Fitting Data to Models

Because of the importance of model fitting in the field of scattering, some hints are presented and sources for further reading are given. 

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. 

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Search for the overall optimum within the available parameter space Factorial, simplex, regression, and brute-force techniques. The classical, the brute-force, and the factorial methods are applicable to the optimization of the experiment. The simplex and various regression methods can be used to optimize both the experiment and fit models to data. 

In the following, an example from Chapter 4 will be used to demonstrate the concept of statistical ruggedness, by applying the chosen fitting model to data purposely corrupted by the Monte Carlo technique. The data are normalized TLC peak heights from densitometer scans. (See Section 4.2) ... 

Although our purpose in introducing the subject of data treatment has been to provide insight into the design of experiments, the technique of least squares (regression analysis) is often used to fit models to data that have been acquired by observation rather than by experiment. In this chapter we discuss an example of regression analysis applied to observational data. 

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... 

Disadvantages arise mainly from the complexity of the statistical algorithms and the fact that fitting models to data is time consuming. The first-order (EO) method used in NONMEM also results in biased estimates of parameters, especially when the distribution of inter individual variability is specified incorrectly. The first-order conditional estimation (EOCE) procedure is more accurate but is even more time consuming. The objective function and adequacy of the model are based in part on the residuals, which for NONMEM are determined based on the predicted concentrations for the mean pharmacokinetic parameters rather than on the predicted concentrations for each individual. Therefore, the residuals are confounded by intraindividual, inter individual, and linearization errors. 

Estimated by fitting model to data of Ghanayem et al. (1990a). 

There are several software packages for modeling chemical and biochemical systems. They simulate the kinetics of systems of chemical and biochemical reactions providing tools to fit models to data, estimate parameters, perform simulations and optimizations among other options. In the general purpose software the task of the user is to develop a mathematical model and rewrite a model in the format expected by the software. 

Ordinary least squares. A technique for fitting models to data which minimizes the sum of the squares of the deviations between the observed values and those predicted from the model. For example, the arithmetic mean is the ordinary least-squares estimator from a set of data values, Xi,X2,---x since the function... 

For further information about the use of least-squares to fit models to data, we suggest the following books, to be studied in the order listed Devore, Draper and Smith, and Bates and Watts. The book Numerical Recipes has a convenient summary as a preface to its discussion of the computational methods used. Volume 210 of this series is devoted to numerical computer methods, most of which use least-squares. 

How can the retailer characterize customer segments based on their propensity to stockpile An empirical study by Iyer and Ye ([50]) takes data regarding retail sales of soup and provides results of a fitted model. Details of this model are provided in Section 4.9, but the key idea is that the characterization of customer segments, their size and propensity to stockpile, can come from a statistically fitted model to data. 

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