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For 1-D signals (spectra) or 2-D signals (images), is there a way to impose the constraint that the data within a group is uncorrelated? I am iteratively applying background correction model fitted to regions that I identify as background (a "segmentation problem") through a mixture model.

The assumption is that background regions should contain exclusively random noise (Gaussian distributed) after appropriate correction, so after many iterations I should be left with a sum of two probability distributions (in signal intensity) - one related to "noise" (yn) in the background and the true signal (ys) which is approximately exponential. However, part the the true signal contribution can "look" Gaussian when computing the probability density without taking sample independence/correlation into account - so it gets lumped in with the background and the separation is not so clean.

For this illustration, I break down the true signal as a sum of two components (ys1 and ys2), but in reality the actual signal comprises countless contributions that appears approximately exponential.

yn <- rnorm(N, 0, .05)   # random 
ys1 <- qnorm(p, 0, 1/5)  # signal
ys2 <- qexp(p, 5)        # signal

To get the desired effect for my illustration, I'll choose x this way:

x <- asin(ys1)

The individual signals (left column) and probability densities (right column) are shown below:

par(mfrow=c(3, 2))
plot(x, yn, type="o")
plot(density(yn, adjust=2))
plot(x, ys1, type="o")
plot(density(ys1, adjust=2))
plot(x, ys2, type="o")
plot(density(ys2, adjust=1))

individual contributions

I want to separate yn from the desired portion of my signal ys1+ys2. However, because ys1+yn "looks" approximately Gaussian, ys1 gets associated with yn rather than ys2 - so the separation is not so clean.

gauss <- ys1+yn  # approximately Gaussian (what I really want is yn)
expon <- ys1+ys2 # approximately exponential (desired)

par(mfrow=c(2, 2))
plot(x, gauss, type="l")
plot(density(gauss))
plot(x, expon, type="l")
plot(density(expon))

combined contributions

I expect that if I can impose a constraint that the points in the Gaussian distribution should not have any autocorrelation, I can get the separation between yn and ys1 that I desire. Is this a reasonable expectation and is it straightforward to implement such a constraint in mixture models?

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