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  1. 1. Introduction
  2. 2. XQML code :
    1. 2.1 cross :
    2. 2.2 auto :
  3. 3. Simulation pipeline :
    1. 3.1 Bicep patch Inverse noise variance :
  4. 4. Simulations variance en fits results :
    1. 4.1 Bicep patch, Nside 128 :
    2. 4.2 Bicep patch, Nside 128 :
    3. 4.3 Litebird patch, Nside 16 :
  5. 5. Error in pixel covariance matrix :
    1. 5.1 Estimation and variance :
    2. 5.2 Ratio of variance
  6. 6. Other :

Previous page XQML

1.  Introduction

2.  XQML code :

2.1  cross :

  • Matrices S and \displaystyle \frac{d S}{d C_b} are computed at the same time.
  • Define matrix \displaystyle P_b = \frac{d S}{d C_b}
  • Diagonal pixel covariance matrix, with V_A = pixel noise variance for map A. Same for map B :
    • C^{AB} = S ,
    • C^{AA} = S + \mathbb 1 \cdot V_A ,
    • C^{BB} = S + \mathbb 1 \cdot V_B .
  • Compute {C^{AA}}^{-1} and {C^{BB}}^{-1}
  • Compute E_b^{AB} = {C^{AA}}^{-1} P_b {{C^{BB}}}^{-1}
  • Compute \displaystyle F_{bb'}^{AB} = Tr_{pix}[ {C^{AA}}^{-1} P_b {{C^{BB}}}^{-1} P_{b'} ] = \sum_{allpix~ij} (E_b^{ij} P_{b'}^{ij} ) (term-by-term matrix product from trace property).
  • Compute {F_{bb'}^{AB}}^{-1}
  • Compute C^{AB}\cdot E_b then G_{bb'}^{AB} = \sum_{allpix~ij} (C^{AB}\cdot E_b)^{ij} ( C^{AB} E_{b'})^{ij}

2.2  auto :

  • Matrices S and \displaystyle \frac{d S}{d C_b} are computed at the same time.
    • S is now computed from \ell = 2 to ell=lmax, whatever the binning.
  • Define matrix \displaystyle P_b = \frac{d S}{d C_b}
  • Diagonal pixel covariance matrix, with V_A = pixel noise variance for map A. Same for map B :
    • C^{auto} = S + \mathbb 1 \cdot V_A ,
  • Compute {C^{auto}}^{-1}
  • Compute E_b^{auto} = {C^{auto}}^{-1} P_b {{C^{auto}}}^{-1}
  • Compute \displaystyle F_{bb'}^{auto} = Tr_{pix}[ {C^{auto}}^{-1} P_b {{C^{auto}}}^{-1} P_{b'} ] = \sum_{allpix~ij} (E_b^{ij} P_{b'}^{ij} ) (term-by-term matrix product from trace property).
  • Compute {F_{bb'}^{auto}}^{-1}

3.  Simulation pipeline :

  1. Get pre-saved S and \displaystyle \frac{d S}{d C_b} matrices computed for defined set of parameters :
    1. patch (bicep or litebird like),
    2. fwhm (beam),
    3. nside,
    4. mask (fksy),
    5. Clth (fiducial spectrum model),
    6. first \ell,
    7. bin size \Delta \ell ,
    8. pixwin (yes/no)
  2. Compute useful matrices for xqml (description above).
  3. Compute 1000 simulations as follow :
    1. Noise maps :
      1. Map noise auto : N = \mathcal N (0, \sqrt {V_A})
      2. Map noise A : N_A = \mathcal N (0, \sqrt {2V_A})
      3. Map noise B : N_B = \mathcal N (0, \sqrt {2V_B})
    2. Map CMB : synfast(Clth, fwhm=fwhmrad, pixwin=True)
    3. Maps :
      1. Map d : d = CMB + N,
      2. Map A : d_A = CMB + N_A,
      3. Map B : d_B = CMB + N_B
    4. cross xqml :
      1. cross pre-estimator \hat y_b = d_A E_b d_B
      2. cross estimator \hat C_b = {F_{bb'}^{AB}}^{-1} \cdot \hat y_{b'}
    5. auto xqml :
      1. auto pre-estimator \hat y_b = d E_b d
      2. spectrum bias term : \hat B_{b} = Tr_{pix} [N \cdot E_b ] = \sum_{allpix~ij} (N^{ij} E_{b}^{ij})
      3. auto estimator \hat C_b = {F_{bb'}^{AB}}^{-1} \cdot (\hat y_{b'} - \hat B_{b} )
    6. Evaluates analytic error : Cov(C_b, C_{b'}) = {F_{bb_1}^{AB}}^{-1} G_{b_1b_2}^{AB} {F_{b_2b'}^{AB}}^{-1} + {F_{bb'}^{AB}}^{-1}
  4. Compute MC spectra means and variances, then compare to the model and the analytic variance.

