Additional Information and History

Fitting History

Prior to the invention of electronic calculation, only manual methods were available, of course - meaning that creating mathematical models from experimental data was done by hand. Even Napier's invention of logarithms did not help much in reducing the tediousness of this task. Linear regression techniques worked, but how to then compare models? And so the F-statistic was created for the purpose of model selection, since graphing models and their confidence intervals was practically out of the question. Forward and backward regression techniques used linear methods, requiring less calculation than non-linear methods, but limited the possible mathematical models to linear combinations of functions.

With the advent of computerized calculations, non-linear methods which were impractical in the past could be automated and made practical. However, the non-linear fitting methods all required starting points for their solvers - meaning in practice you had to have a good idea of the final equation parameters to begin with!

If however a genetic or monte carlo algorithm searched error space for initial parameters prior to running the non-linear solvers, this problem could be strongly mitigated. This meant that instead of hit-or-miss forward and backward regression, large numbers of known linear *and* non-linear equations could be fitted to an experimental data set, and then ranked by a fit statistic such as AIC or SSQ errors.

Note that for an initial guesstimate of parameter values, not all data need be used. A reduced size data set with min, max, and (hopefully) evenly spaced additional data points in between are used. The total number of data points required is the number of equation parameters plus a few extra points.

Reducing the data set size used by the code's genetic algorithm greatly reduces total processing time. I tested many different methods before choosing the one in the code, a genetic algorithm named "Differential Evolution".

I hope you find this code useful, and to that end I have sprinkled explanatory comments throughout the code. If you have any questions, comments or suggestions, please e-mail me directly at zunzun@zunzun.com or by posting to the user group at the URL https://groups.google.com/g/findcurves

I will be glad to help you.

Author History

This is James Phillips, author of zunzunsite3. My background is in nuclear engineering and industrial radiation physics, as I started working in the U.S. Navy as a submarine nuclear reactor operator many, many neutrons ago.

I have quite a bit of international experience calibrating industrial metal thickness and coating gauges. For example the thicker a piece of steel the more radiation it absorbs, and measuring the amount of radiation that passes through a sheet of steel can tell you how thick it is without touching it. Another example is that the thicker a zinc coating on steel sheets, the more zinc X-ray fluorescence energy it can emit - again allowing accurate thickness measurement for industrial manufacture.

My post-Navy employer originally used ad-hoc spreadsheets to very tediously create 4th-order polynomials calibrating to readings from known samples. So I started writing my own curve-fitting software in C.

When X-rays pass through aluminium, the atomic number of the alloying elements is much greater than that of the aluminium itself such that small changes in alloy composition lead to large changes in X-ray transmission for the same thickness. Alloy changes look like thickness changes, egad! However, alloy changes also cause changes to the X-rays that are scattered back from the aluminium, so that if both the transmitted and backscattered radiation is measured a more alloy-insensitive thickness measurement can be made - but this is now a 3D surface fit, and I started writing surface fitting software. I began to do considerable international work.

This finally led to the development of my Python fitting libraries, and this code. I also have wxPython and tkinter desktop versions on Bitbucket.

James