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1. planck data

Three methods :
- N-R = Newton Rwphson Likelihood using the full covariance {$ 3 \times 2 \times npix $} matrix (CMB + noise)
- xLR : cross linear regression using the full covariance {$ 2 \times npix $} matrix (CMB + noise)
- oLR : ordinary linear regression, map pre-smoothed by a 3° fwhm beam.
Maps pre-downgrading :
- loaded at nside 2048
- smoothed by a cosine beam
  - {$ b_\ell = 1 \quad \ell \leq \ell_1$},
  - {$ b_\ell = 0.5(1 + \cos((\ell-\ell_1) \pi/(\ell_2 - \ell_1))] \quad \ell_1 < \ell \leq \ell_2$},
  - {$ b_\ell = 0 \quad \ell_2 < \ell$}
- downgraded at nside 16
There is two types of dataset-split : half mission, and odd/even rings
For each dataset split, there is 8 possible combinations to estimate the coefficient.
- P1 or P2 are the map to clean
- D1 or D2 are the 353GHz dust template
- S1 or S2 are the 30GHz synchrotron template
We consider four masks : from fsky=0.9 to fsky=0.5
Two types of error-bar :
- smaller ones computed from simulation with noise generated from planck cov maps
- bigger ones computed from simulation with public planck noise including sistematics.

1.1 alpha VS freq ; full by full

planck data

simus (FFP10, white noise)

planck data

143 seesm biased on simulations. Do not known why. (pySM models ? )
N-R performs best !
grey bands indicate 1, 2, and 3 sigma dispersion of the input coeffidients maps (from pySM) {$ \alpha(\hat n) $}