although it's not in Matlab, but in Python - Numpy - Scipy, you'll find this explanation quite useful: 

http://www.scientificpython.net/pyblog/fitting-of-data-with-exponential-functions

and it's anyhow recommended to move (forward :)) to Python ecospace

Hi Alex,

Thank you for the link.  It is indeed useful.  Is it critical to use Legendre polynomials or would any other orthogonal polynomial basis be just as good?

Thanks, Charles

I do not know if other orthogonal polynomial basis would work. Legendre basis was shown to be useful (at least) in a recent PLoS one http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0090500

Alex:

Excellent reference.  This is perfect for my application. Thank you very much!

Charles

I have never had to do it myself, but the fitting of a sum of exponentials to noisy data is well known to be an ill conditioned problem.  Therefore it needs a good initial guess, and not too many unknown parameters to be fitted.  See for example:

W.J. Wiscombe, J.W. Evans

Exponential sum fitting of radiative transmission functions

J. Comput. Phys., 24 (1977), pp. 416–444

and many more recent papers.

If the data is noisy, then integrate and differentiate to smooth out. Begin with one-component model and obtain the best fit. A two component model will have time constants either side of the one component model. Like shampoo, lather, rinse repeat. Progressive complexity analysis is fine. If at some point the F test has a P>=0.05, then no further complexity can be supported by the data. I have applied this to exponential data with three components and it works well--if sufficient data points are present in each range of time constants. Bon chance!

James: I guess by "integrate and differentiate" you mean some kind of low-pass filtering, which can be implemented in many ways.  The Legendre polynomial technique that Alex pointed to is one of them.  The approach of incrementally increasing the complexity of the model until the noise level indicates you're done makes perfect sense and I will go that way. 

John: You're right too. Especially the ill conditioning gets worse if the time constants of the exponentials are close together.  And trying too many exponentials in the sum will undoubtedly lead to trouble, as any over-parameterization will.

Thanks to everyone for the good commentary.

A follow up: the Legendre polynomial technique for smoothing is susceptible to the Runge phenomenon.  Even without any noise in the samples, for uniform sampling intervals, the polynomial fit can "ring" at the edges of the sampling window.  A smoothing spline is a better alternative.

The space of sums of exponentials can be shown to be dense for distribution functions on (0, infinity). In other words, an arbitrary distribution (monotonic decreasing, bounded above and below) can be approximated arbitrarily well by a finite sum of exponentials - this can be achieved through the use of partial fractions on the Laplace transform. This methodology will therefore cover all linear transfer functions etc., and any problems that can be transformed into this (convolution-type) format. Inevitably there is noise in the data (possibly non-linear), and the truncation of the exponential series is an effective smoothing method. I don't have the reference at hand, but I have used this technique for a long time for obtaining time constants for rainfall-runoff hydrographs.

This sounds very interesting.  If you could recommend a reference, that would be great. Complicating factors are: the presence of noise, a signal of finite length, and the possibility of an unknown additive constant.  I'd love to see a discussion of how these are treated in the approach you describe.  I'm also interested in the density theorem you mention.  Thanks!

The classic algorithm for fitting sums of exponentials is Varpro.  Varpro is generally better conditioned than other optimization techniques because it eliminates the linear parameters.  There is a 2010 book titled "Exponential Data Fitting and its Applications". http://ebooks.benthamscience.com/book/9781608050482/  You can find more in a paper and also matlab code at https://www.cs.umd.edu/users/oleary/software/  by Diane O'Leary and Burt Rust.   This can probably do on the order of 1000's of exponentials.   A recent (2013) dissertation from Rice University titled "Numerically Stable and Statistically Efficient Algorithms for Large Scale Exponential Fitting" is also a good reference.  Matlab code can be found at https://github.com/jeffrey-hokanson/exponential_fitting_code/tree/master/Matlab.   good luck

I just noticed this discussion.

You can consider the so-called Padé-Laplave method (within which a series of other methods can be reformulated as particular, or sub-optimal cases, such as Prony, the method of moments etc). The basic reference is:

Analysis of multiexponential functions without a hypothesis as to the number of components  E. YERAMIAN* & P. CLAVERIE† Nature 326, 169 - 174 (12 March 1987)

regards

Can each exponential term be expanded as a series and the equations solved for as many terms as best fits the data ?

Also a0 shifts the curve up/down that can be done later after fitting data with exponential terms first.

@Eduard Is there a Python library which implements the  Padé-Laplave method?

Hi,

I some time ago haved the same problem. And presented a simple calculation method in paper from BHR 2012 conference titled:

"New approximation of unsteady friction weighting functions "

Usness of this method is verified in paper from 2016 titled:

"Comparing convolution-integral models with analytical pipe- flow

solutions"

Also in this last paper a small correction is presented to the earlier version of this method.

Hopes it helps You and the others, as for me it works very well