7 - Fitting functions to data

Summary

If you want to fit parameters in a functional relation to experimental data, the best method is a least-squares analysis: Find the parameters that minimize the sum of squared deviations of the measured values from the values predicted by your function. In this chapter both linear and nonlinear least-squares fits are considered. It is explained how you can test the validity or effectiveness of the fit and how you can determine the expected inaccuracies in the optimal values of the parameters.

Introduction

Consider the following task: you wish to devise a function y = f(x) such that this function fits as accurately as possible to a number of data points (x_i, y_i), i = 1, …, n. Usually you have – based on theoretical considerations – a set of functions to choose from, and those functions may still contain one or more yet undetermined parameters. In order to select the “best” function and parameters you must use some kind of measure for the deviation of the data points from the function. If this deviation measure is a single value, you can then select the function that minimizes this deviation.

This task is not at all straightforward and you may be lured into pitfalls during the process. For example, your choice of functions and parameters may be so large and your set of data may be so small that you can choose a function that exactly fits your data.