# Confidence intervals on plots with correlated error bars?

I apologize in advance for any imprecise language I might use here, as I don't have much formal training in statistics.

I have a plot of intensity versus wavelength (an intensity profile) with associated error bars on the intensity at every wavelength; these errors are assumed to be independent for now. I then applied Gaussian smoothing to the intensity profile (the process is similar to the one described here: http://imaging.mrc-cbu.cam.ac.uk/imaging/PrinciplesSmoothing), and I know how to calculate the "Gaussian PDF weighted" errors for the smoothed intensity profile. However, the overlap between the Gaussian smoothing windows means that the error in the smoothed intensity at a given wavelength is no longer independent of the errors in the intensities at nearby wavelengths.

Let $I_k$ and $\sigma_k$ denote the smoothed intensity at wavelength $k$ and its associated error respectively; because of this correlation with the nearby errors, I am guessing this means that it is inappropriate to say that the 68% confidence interval is still $I_k \pm \sigma_k$. If so, how can I compute an "error" in $I_k$ which I can use to quote confidence intervals?

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The formula is okay if the I$_k$ are still approximately Gaussian but what changes is the calculation for σ$_k$. You would need to estimate to covariances due to the overlapping values that are smoothed. That will depend on the weight function and band width of your smoother.

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Ok, I was thinking this would involve covariances as well. I've been looking around, but haven't found any online references that give me the expressions I need to compute covariances and how to combine them into a a single $\sigma$ which I can use to quote confidence intervals; do you happen to know of any texts or documents that do? –  Jonas Jul 27 '12 at 20:29