Unrealistic fitting confidence levels in noisy data

The fitting of simple exponential is giving a confidence level which is not realistic providing how noisy are the data. Any explanation please ?

Thank you. Code:

Data = xlsread('data_test.xlsx','Sheet1','A1:B3990'); 
Time = Data(:,1) - 2.6E-6 ;   %  time offset
Signal = Data(:,2)-0.078;     % amplitude offset

x = Time;
y = Signal;
y = y/(0.8*max(y));  % data normalization to about 1
format long   % just to get more precision digits
mdl = fittype(' a*(1-exp(-b*x)) ','indep','x')

mdl = 
     General model:
     mdl(a,b,x) = a*(1-exp(-b*x))

fittedmdl = fit(x,y,mdl,'StartPoint', [max(y) 1E5])

fittedmdl = 
     General model:
     fittedmdl(x) = a*(1-exp(-b*x))
     Coefficients (with 95% confidence bounds):
       a =      0.9589  (0.955, 0.9628)
       b =   5.557e+06  (5.373e+06, 5.74e+06)

figure
plot(x,y, 'bp', 'DisplayName','data')
hold on
plot(fittedmdl)
grid
xlabel('Time /s') 
ylabel('Intensity /a.u') 
ax = gca;
ax.FontSize = 15;

As you can see the error on (b) is very small (only 3% !) which is not realistic looking how noisy are the data:

coefficientValues = coeffvalues(fittedmdl);
a = coefficientValues(1);
b = coefficientValues(2);
Tau = (1/b)

Tau = 
     1.799653625749003e-07

ConfIntervals = confint(fittedmdl);
b_err = (ConfIntervals(2,2) - ConfIntervals(1,2))/2;
DeltaTau = Tau * (b_err/b)  

DeltaTau = 
     5.935716982509685e-09

Error = (DeltaTau / Tau)*100

Error = 
   3.298255229552455

Answer 1

[Bjorn Gustavsson]

This is most likely due to the large number of data-points you have. Compare with the simpler case of the uncertainty of the average of a number of random samples from a normal-distribution with a large standard deviation - it decreases roughly as the square-root of the number of samples. Here you have a similar situation. One thing you can do is to plot the distribution of the residuals, and investigare how it varies as you vary the parameters - and check how much it starts to skew off zero-centred and symmetric. Another thing you should look at are the residuals - to my naked eye there seems to be some systematic variation in the signal that aren't accounted for - some higher-frequency variations that do not look like random noise.

HTH