Calculating the error on a non-linear least squares fit

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Karl 2014-4-23

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https://ww2.mathworks.cn/matlabcentral/answers/126855-calculating-the-error-on-a-non-linear-least-squares-fit

Hi, so I'm currently running a program which fits some noisy data to a monoexponential function using a non-linear least squares fit. I know that the equation can be linearized by taking a logarithm, but since the data is noisy and the curve decays to zero I've found that a non-linear least squares fits the best (to avoid taking the logarithm of very small and/or negative values). Here is my fitting code:

        s = fitoptions('Method','NonlinearLeastSquares',...
            'Lower',[0.00001,1E-3],...
            'Upper',[100,1],...
            'Startpoint',[0.8 1/40]);
        f = fittype('a*exp(-x*b)','options',s);
        fitresults = fit(t',S',f);
        ci = confint(fitresults,0.6827);
        R2Values(counterTE,counterTR) =  fitresults.b;
        R2Error(counterTE,counterTR) = (ci(2,2)-ci(1,2))/2;

I believe this is right and I get reasonable values for both R2 (relaxation rate) and the error on it. But what I am curious is how exactly matlab calculates the error on the fit parameters. This is quite important for my particular application because I'm interested in the noise properties. If possible I was wondering if anyone knew the equation matlab used to calculate the error (intuitively I feel like it must use the residual), so that I could actually calculate the error myself. Thanks very much in advance!