2012-2013
HiLLiPOP 20 pars (ACib, Asz, Adust)
TT vs TT+pol
- mask ext
TT only
mask ext
\omega_b | 0.022379057 | 0.00012035533 |
\omega_{cdm} | 0.11845001 | 0.00098530883 |
H_0 | 67.992029 | 0.47126828 |
z_{reio} | 11.404633 | 0.83129284 |
n_s | 0.96165072 | 0.0034102097 |
log[10^{10}A_s] | 3.0945009 | 0.019477673 |
c_0 | 0.99702925 | 0.0014987291 |
c_1 | 0.99958261 | 0.0014908308 |
c_3 | 0.99959829 | 0.0011732993 |
c_4 | 1.0018138 | 0.0015714879 |
c_5 | 1.0024945 | 0.0015972064 |
A_{ps}^{100x100} | 0.00011124794 | 2.7883902e-05 |
A_{ps}^{100x143} | 6.6166294e-05 | 9.2137408e-06 |
A_{ps}^{100x217} | 5.6762334e-05 | 8.9679926e-06 |
A_{ps}^{143x143} | 2.4252632e-05 | 4.9369440e-06 |
A_{ps}^{143x217} | 2.4074980e-05 | 3.5883079e-06 |
A_{ps}^{217x217} | 5.4345085e-05 | 1.0765712e-05 |
A_{SZ} | 0.98915093 | 0.096186627 |
A_{CIB} | 0.98896531 | 0.17944047 |
A_{dust} | 0.93456305 | 0.20207269 |
- mask ext
TT+pol
mask ext
\omega_b | 0.022175680 | 7.2685789e-05 |
\omega_{cdm} | 0.11922543 | 0.00062694041 |
H_0 | 67.459684 | 0.29084713 |
z_{reio} | 10.390123 | 0.73880373 |
n_s | 0.96091414 | 0.0025132256 |
log[10^{10}A_s] | 3.0739369 | 0.016387454 |
c_0 | 0.99481810 | 0.0014291316 |
c_1 | 0.99728873 | 0.0014158846 |
c_3 | 0.99887959 | 0.0011659994 |
c_4 | 0.99680509 | 0.0014886902 |
c_5 | 0.99636673 | 0.0014644944 |
A_ps_100x100 | 0.00011469479 | 2.6547886e-05 |
A_ps_100x143 | 5.9890144e-05 | 8.5499343e-06 |
A_ps_100x217 | 6.7697991e-05 | 8.4064065e-06 |
A_ps_143x143 | 1.8669217e-05 | 4.6595079e-06 |
A_ps_143x217 | 2.0767009e-05 | 3.4791328e-06 |
A_ps_217x217 | 4.3328786e-05 | 1.0682537e-05 |
A_{SZ} | 0.95702884 | 0.098479700 |
A_{CIB} | 1.2170840 | 0.15270657 |
A_{dust} | 0.97233877 | 0.066120385 |
mask no ext
\omega_b | 0.022263755 | 8.9475284e-05 |
\omega_{cdm} | 0.11867956 | 0.00076152065 |
H_0 | 67.733489 | 0.35821854 |
z_{reio} | 10.828718 | 0.82768626 |
n_s | 0.96362996 | 0.0031400847 |
log[10^{10}A_s] | 3.0857178 | 0.018836652 |
c_0 | 0.99459983 | 0.0016563812 |
c_1 | 0.99719222 | 0.0016400131 |
c_3 | 0.99898264 | 0.0014217997 |
c_4 | 0.99631891 | 0.0017455929 |
c_5 | 0.99582094 | 0.0018064114 |
A_{ps}100x100 | 0.00011695351 | 3.1376064e-05 |
A_{ps}100x143 | 5.6187471e-05 | 1.0325234e-05 |
A_{ps}100x217 | 6.4567769e-05 | 1.0419211e-05 |
A_{ps}143x143 | 1.7279342e-05 | 6.1229137e-06 |
A_{ps}143x217 | 2.0991061e-05 | 4.4999899e-06 |
A_{ps}217x217 | 4.7834940e-05 | 1.2718610e-05 |
A_{SZ} | 0.98676201 | 0.096564312 |
A_{CIB} | 1.1502160 | 0.17659439 |
A_{dust} | 0.98812484 | 0.10076243 |
cosmo
- mask ext
- mask no ext
astro nuicance
- mask ext
- mask no ext
calib
- mask ext
- mask no ext
Correlation matrix
- mask ext
- mask no ext
- mask ext
- mask no ext
Camspec PLANCK +WP (fixed beams but b11)
this results are obtained directly calulating moments.
