Assessing Regression Quality

The chapters presented in Part El describe development of mathematical models that can be used for interpretation of impedance measurements. These models may be regressed to data using the approaches presented in Chapter 19. A systematic approach is presented in this chapter to determine whether the model provides a statistically adequate description of the data. 

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For the purpose of demonstrating the means of assessing regression quality, three models were applied for the aiialysis of the impedance data ... 

The coefficients in Eq. [1] are determined with the aid of regression analysis by solving a set of n simultaneous equations. In addition to providing coefficient values, aj, these calculations provide statistical parameters that assess the quality of the equation derived. 

The statistical measure of the quality of the regression is used to determine whether the model provides a meaningful representation of the data. The parameter estimates are reliable only if the model provides a statistically adequate representation of the data. The evaluation of the quality of the regression requires an independent assessment of the stochastic errors in the data, information that may not be available. In such cases, visual inspection of the fitting results may be useful. Issues associated with assessment of regression quality are discussed further in Section 19.7.2 and Chapter 20. 

The use of equation (19.35) to assess the quality of a regression is valid only for an accurate estimate of the variance of the stochastic errors in the impedance data. 

Both quantitative and qualitative approaches may be used to assess the quality of a regression. 

Remember 20.1 If the experimental error structure is not known, the numerical value of the statistic cannot be used to assess the quality of a regression. 

In this age of powerful computers, it is no longer even necessary to find analytical solution to differential equations. There are many software packages available that cany out numerical integration of differential equations followed by non-linear regression to fit the model and assess its quality by comparing with experimental data. In this study we have used a numerical integration approach to compare kinetic properties of Photinus pyralis and Luciola mingrelica firefly luciferases. 

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

The main goal of this chapter is to present the theoretical background of some basic chemometric methods as a tool for the assessment of surface water quality described by numerous chemical and physicochemical parameters. As a case study, long-term monitoring results from the watershed of the Struma River, Bulgaria, are used to illustrate the options offered by multivariate statistical methods such as CA, principal components analysis, principal components regression (models of source apportionment), and Kohonen s SOMs. 

The quality of a multiple linear regression is usually assessed using the squared correlation coefficient, written r2 [44]. r2 is calculated using the following formula ... 

The quality of a regression can also be assessed by visual inspection of plots. Of course, some plots are more sensitive than others to the level of agreement between model and experiment. As will be demonstrated in this chapter, the plot types can be categorized as given in Table 20.1. The comparison of plot types is presented in the subsequent sections for regression of models to a specific impedance data set. 

As described in Sections 20.2.1 and 20.2.2, the quality of the regressions can be assessed to varying degrees of success by inspection of plots. The Nyquist or complex-impedance-plane representation given in Figure 20.13 reveals the difference between the finite-diffusion-length model and the models based on numerical solution of the convective-diffusion equation, but cannot be used to distinguish the models based on one-term, two-term, and three-term expansions. 

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