Confidence intervals from fit

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Karl 2014-8-14

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https://ww2.mathworks.cn/matlabcentral/answers/150882-confidence-intervals-from-fit

评论： Karl 2014-8-14

Hi, I'm currently trying to fit my data to the model y = a*(1-exp(x/b)), and extract the two constants a and b. I'm currently using simulated data where I set a = 1 and generate some points y and add gaussian white noise to them. If I then put these data points into the curve fitting toolbox and fit to y = 1-exp(x/b) I get a reasonably good fit I get b = 1.328 (I put in 1.1) with a 95% confidence interval of (1.035,1.621) and an R^2 of 0.9066. If I allow a to vary with the same data I get a = 0.9821, with a 95% CI of (0.8541,1.108), and b = 1.261 with a CI of (0.7102,1.811) and an R^2 of 0.91, so the fit got a little better but it's pretty close, since a only changed by about 2%.

What you'll notice now though is the CI for b has about doubled, even though a is about the same. I remember reading a long time ago about how the CIs were correlated and it could give artificially large confidence intervals when fitting with multiple parameters, but I don't know where I read that (or if that's even true!). Does anyone have any information on where this large discrepancy comes from, and how I could fix it? The reason why I ask is because with my actual data I will not "know" a, as i did with my generated data, so I will have to fit it into my model. An easy solution would be to fit it, extract a and then simple set a in the fit instead of allowing it to vary, but I worry this isn't 100% justified.

In short I believe that I am getting larger CIs when fitting to multiple parameters than I would expect and was wondering if there is a way to correct for this