October 2003

Principles and Applications of Non-Linear Least Squares

Least squares is a popular and powerful technique for fitting equations to experimental data. When the technique is applied in the physical sciences, an equation to be fitted is often derived from the application of physical or chemical principles. As a consequence, estimates of useful physical/chemical parameters emerge through the process of fitting the equation to data.

When an equation to be fitted to data is linear in the parameters  (such as y = A + Bx or y = A + Bx +C/x) then linear least squares may be performed in order to find ‘best’ estimates of the parameters A, B and C.

Linear least squares can be carried out by most statistical packages, including the Analysis ToolPak provided in Excel. There are situations, encountered commonly in the physical and chemical sciences, in which an equation to be fitted to data is not linear in the parameters, such as y = Aexp(Bx) + C. In such a situation we must resort to using non-linear least squares. While many advanced statistical packages allow for the fitting of equations to data using non-linear least squares, Excel does not offer this as a standard option.

The downloadable pdf file, Nonlin2003v5.pdf, discusses issues associated with non-linear least squares from the perspective of physics and chemistry. The file shows how an add-in available as standard in Excel (the add-in is called ‘Solver’) may be used to perform nonlinear least squares. Included are examples, exercises and problems (with answers) drawn from a range of areas in physics and chemistry.

The material presented in the file assumes some basic exposure to data analysis at undergraduate level. The level of the discussion is consistent with that usually found at 2nd/3rd year undergraduate level, though graduate students might also find some of the material useful.

Les Kirkup,  Department of Applied Physics, Faculty of Science, University of Technology, Sydney, New South Wales 2007, Australia.

email: Les.Kirkup@uts.edu.au.

Downloadable PDF File

Version dated: October 2003