Setting good lower and upper bounds on and exponential fit.

Hi all, I am trying to get an optimal fit for some data. It seems it fits a double exponential fit quite well, however......the confidence intervals seem too big. How can I set appropriate upper and lower bounds? I have seen it said many times while searching on websites that the bounds should be set "according to your data." What does this mean? How can I actually decide what good bounds are? Here is my data:

x are time points, and y is the data

x =

           0
           1
           3
           5
           7
          10
          20
          30
          40
          50
          60
         180
         300
        1080
        1440
y =

   14.7311
   16.4375
   16.9743
   17.6971
   17.8000
   17.9643
   18.1380
   18.4612
   18.4525
   19.1260
   19.0100
   19.9080
   20.0220
   21.5480
   21.1938

the fit I used was:

s2 = fitoptions('Method','NonlinearLeastSquares','StartPoint',[3 1 15 2 0.1])

pmexp2 = fittype('a*(1-exp(-x/b))+c+d*(1-exp(-x/e))','options',s2);

figure(2) [fo1 go1] = fit(x,y,pmexp2) hold on plot(x,y, 'o')

A professor gave me this equation and the startpoints(I dont totally understand these). Like I said the fit does look pretty good.

Result of fit:

fo1 =

     General model:
     fo1(x) = a*(1-exp(-x/b))+c+d*(1-exp(-x/e))
     Coefficients (with 95% confidence bounds):
       a =       3.085  (2.395, 3.775)
       b =       2.362  (1.029, 3.696)
       c =       14.93  (14.32, 15.55)
       d =       3.308  (2.727, 3.888)
       e =       231.7  (107.5, 355.9)

go1 =

           sse: 0.8727
       rsquare: 0.9801
           dfe: 10
    adjrsquare: 0.9721
          rmse: 0.2954

the e term is what I am most concerned about. That confidence interval seems very large? Any help much appreciated. Thanks!

Answer 1

[Walter Roberson]

You have exp(-x/e) with x as a maximum of 1440. exp(-1440/231.7) is only on the order of 0.002, not a very big contribution. exp(-1440/107.5) has several more leading zeros; exp(-1440/355.9) is about 0.02 which is still not a big contribution compared to other parts.

[Harold Bell]

So how do I decide how to set the bounds?