Model-based analyses

Very rarely are measurements themselves of much use or of great interest. The statement the absorption of the solution increased from 0.6 to 0.9 in ten minutes , is of much less use than the statement, the reaction has a half-life of 900 sec . The goal of model-based analysis methods presented in this chapter is to facilitate the above translation from original data to useful chemical information. The result of a model-based analysis is a set of values for the parameters that quantitatively describe the measurement, ideally within the limits of experimental noise. The most important prerequisite is the model, the physical-chemical, or other, description of the process under investigation. An example helps clarify the statement. The measurement is a series of absorption spectra of a reaction solution the spectra are recorded as a function of time. The model is a second order reaction A+B- C. The parameter of interest is the rate constant of the reaction. 

The purpose of this chapter is to develop a collection of methods that allow the determination of the best set of parameters for a particular given model and one or a collection of measurements. In other words we fit the parameter(s) to the measurement(s). 

The tools we created in Chapter 3, Physical/Chemical Models, form the core of the fitting algorithms of this chapter. The model defines a mathematical function, either explicitly (e.g. first order kinetics) or implicitly (e.g. complex equilibria), which in turn is quantitatively described by one or several parameters. In many instances the function is based on such a physical model, e.g. the law of mass action. In other instances an empirical function is chosen because it is convenient (e.g. polynomials of any degree) or because it is a reasonable approximation (e.g. Gaussian functions and their linear combinations are used to represent spectral peaks). 

A crucial point, not mentioned so far, is the question about the meaning of the expression best parameters. Intuitively it seems to be clear they are the parameters for which the calculated data match the measured data as closely as possible. Almost invariably the sum of the squares of the differences between the measured data and the calculated model function is minimised and is the measure for the quality of the fit. 

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Chapter 4, Model-Based Analyses, is essentially an introduction into least-squares fitting. It is crucial to clearly distinguish between linear and nonlinear least-squares fitting linear problems have explicit solutions while non-linear problems need to be solved iteratively. Linear regression forms the base for just about everything and thus requires particular consideration. 

The algorithms developed in this chapter can model any situation, e.g. they can serve to demonstrate the effects of initial concentrations and rate constants in kinetics and of total concentration and equilibrium constants in equilibrium situations. Very importantly, these algorithms further form the core of non-linear least-squares fitting programs for the determination of rate or equilibrium constants, introduced and developed in Chapter 3, Model-Based Analyses. 

Generally, the multivariate data analysis attempts to find the best matrices C and A for a given measured Y. We discuss a wide range of methods for this task, in depth, in the two Chapters 4 and 5, Model-Based Analyses and Model-Free Analyses. 

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