Study of the Dispersion of the Covariance Matrix Diagonal


date:25/05/09

What does the Diagonal look like?
Check that the Diagonal dispersion is linked to the number of MC simulation.

CONCLUSION:
The dependence of the Diagonal dispersion with respect to the number of MC simulation is the one expected:
Standard Deviation is proportionnal to one over the square root of the number of simu.

See also study of the first under-diagonal: MC UnderDiagonal 1 and the antidiagonal: MC AntiDiagonal

Caracteristics of the data:

Data simulated @ CC

group/runnamesimupapier6
bolo143-5
nSide512
IMO'1127593'
signal name'cmb_gal_dipcosmo_gauss'
first ring580
last ring13080
NoiseWhite Noise (WN) sigma=62e-6

Commentary:
- simupapier6 corresponds to the new PPL (with the long ring) and in LSCmission mode_phase_pointing="random" (not "ideal")

A Diagonal:

Hereafter the diagonal of the covariance matrix of Offsets obtained with the MC simulation.
The results below are for 1000 simulations of noise.

Diagonal
Diagonal normalised wr2 its mean
Difference between the MC Diagonal and the PolkaCov Diagonal
Histogram of the dispersion

Commentary:
- The diagonal has the same shape as the weight per ring (see graph below). It's an image of the PPL! You can see for example the long ring in the middle. (see nside dependence for more explanation).
On the graph below, the polkapix covariance diagonal is also plotted in red.

- On the first plot, the polkapix covariance diagonal can't be observed because it's higher (~10e-6). That is why it's normalised after (second plot). The mean value of the Polkapix Covariance Diagonal and the MC Diagonal are correlated to the amplitude of noise. Indeed, the Polkapix Covariance Diagonal is in sigma-unity.

PolkaCov Diag mean2.2214088e-06
MC Diag mean1.4702191e-12
ratio6.61841e-07
sqrt(ratio)8.13536e-04
sigma62e-06
sample freq172.18
sigma/sample = sigma*sqrt(sample_freq)8.13548e-04

Ok, sigma/sample and sqrt(MC Diag mean / PolkaCov Diag mean) are close.
The level of the MC diag is correct !!!

- The noise on the Diagonal (see the third plot: Difference between the MC Diagonal and the PolkaCov Diagonal) isn't uniformly gaussian! There is still a correlation between the amplitude of this noise (or the standard deviation) and the local mean value of the diagonal.

Correlation between the diagonal dipersion and the number of MC simu:

Diagonal
Diagonal normalised wr2 its mean
Difference between the MC Diagonal and the PolkaCov Diagonal
Histogram of the dispersion
Gaussian Fit of the Histogram
Standard Deviation VS 1/sqrt(nb simu)

Commentary:
- The diagonal dispersion is proportionnal to the inverse of the square root of the number of MC simu used.
OK! This is the expected behaviour.