysis of a carbon—hydrogen bond ... 

Fig. 6. Pilot-scale kiln results for a fill fraction of 0.08% at 0.5 rpm and an initial toluene loading, on a dry, calcined, montmorillonite clay adsorbent, of 0.25 wt %, at A, 790°C B, 330°C and C, 190°C. The soHd lines are model fits using equation 24. The model simultaneously fits to all of the data (24).

Flow Models. Many flow models have been proposed (10,12), which are useful for the treatment of experimental data or for describing flow behavior (Table 1). However, it is likely that no given model fits the rheological behavior of a material over an extended shear rate range. Nevertheless, these models are useful for summarizing rheological data and are frequently encountered in the Hterature. 

Evaluating the model in tenns of how well the model fits the data, including the use of posterior predictive simulations to determine whether data predicted from the posterior distribution resemble the data that generated them and look physically reasonable. Overfitting the data will produce unrealistic posterior predictive distributions. 

Fig. 19-13. Three-parameter averaging-time model fitted through the arithmetic mean and the second highest 3-hr and 24-hr SOj concentrations measured in 1972 a few miles from a coal-burning power plant. Source From Larsen (21).

The model-dependent aspect of ellipsometric analysis makes it a difficult technique. Several different models fit to one set of data may produce equivalendy low MSEs. The user must integrate and evaluate all available information about the sample to develop a physically realistic model. Another problem in applying ellip-sometry is determining when the parameters of the model are mathematically correlated for example, a thicker film but lower index of refraaion might give the same MSE as some other combinations of index and thickness. That is, the answer is not always unique. 

The validity of least squares model fitting is dependent on four prineipal assumptions eoneerning the random error term , whieh is inherent in the use of least squares. The assumptions as illustrated by Baeon and Downie [6] are as follows ... 

When considering pressure drop models based only on water, hydrocarbons system capacity can be significantly overstated. For Nutter random ring packings the pressure drop/capacity models fit the data within +10% over the range of commercial interest, i.e., 0.1 to 1.0 in. water/ft of packing. Pressure drop alues for design operation should... 

Feff, P., Dougall, I. G., and Harper, D. (1993). Estimation of partial agonist affinity by interaction with a full agonist A direct operational model-fit approach. Br. J. Pharmacol. 110 239-244. 

Fig. 8. X-ray reflection diagram of a thin polystyrene film on float glass [160]. The reflectivity R is plotted against the glancing angle . The film is spin coated from solution. A model fit (dashed line) to the reflectivity data is also shown where the following parameters are obtained film thickness = 59.1 0.1 nm, interface roughness glass-polymer = 0.4 0.1 nm, surface roughness polymer-air = 0.6+1 nm, mean polymer density = 1.05 + 0.01 g/cm-3. The X-ray wavelength is 0.154nm...

The curve drawn illustrates how the model fits measured data. The first derivative of Equation 30.4 allows calculating the slope at any strain. The same model can be used to fit any relative torque harmonic, for instance the 3rd one, T(3/l). Note that in using Equation 30.4 to model harmonics variation with strain, one may express the deformation (or strain) y either in degree angle or in percent. Obviously all parameters remain the same except C, whose value depends on the unit for y. The following equality applies for the conversion C(y,deg) = x C(y,%), where a = 0.125 rad. 

Copolymer sequence analysis follows the same procedure. A computer program (HIXCO.TRIAD) was previously written for the two-state B/B model-fitting of triad sequence distributions and applied to (unfractionated) propylene-butylene copolymers and... 

Initiator 1 (fasti solvent A solvent B Temp.°C (benzene) (n-decane) solvent C (dodecane) model fit Yl = 0.6346... 

The final values of the rate constants along with their temperature dependencies were obtained with nonlinear regression analysis, which was applied to the differential equations. The model fits the experimental results well, having an explanation factor of 98%. Examples of the model fit are provided by Figures 8.3 and 8.4. An analogous treatment can be applied to other hemicelluloses. 

