MATLAB: Unexpected results from ‘exp2’ least squares fit

fit exp2 exponential curve fitting

For most of my data this code works fine, but I am having a very poor fit for the following section of data. Does anyone know what I did wrong?
 x = [1.7125    5.0430    9.1750   15.4250   22.7550   30.3800   40.4500   52.4150   63.9400   74.5300   89.5500  105.3000  119.8000 135.9000  153.4500  167.5000  185.4000  205.9000  223.9000  244.5000  267.7000  290.8500  317.6000  341.9500  367.9000  399.1000  432.7500  461.8000  493.4500  521.5000  555.6000  588.4500  617.5500  654.2000  688.9000  729.6000  772.2000  820.5000  862.5000]';
y = [3.7148    3.3178    2.7135    2.4753    2.2017    1.9689    1.6732    1.5649    0.6837    0.4081    0.3183    0.2845   -0.3022  -0.1981   -1.1470   -1.7174   -1.6783   -1.2277   -2.3289   -2.5576   -2.4626   -2.5281   -3.4626   -2.8697   -4.7081   -3.6971   -3.7245   -4.7752   -4.7768   -4.6965   -5.3429   -6.0585   -5.4439   -6.2222   -6.2646   -6.1612   -6.4458   -6.8387   -6.9563]';
scatter(x,y,5,[0 0 0],'fill');
hold on
f = fit(x,y,'exp2');
x2=linspace(min(x),max(x));
y2=f(x2);
plot(x2,y2,'color',[0 0 0],'linewidth',2);

Best Answer

Sums of exponential fits are difficult to solve. They are HIGHLY susceptible to starting value choices.
1. When you start with the wrong starting values, then expect crap.
2. Fit by default starts from some very arbitrary starting values. So what should you expect, at least some of the time? (See 1.)
A better scheme to estimate that nonlinear model is to use my fminspleas . It is on the file exchange. Follow the link.
mdlterms = {@(coef,X) exp(-coef(1)*X), @(coef,X) exp(-coef(2)*X)};
format long g
[nlcoef,lincoef] = fminspleas(mdlterms,[0.001,0.002],x,y)
nlcoef =
     -0.000869991132855897       0.00532565665197467
lincoef =
          -3.4116826458322
          6.61061671877021
Fminspleas survives here because it uses a completely different scheme to estimate the linear versus nonlinear coefficients in the model. It reduces the nonlinear search space.
yhat = lincoef(1)*exp(-nlcoef(1)*x) + lincoef(2)*exp(-nlcoef(2)*x);
plot(x,y,'ro',x,yhat,'b-')
To be honest, I really don't like the choice of model here at all. What did I say above? They are difficult to fit. But worse, look at the coefficients!!!!!!!!! Look at them carefully!
One of the nonlinear rate coefficients is positive, the other negative. So if we tried to extrapolate this model, see what happens.
xextrap = linspace(-150,3000,500);
yextrap = lincoef(1)*exp(-nlcoef(1)*xextrap) + lincoef(2)*exp(-nlcoef(2)*xextrap);
plot(x,y,'ro',xextrap,yextrap,'b-')
So, the question is, do rate coefficients of completely different sign make ANY sense in context of why you chose that model? Be honest. Why did you choose that model in the first place? If your only reason was because you thought it might have the right shape, that is a bad reason.
An alternative model is just to use a spline fit. Using my SLM toolbox (also on the file exchange)
slm = slmengine(x,y,'decreasing','on','concaveup','on','plot','on');
The fit looks at least as good a fit as the sum of exponentials.
Related Question