Fitting Exponential Equations to Data - John Dunn, Consultant, Ambertec, P.E., P.C.

Most physical processes in nature have exponential properties. Radiation half-life is one example, thermal transitions from hot to cold or cold to hot are another example.

Often, a set of data from some process can be a little messy, a little noisy. However, if the data points look exponential, the underlying process is probably exponential and its fairly easy to find an equation for that exponential behavior:

We can go from this:  

to something that looks like this:

 

Assuming an equation of the form Y = A1 + A2 * exp ( - x / A3 ), we have three unkknowns, the A1, A2 and A3 for which we take three x-and-y pairs from our data and solve three equations in three unknowns to obtain the A1, A2 and A3 coefficients.

The resulting equation may not be the mathematical best fit, but it can be pretty darn close. The following algorithm serves well: 

 
It may be that there's someone smart enough to find the value of A3 by some one-step, explicit method, but I couldn't do it. Therefore, the value of A3 is found here using an iterative loop. Then with A3 in hand, the values of A2 and A1 are one-step finds.

I wrote the code for this in GWBASIC many years ago (Please continue reading after you stop laughing.) and just recently, I put it into an Excel spreadsheet. (Ah, these modern times.)

If anybody would like a copy of either, just get in touch with me and I'll send it to you.

 

Comments

[Fred Katz]

John;

After you have a rough solution, list all of the points vertically on a spreadsheet and use Excel to find the standard deviation between the equation fit and the actual points. Write the equation with each of the exponents as individual cells. Tweak each cell value trying to minimize the sum of the standard deviations. This will give you a tighter fit.

Fred Katz

[Ray Keefe]

Hi John,

nice method. A pity about the iteration but that is often the way of these things.

I assume this method relies on the data being at uniform x axis intervals?

And I'm always interested in useful spreadsheet examples so I'd like to take you up on you offer and get a copy.

Thanks for a useful post

Ray Keefe
Successful Endeavours Pty Ltd
Casey Business of the Year 2010
Award Winning Electronics Design and Embedded Software Development

[John D.]

Hi, Ray.

The intervals for "x" don't bneed to be evenly spaced. The x-values can be chosen arbitrarily.

I will e-mail the spreadsheet to you in just a little while.

john

[Jim Anderson]

Hi Don,

I would appreciate a copy of the spreadsheet also.