3.1  Bicep patch Inverse noise variance :

My version (wrong)

Let w be the apodization weighing mask map of bicep ( [0,..,1] \in w ). We want to build a noise map with variance inversely proportional to the weighing map. If we choose the weighed mean variance per pixel of the map to be \sigma^2 [\mu K\cdot arcmin] , and V_i the variance of pixel i, then we have \displaystyle \sigma = \frac{\sum_i V_i w_i}{\sum_i w_i} . Since we want to weight the variance for each pixel, we take V_i = \alpha \sigma^2 w_i^{-1} with \alpha the normalisation factor that assures the mean variance of the map V to be \sigma^2. Thus, \displaystyle \sigma^2 = \frac{\sum_i \sigma \alpha}{\sum w_i} = \frac{n_{pix} \sigma^2 \alpha}{\sum w_i} , which gives \displaystyle \alpha = \frac{\sum_i w_i}{n_{pix}} , and the final variance per pixel read : \displaystyle V_i = \frac{\sigma^2}{ w_i} \frac{\sum_j w_j}{n_{pix}}

Matt's version :

We demand the total variance of the map to be \displaystyle \frac{\sigma^2}{n_{pix}} [\mu K\cdot arcmin] = \frac 1 {\sum_i 1/ V_i} which is equal to \displaystyle \frac{\sigma^2}{n_{pix}} if V_i = \sigma^2 (white noise case). Now we demand a weighting of the variance such that V_i = \beta w_i^{-1} with \beta the normalisation factor. Injecting V_i gives \beta = \displaystyle \frac{\sigma^2}{n_{pix}} \sum_i w_i ad finally \displaystyle V_i = \frac{\sigma^2}{ w_i} \frac{\sum_j w_j}{n_{pix}} .

4.  Simulations variance en fits results :

  • Compute std for two patches : Full and small (litebird and bicep like).
  • Xqml spectra estimators are computed for 1000 simulation. Theoretical std is compared to MC std.
  • Upper plot corresponds to the mean of the 1000 xqml simulations spectrum estimation (blue dot) compared to C_\ell model (dashed black).
  • Down plot are standard deviations (error bars) compared to Fisher error (dasher red) and C_\ell model (dashed black).
  • The cross Fisher variance estimate follows :

\displaystyle Cov( C^{AB}_\ell, C^{AB}_{\ell} ) = \frac{1}{ f_{sky} (2 \ell+1) \Delta \ell} \left[ C_\ell^{AA} C_\ell^{BB} + C_\ell^{AB}C_\ell^{BA} \right]

\displaystyle Cov( C^{AB}_\ell, C^{AB}_{\ell} ) = \frac{1}{ f_{sky} (2 \ell+1) \Delta \ell} \left[ 2 C_\ell^2 + C_\ell ( N_\ell^A + N_\ell^B )b_\ell^{-2} + N_\ell^A N_\ell^B b_\ell^{-4} \right]

where \displaystyle f_{sky} = \frac 1{n_{pix}} \frac{(\sum_i W_i^2)^2}{\sum_i W_i^4} (from MASTER paper)

  • While the auto- Fisher variance estimate follows :

