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When the data are spatially correlated, is the usual GMM likelihood function overweighted?

The data. Scattering experiment, sensor is like a CCD. Can't see individual events, only density estimate per pixel (see pic below). As you can see it looks like I'm imaging a connected structure - it's actually a polymer. Goal: determine the most likely atomic positions of that polymer.

Current approach Model each atom as a 2D Gaussian, so the problem is fitting a GMM to the data (with additional priors, e.g. linearity). For simplification, assume the atoms are all spherical (variance $\sigma^2$) and identical weight. This is the total density for $K$ atoms:

$$ I_{model}(x,y| \boldsymbol \mu,\sigma)=\sum_{i=0}^K \frac{1}{2\pi\sigma K}\exp\left[-\frac{(x-\mu_{i_x})^2+(y-\mu_{i_y})^2}{2\sigma^2}\right] $$

Assume independence of the pixels (see below), and Gaussian noise per pixel, here is the likelihood based on a $\chi^2$ statistic, where the data is $I_{exp}$:

$$ \begin{align} \\ \mathcal{L}(I_{exp};I_{model})&=\prod_{x,y}\frac{1}{\sqrt{2\pi}\lambda}\exp\left[-\frac{(I_{exp}(x,y)-I_{model}(x,y| \boldsymbol \mu,\sigma))^2}{2\lambda^2}\right] \\ &=\left(\frac{1}{\sqrt{2\pi}\lambda}\right)^{N_{pix}}\exp\left[-\frac{1}{2\lambda^2}\sum_{x,y}\left(I_{exp}(x,y)-I_{model}(x,y| \boldsymbol \mu,\sigma)\right)^2\right] \end{align} $$

Here's the problem. This value has a huge probability weight because there are tons of pixels. The problem is that the data are highly spatially correlated! Scatter events are due to a small number of physical objects. Is there a way to take into account this spatial correlation in the likelihood function? (Maybe that would also help the likelihood be less dependent of the number of pixels in the sensor.)

One solution I've considered is fitting a Gaussian Process to the data to smooth it out. And I suppose compare the GMM to a points sampled from the GP. Though I don't know how many points are needed for that.

density picture

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  • $\begingroup$ "Spatial correlation" means that, even when all of the Gaussian parameters $(\mu, \sigma)$ are known, the measurement at one pixel is related to the measurement at another pixel. Can you explain how this is possible? A normal CCD does not have this. $\endgroup$ – Tom Minka Nov 24 '14 at 19:10
  • $\begingroup$ Well, each scatterer leads to a chunk density on the sensor, with some amount of spread (parameterized by $\sigma$). Doesn't that mean the pixels must have a certain amount of smoothness (=local correlation)? $\endgroup$ – cgreen Nov 24 '14 at 19:17
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If the image has been blurred by a point spread function, then this needs to be taken into account in the likelihood. The sampling process turns the mixture of Gaussians into a noisy density, which is then convolved by the point spread function.

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  • $\begingroup$ Sure, it would probably be more accurate to represent the system as hard spheres that are blurred with a PSF. I will do that. But for the other question--is it correct to consider each pixel an independent observation? I.e., if you swap out for a higher-density sensor you have a much more highly weighted likelihood, even though the underlying signal is the same (and much coarser). $\endgroup$ – cgreen Nov 24 '14 at 19:40
  • $\begingroup$ If there is no point spread function, then each pixel on a CCD is an independent observation. Keep in mind that when swapping out for a higher-density sensor, the noise on each pixel will typically increase. $\endgroup$ – Tom Minka Nov 26 '14 at 12:06
  • $\begingroup$ Ah ok. Makes sense that the PSF is the source of dependence between pixels. The first equation in my question is the PSF (assuming point sources). Do you have a reference on how to better incorporate it into the likelihood? $\endgroup$ – cgreen Nov 26 '14 at 13:36

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