omega_b | 0.0220678 | 0.000257888 |
omega_cdm | 0.120065 | 0.00240494 |
H_0 | 67.2408 | 1.07455 |
z_reio | 10.9514 | 0.991644 |
n_s | 0.959535 | 0.00659704 |
A_s | 3.08640 | 0.0223076 |
A_ps_100 | 168.139 | 55.6082 |
A_ps_143 | 54.1184 | 11.9975 |
A_ps_217 | 110.345 | 14.1698 |
A_cib_143 | 8.23319 | 4.60434 |
A_cib_217 | 27.7484 | 6.32935 |
A_sz | 4.90709 | 2.60693 |
r_ps | 0.893450 | 0.0678707 |
r_cib | 0.355640 | 0.179416 |
n_Dl_cib | 0.538330 | 0.109830 |
cal_100 | 1.00058 | 0.000362591 |
cal_217 | 0.996396 | 0.00125990 |
xi | 0.482154 | 0.273575 |
A_ksz | 4.48624 | 2.72452 |
Bm_1_1 | 0.527794 | 0.506504 |
Camspec PLANCK +WP (fixed beams but b11)
chain not completely converged!
omega_b | 0.0220874 | 0.000285975 |
omega_cdm | 0.119937 | 0.00236612 |
H_0 | 67.3167 | 1.10474 |
z_reio | 11.0348 | 0.900486 |
n_s | 0.959881 | 0.00879021 |
A_s | 3.08785 | 0.0287007 |
A_ps_100 | 157.032 | 54.1907 |
A_ps_143 | 55.1763 | 11.6363 |
A_ps_217 | 117.428 | 13.1730 |
A_cib_143 | 6.19464 | 3.67448 |
A_cib_217 | 23.4497 | 5.62208 |
A_sz | 5.80824 | 2.47774 |
r_ps | 0.845278 | 0.0796822 |
r_cib | 0.444556 | 0.192859 |
n_Dl_cib | 0.388329 | 0.157984 |
cal_100 | 1.00053 | 0.00662747 |
cal_217 | 0.996448 | 0.00670545 |
xi | 0.487896 | 0.273396 |
A_ksz | 4.60022 | 2.68446 |
Bm_1_1 | 0.496810 | 0.501291 |
Plik 7N 18 vs 20 parameters
- plik with n_cib and A-ksz fixed
67.051564 1.3110665
0.022002617 6.6695976e-08
0.12165493 7.0877670e-06
3.2303953 0.00091294046
0.95627828 4.7465556e-05
10.824911 0.84220162
3.1525278 0.41996407
56.035293 6.1686063
0.86380524 0.0010006399
2.3372633 0.93552483
0.43150753 0.059639434
157.21067 371.41839
100.25894 142.66303
76.754522 90.939383
79.284954 54.392776
65.829367 29.172489
63.092615 53.651448
4.1879759e-05 7.6629848e-13
- plik with the 20 parameters
67.025936 0.94635801
0.021977616 5.6594002e-08
0.12163599 5.2858761e-06
3.2331443 0.00066098884
0.95361424 3.2663754e-05
10.638049 0.72229094
2.0421712 0.11818649
43.914058 15.579249
0.89361855 0.00066491921
1.5051065 0.75769922
0.34283761 0.045385609
152.71505 413.90302
90.637823 207.09059
61.316836 170.08268
68.102259 136.49250
52.622826 106.35467
73.738066 154.37935
3.8085516e-05 2.7984361e-12
-1.4661110 0.0027926783
7.4354695 27.992429
we cleary see that fixing n_cib shifts all the nuisances.
