Data Fitting

Abstract

A universal problem of the experimental, physical sciences consists of asking and answering two questions. The first question is, “Given a set of experimental data, y _i , and a theoretical model that establishes a connection between the data and a set of parameters, x _j , what are the values of the parameters that give the best fit to the data?” The second question is, “Having found the best fit, what can we say about the adequacy of the model in describing the data, and within what ranges do the true values of the parameters lie?” In order to establish a practical procedure for answering these questions, we must first find the answers to several auxiliary questions. The first, and most important, of these is, “What do we mean by the best fit?” We shall assume, in the following discussion, that the best fit corresponds to a minimum value of some function, S(y,x), of all data points and all parameters. In this chapter we shall begin with a discussion of the form of the function S in somewhat greater detail than usually appears in treatments of model fitting, in order to highlight some of the assumptions that are made implicitly when a particular procedure is used. We shall then discuss various approaches to the numerical analysis problem of finding the minimum of this function. In subsequent chapters we shall discuss the connected problems of assessing the precision of the results and constructing and comparing models that obey the laws of physics and chemistry.