GP 1] [R 1] A kinetic model for the oxidation of ammonia was coupled to a hydro-dynamic description and analysis of heat evolution [98], Via regression analysis and adjustment to experimental data, reaction parameters were derived which allow a quantitative description of reaction rates and selectivity for all products trader equilibrium conditions. The predictions of the model fit experimentally derived data well. 

The two preceding applications showed that our hydrogenic model fits well with the helium atom and the dihydrogen molecule for the determination of the polarization functions except that their exponent ( is different from Co which is the exponent of the genuine basis set It is obvious that the hydrogenic model will fit less and... 

The form of the mathematical model fitted to the consensus factorial table must be reassessed after the hierarchical tree Is pruned and the experimental design has been revised by the statistician. 

Figure 14. I-V curves measured at 4.2 K (open squares) and at 295 K (solid squares). The solid curves denote fits of the KN model. Fitting parameters for these curves are Fc = 55mV, Ro=11x10 Q, 0 = 0.15c (offset charge) and a = E (cj h) = 0.5. The dashed curve (a = 0) represents the conventional model, which assumes a voltage-independent tunnel barrier. (Reprinted with permission from Ref. [29], 1997, American Institute of Physics.)...

It is usual to have the coefficient of determination, r, and the standard deviation or RMSE, reported for such QSPR models, where the latter two are essentially identical. The value indicates how well the model fits the data. Given an r value close to 1, most of the variahon in the original data is accounted for. However, even an of 1 provides no indication of the predictive properties of the model. Therefore, leave-one-out tests of the predictivity are often reported with a QSAR, where sequentially all but one descriptor are used to generate a model and the remaining one is predicted. The analogous statistical measures resulting from such leave-one-out cross-validation often are denoted as and SpR ss- Nevertheless, care must be taken even with respect to such predictivity measures, because they can be considerably misleading if clusters of similar compounds are in the dataset. 

Fig. 37.1. Quadratic Hansch model fitted to the bactericidal activities (log 1/C) of 10 doubly substituted phenols in Table 37.2 as a function of lipophilicity (log P) [20].

Figure 13. Electrostatic model fitted to partition coefficients for cations entering the M2-site in orthopyroxene, based on the experiments of McDade et al. (2003a,b). The curves are fits to Equation (7) and can be used to estimate and Do(m2) > from which D-p ui) can be calculated via the lattice strain model. The fit parameters are given in the legend.

Table 5. Electrostatic model fit parameters for garnet and estimates of DpJD ...

Figure 17. Electrostatic model fitted to partition coefficients for cations entering the M4-site in amphibole, based on the experiments of Brenan et al. (1995) and La Tourrette et al. (1995). A single M4-site is assnmed, rather than M4 and M4, as proposed by Botlazzi et al. (1999). The curves are fits to Eqnation (7) and can be nsed to estimate Do(M4), from which Z)pa(M4) can be calculated via the lattice strain model. Becanse of the mnltiplicity of sites in amphibole, it is unlikely that extrapolation of the curves to zero charge gives a reliable estimate for Dr . The fit parameters are Zo(M2> = 1-87 and ps = 38.1 A (La Tonrrette et al. 1995), and 2.31, 36.1 A (Brenan et al. 1995).

Figure 18. Electrostatic model fitted to partition coefficients for cations entering the large M-site in plagioclase, based on the experimental results of Bindeman and Davis (2000) and Blundy and Brooker (2003). The solid curve is a fit of the and Dll data of Blundy and Brooker (2003) to...

Parameter estimation is one of the steps involved in the formulation and validation of a mathematical model that describes a process of interest. Parameter estimation refers to the process of obtaining values of the parameters from the matching of the model-based calculated values to the set of measurements (data). This is the classic parameter estimation or model fitting problem and it should be distinguished from the identification problem. The latter involves the development of a model from input/output data only. This case arises when there is no a priori information about the form of the model i.e. it is a black box. 

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