Fitting an exponential mixture model with interval constraints on the mixture weights

Question

What methods are there to fit a model of the form $y=A\mathrm e^{Bx}+C\mathrm e^{Dx}+E$?

Here is the actual scientific data to be fitted: http://dl.dropbox.com/u/39499990/Ben%2C%20real%20data.xlsx

B should be in the range of -1 to -100.

D should be in the range of -100 to -500.

E is a constant.

This specific model is of interest as it is the acceptable one in the scientific community for describing the biological proccess in hand- Inactivation of a voltage dependent calcium channel.

Best would be a solution that I could implement in Excel, but anything else is fine too (preferably one that could be rutinely imported into Excel).

Answer 1

Best is not a solution in Excel -- spreadsheets are not a good environment for data analysis: http://www.burns-stat.com/pages/Tutor/spreadsheet_addiction.html

The 'nls' function in R would be one choice.

Answer 2

I'll show you how to analyze your data with Mathematica. First I'll use your model as requested.

data = Import["Desktop/data.csv"];
y = NonlinearModelFit[data, 
  a Exp[b x] + c Exp[d x] + e, {a, b, c, d, e}, x];

Mathematica returns this error:

NonlinearModelFit::cvmit: Failed to converge to the requested accuracy or precision within 1000 iterations

This means that your model might not be ideal. Let's consider the data to see if we can come up with a better one. The log-plot of the negative of your series (so we can take the logarithm) shows a fairly polynomial curve:

(Generated using ListLogPlot[# {1, -1} & /@ data])

This suggests that instead of an exponential mixture, we should use a log-linear model:

$ \hat y = -\exp(a+bx+cx^2 + \dots)$

Let's try a cubic polynomial:

nlm = LinearModelFit[{#[[1]], Log[-#[[2]]]} & /@ data, {x, x^2, x^3}, x]

Mathematica returns 71.6838 - 391.293 x + 764.791 x^2 - 501.198 x^3

In other words,

$ \hat y = -\exp(71.6838 - 391.293 x + 764.791 x^2 - 501.198 x^3)$.

Here is a plot of the data, the mixture model (red), and the log-linear model (green):

(Generated using Show[{ListPlot@data, Plot[y[x], {x, 0.38, 0.57}, PlotStyle -> {Thick, Red}], Plot[-Exp@nlm@x, {x, .35, 0.6}, PlotStyle -> {Thick, Green}]}])

You can get an ever better fit with a quartic polynomial, but you get the idea. The residuals show structure, which means that the model has not squeezed all the information out of the data:

(Generated using ListPlot@nlm["FitResiduals"])

Fortunately, the order virtually disappears by the time you raise the order to six:

Answer 3

Are you looking for methods or software to implement it? It looks like a typical nonlinear regression problem. In SAS this is implemented using proc nlin.