Fitting models to data

Abstract

In Chapters Bl, B2 and B4 some examples involving iteration were considered. This Chapter considers a third kind of iteration. Suppose we have a model which we think is appropriate and a method of solving the equations that constitute the model. Suppose further that we have a set of data and we want to test whether the model can, in fact, describe that data. The data might consist of a single response variable (or dependent variable) — such as the amount of adsorption. Or there might be more than one response variable — such as, the amount of adsorption, the charge on the surface, and the zeta potential. There might also be several explanatory (or independent) variables — such as the pH, the concentration of adsorbate, and the ionic strength of the medium. The problem is then to find appropriate values of the parameters. Examples of parameters are: the maximum adsorption possible for a given ion; the capacitance of the regions between the mean planes of adsorption; and the binding constants for the individual ions. The solution to this problem depends on the modellers’ definition of the word “appropriate”. He might argue that the parameters should be chosen from independent knowledge of the system — for example, the maximum adsorption from the geometry of the surface and of the adsorbing ion; the capacitance from the distances between the layers and from the dielectric properties; and the binding constants from knowledge of the bonds formed. An alternative argument is that the knowledge for this ideal solution is not adequate. Hence the appropriate values are those which best tune the model to the data. In this case the level of knowledge required is obviously less and the hypothesis being tested is whether the model is indeed capable of describing the data.