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I performed an experiment where I measured diffusion in a lipid bilayer. I performed the measurement by fluorescently labeling some molecules in the bilayer, and the irradiating them with light until they become nonfluorescent (bleached).

This results in a dark spot in the field of view which gradually regains fluorescence as time progresses.

The data here represents the normalized fluorescence recovery. at t = 0 we have no recovery so the value is zero. As time progresses the fluorescence recovery increases until about 70% of the initial value.

I am trying to fit the following two functions to my data. However they are not converging. For fitequation1, I can get convergence if I specify constraints. For fit equation2 I cannon get convergence. Any suggestions?

normalized = {0., 0.136813, 0.260859, 0.426885, 0.469779, 0.505445, 0.543104, \
0.566412, 0.579988, 0.595328, 0.617525, 0.644393, 0.647385, 0.665809, \
0.670076, 0.673314, 0.675727, 0.665924}

fitEquation = a (1 - Exp[-b t]) + c (1 - Exp[-d t]);
fitEquation2 = y0 + a (1 - Exp[-k t]);

soln = FindFit[
normalized, {fitEquation, {a > 0 , b > 0, c > 0, d > 0, a <= 1, 
b <= 1, c <= 1 }}, {a, b, c, d}, t] 
soln2 = FindFit[
normalized, {fitEquation2 , {y0 + a < 1,  k <= 1}}, {y0, a, k}, t]
share|improve this question
    
It is not quite clear what are the sequence of times in your data. It should have the format {{time1, intensity1},{time2, intensity2},...}, otherwise Mma understands it as {{1, intensity1},{2, intensity2},...}. Did you have this latter case in mind? –  Alexei Boulbitch Apr 7 at 13:09

2 Answers 2

up vote 2 down vote accepted

You can try to improve it "by hand", as follows.

Assume that the model is like this: model = (a*(Exp[c*t] - 1))/(1 + b*Exp[c*t]) and assume that you have in mind that your data correspond to {{1, intensity1}, {2, intensity2},...} (see my comment above) and assume that norm is the name of your data.

Try the following:

    Clear[model, a, b, c];
model = (a*(Exp[c*t] - 1))/(1 + b*Exp[c*t]);
ff = FindFit[norm, model, {a, b, c}, t]

returning

(* {a -> 0.180745, b -> 0.276041, c -> 0.47336} *)

Then let us try this:

    Manipulate[

 Show[{
   ListPlot[norm, PlotStyle -> {Blue, PointSize[0.01]}, 
    PlotRange -> {{0, 20}, {0, 0.7}}],
   Plot[((a + da)*(Exp[(c + dc)*t] - 1))/(
     1 + (b + db)*Exp[(c + dc)*t]) /. ff, {t, 0, 20}, PlotStyle -> Red]

   }, Epilog -> 
   Inset[Mean[(# - (((a + da)*(Exp[(c + dc)*#] - 1))/(
            1 + (b + db)*Exp[(c + dc)*#]) /. ff)) & /@ norm /. 
      x_ -> x^2], Scaled[{0.8, 0.2}]]], {da, -0.03, 0.03}, {db, -0.05,
   0.05}, {dc, -0.1, 0.1}]

with this you may play with the values of parameters da, dband dc and watch the mean square error (in the corner of the screen). This: enter image description here should appear on the screen. This does not bring too much, however. The variation of the mean square error is in the third position after comma. As much as I remember, the experimental method you use is itself not too precise. If I am right, it is not that much the problem of a model as that of the method that matters.

share|improve this answer
ff = FindFit[normalized, {fitEquation2}, {y0, a, k}, t]
Show[ListLinePlot@normalized, Plot[fitEquation2 /. ff, {t, 0, 20}, Evaluated -> True]]

Mathematica graphics

share|improve this answer
    
can anything be done to tune the accuracy of the parameters for either equation? They don't look like they fit the data very well. –  olliepower Apr 7 at 6:06
    
While that improves the appearance of the fit, what is the physical significance of the sin term? Is it possible to improve just the fit parameters themselves? –  olliepower Apr 7 at 6:20

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