Nature of the Wiener Filtering

[lawrennd]

The Wiener filtering on the CMB, is it done as a preprocessing? And would that noise be considered additive or is it on the sensors i.e. after other nonlinear corruptions have been placed on.

From an ML perspective I'm thinking are we observing

$y = f(T; \theta) + \epsilon$

where $\epsilon \sim N()$ is distributed like a Wiener LTI process

or are we observing $y = f(T + \epsilon; \theta)$ ($\theta$ here the parameters of the CMB plus distortions.

[ixkael]

The Wiener filtering is typically used to make full sky CMB maps from partial sky noisy observations, by producing the MAP solution under a Gaussian assumption (which is very good). The typical reference for this is https://arxiv.org/abs/1210.4931. This is also where the messenger field trick is described, to iteratively find the inverse covariance matrix by exploiting the sparsity of the signal and the noise in different bases via a messenger field. I will write a demo of this in a notebook asap.

The actual data is $y = f(T; \theta) + \epsilon$ but takes a peculiar form: multiple full-sky maps of the microwave sky, in different frequencies. The CMB is the same in all frequencies, but the foreground contamination is not, so we end up having to deal with a component separation problem. If the frequency is indexed by $i$ (e.g., Planck has $i=1, \cdots, 9$) we have $y_i = f_i(T; \theta) + \epsilon_i$. The CMB signal $T(\theta)$ is the same noiseless Gaussian random field. The noise amplitude and spectrum changes with $i$. And the transfer function $f_i$ is some additive function of the CMB and other galactic contributions: $f_i(T(\theta)) = T(\theta) + \sumj \alpha{ij} F{ij}$ where $F{ij}$ is the j-th foreground observed in the i-th channel. We have physical models for those $F{ij}$s as well as priors on $\alpha{ij}$ (at least for CMB temperature, not polarization. As a side note, it would be amazing if we could learn a joint model for temperature and polarization foregrounds...) The state of the art for this joint component separation and parameter estimation is https://commander.bitbucket.io/ for low-resolution pixel-space analysis. For higher resolution this is done directly with the power spectra, with similar models. I will write this in more detail in a notebook too.

[lawrennd]

Linking in @adamian because this model has the feel of some 'multi fidelity' models that are used in the area of surrogate modelling.

Just trying to unpick what you're saying about Planck. Does this mean that Planck has nice sensors, each sensitive to a different area of the spectrum? I.e to use a simple analogy, just as our eyes have r,g,b can I think of planck having r,g,b,u,v,s,p,q,r sensitivity? Like nine colour? Or have I misunderstood.

If the nine colour idea is correct, then (ignoring the effect of gravitational lensing) are you picking up convolved versions of the CMB? Or am I way off base here?

On Sun, 21 May 2017 at 17:19 Boris Leistedt notifications@github.com wrote:

The Wiener filtering is typically used to make full sky CMB maps from partial sky noisy observations, by producing the MAP solution under a Gaussian assumption (which is very good). The typical reference for this is https://arxiv.org/abs/1210.4931. This is also where the messenger field trick is described, to iteratively find the inverse covariance matrix by exploiting the sparsity of the signal and the noise in different bases via a messenger field. I will write a demo of this in a notebook asap.

The actual data is $y = f(T; \theta) + \epsilon$ but takes a peculiar form: multiple full-sky maps of the microwave sky, in different frequencies. The CMB is the same in all frequencies, but the foreground contamination is not, so we end up having to deal with a component separation problem. If the frequency is indexed by $i$ (e.g., Planck has $i=1, \cdots, 9$) we have $y_i = f_i(T; \theta) + \epsilon_i$. The CMB signal $T(\theta)$ is the same noiseless Gaussian random field. The noise amplitude and spectrum changes with $i$. And the transfer function $f_i$ is some additive function of the CMB and other galactic contributions: $f_i(T(\theta)) = T(\theta) + \sumj \alpha{ij} F{ij}$ where $F{ij}$ is the j-th foreground observed in the i-th channel. We have physical models for those $F{ij}$s as well as priors on $\alpha{ij}$ (at least for CMB temperature, not polarization. As a side note, it would be amazing if we could learn a joint model for temperature and polarization foregrounds...) The state of the art for this joint component separation and parameter estimation is https://commander.bitbucket.io/ for low-resolution pixel-space analysis. For higher resolution this is done directly with the power spectra, with similar models. I will write this in more detail in a notebook too.

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[ixkael]

Yes, Planck has mapped the full sky/sphere in nine frequency channels, from a 1e1 to 1e3 GHz. The (mean) CMB is a black-body spectrum to an excellent approximation, and the temperature fluctuations are the same in all frequencies. However, the foregrounds are different. A view of the sky maps is http://www.esa.int/spaceinimages/Images/2013/04/Planck_all-sky_frequency_maps. This is the data we use to constrain cosmological parameters. The simplest way to do that (but incorrect because not accurate enough) is to mask the areas with strong foregrounds and measure the CMB fluctuations everywhere else. In this case the problem statement is simple: constrain the CMB power spectrum given nine noisy partial-sky observations. So I think the color analogy is correct: we observe the same CMB in nine channels, with different resolution (convolution beam), noise, and foregrounds.

[ixkael]

Tagging @davidwhogg