\displaystyle Cov( C^{A=B}_\ell, C^{A=B}_{\ell} ) = \frac{1}{ f_{sky} (2 \ell+1) \Delta \ell} \left[ C_\ell^{AA} C_\ell^{BB} + C_\ell^{AB}C_\ell^{BA} \right]

\displaystyle Cov( C^{A=B}_\ell, C^{A=B}_{\ell} ) = \frac{1}{ f_{sky} (2 \ell+1) \Delta \ell} \times 2 \times \left[ C_\ell + N_\ell^A b_\ell^{-2} \right]^2

  • Right tab describes simulation inputs :
    • Patch and nside. (BICEPivw means that I used inverse variance weighting (ivw) for noise maps. Otherwise, "BICEP" alone means that I used white noise.).
    • 1bins40 means that the minimum \ell taken in the binning starts at \ell = 40
    • pixwin (1 = True, always the case in simulations so far)
    • fwhmdeg (beam in degrees, always used 0.5 deg in simulations so far)
    • wiEE corresponds to C_\ell^{EE} \neq 0. While noEE is the opposite ( C_\ell^{EE} = 0).
    • r0.1 corresponds to r=0.1 (always used r=0.1 case in simulation so far).
    • fsky
    • noise \sigma = 0.1 [\mu K\cdot arcmin ] , gives the following variance noise per pixel : \displaystyle \sigma_{pix}^2 = \frac{\sigma^2 \cdot 10^{-12} }{60^2 \cdot PixelArea [deg]}
    • mean value of N_\ell is boxed in magenta, and \ell (\ell+1)/2/\pi N_\ell is plotted in magenta.
    • yellow triangles are Xpol fits and variance.

4.1  Bicep patch, Nside 128 :

no ponderation (white noise)

4.2  Bicep patch, Nside 128 :

  • Avec ponderation du bruit par pixel !
fwhm=0.5deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
fwhm=0.5deg, \sigma=0.1 \mu Karcm
fwhm=0.5deg, \sigma=1.0 \mu Karcm
fwhm=0.5deg, \sigma=5.0 \mu Karcm

4.3  Litebird patch, Nside 16 :

fwhm=1.0deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
fwhm=1.0deg, \sigma=0.1 \mu Karcm
fwhm=1.0deg, \sigma=1.0 \mu Karcm
fwhm=1.0deg, \sigma=5.0 \mu Karcm

5.  Error in pixel covariance matrix :

5.1  Estimation and variance :

We change C = S + N by C = S + p \cdot N with p = [0.5, 1.0, 2.0.]

  • Note : Le biais en auto sur Bicep pondéré est dû à une erreur dans mes jobs : Je lui donne une matrice de bruit diagonale constante pour débiaiser l'estimateur, alors que dans ce cas la diagonale n'est justement pas constante. C'est en cours de correction.

Bicep (ponderé)

cross :

fwhm=0.5deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
\sigma=1.0 \mu Karcm
\sigma=5.0 \mu Karcm

auto :

fwhm=0.5deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
\sigma=1.0 \mu Karcm
\sigma=5.0 \mu Karcm

Litebird :

Cross :

fwhm=0.5deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
\sigma=1.0 \mu Karcm
\sigma=5.0 \mu Karcm

Auto :

fwhm=0.5deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
\sigma=1.0 \mu Karcm
\sigma=5.0 \mu Karcm

5.2  Ratio of variance

Again, we change C = S + N by C = S + p \cdot N with p = [0.5, 1.0, 2.0.], and plot the ratio \displaystyle R(p) = \sqrt{ \frac{VarCl (C = S + p \cdot N)}{VarCl (C = S + N)} } of the MC standard deviations. Note that for each value of p, the 1000 cmb + noise simulations are the same.

Bicep (ponderé)

Cross and auto :

fwhm=0.5deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
\sigma=1.0 \mu Karcm
\sigma=5.0 \mu Karcm

Litebird :

Cross and auto :

fwhm=0.5deg C_\ell^{EE} \neq 0 C_\ell^{EE} = 0
\sigma=1.0 \mu Karcm
\sigma=5.0 \mu Karcm

6.  Other :

  • My to do list :
    • Test and compare Pure B-mode