Effect of the final lensing likelihoods on cosmological parameters
- In the first plot here, using Plik7N+WMAP9, we show the results for the six cosmological parameters LCDM obtained with the final lensing likelihood, namely the combination 143GHzX217GHz (cleaned with 857Ghz map) with 30% galactic mask and linear binning (in light blue).
This final version is compared to the 40% galactic mask and equal variance binning case (in blue).
They are very well in agreement. The black curves in the no lensing case.
- In the second plot we show the same results using Plik7N only, without WMAP9 .
We clearly see the degeneracy for A_s and z_reio in the no lensing case (black curve).
We notice also that the way the lensing breaks this degeneracy is slightly different for the two lensing likelihood (again final one in light blue and previous one in blue).
PLANCK ONLY: Adaptive Algo with only cosmological parameter, no lensing
var | mean |
H0 |
omega_b |
omega_cdm |
log(10^10A_s) |
n_s |
z_reio |
Plik 7N: Adaptive Algo with only cosmological parameter, no lensing
var | mean | err |
H0 | 67.05 | 1.24 |
omega_b | 0.0220 | 0.0003 |
omega_cdm | 0.1216 | 0.0029 |
log(10^10A_s) | 3.22 | 0.03 |
n_s | 0.9569 | 0.0069 |
z_reio | 10.76 | 0.98 |
Plik 7N: Metro Algo with only cosmological parameter, no lensing (only one run)
67.606514 1.3236871 -6.6760170 396.72404
0.022061285 8.3364645e-08 -14.783787 1135.0212
0.12032863 6.7663619e-06 -3.3118547 152.13659
3.2181406 0.0012328797 -25.709822 2350.5108
0.96005083 5.9512680e-05 -64.096896 7992.6825
10.913159 1.2462165 0.017568006 0.41391798
problems with n_s
MCMC for studing lensing
M. Spinelli & S. Plaszczynski
Effect of new lensing likelihoods on cosmological parameters
we study the effect of different lensing candidate likelihoods on LCDM parameters determination.
Our CMB data consists in PLIKv7N likelihood+ WMAP9low-ell.
Chains are run within the CAMEL.fr software, using an Adaptive Metropolis (AM) algorithm.
All PLIK nuisance parameters have been fixed to their best fit values, to avoid hiding any potential effect into these classes.
CMB w/o lensing (different channels, default binning: [40,400])
- We compare the results for the cosmological parameters obtained using the lensing likleihood for 143GHz, 217GHz and the combination 143GHzX217GHz (cleaned with 857Ghz map and using 40% galactic masking).
var | no lens | 143x217GHz | 143GHz | 217GHz |
H0 | 67.05+-1.24 | 68.07+-1.00 | 68.08+-1.05 | 68.12+-1.03 |
omega_b | 0.0220+-0.0003 | 0.0221+-0.0003 | 0.0221+-0.0003 | 0.0221+-0.0025 |
omega_cdm | 0.1216+-0.0029 | 0.1193+-0.0023 | 0.1192+-0.0024 | 0.1190+-0.0023 |
log(10^10A_s) | 3.22+-0.03 | 3.21+-0.03 | 3.21+-0.03 | 3.20+-0.03 |
n_s | 0.957+-0.007 | 0.961+-0.006 | 0.962+-0.006 | 0.962+-0.006 |
z_reio | 10.76+-0.98 | 10.71+-0.99 | 10.66+-0.97 | 10.64+-1.0 |
- clear shift of H0 by 1 sigma (omega_cdm and ns too) for both 143 and 217 lensing reconstruction
- as expected improvement on H0 and omega_cdm errors
CMB w/o lensing (different channels, adding external bins: [10,1000])
- We compare the results for the cosmological parameters obtained using lensing files for 143GHz, 217GHz and the combination 143GHzX217GHz (cleaned with 857Ghz map and using 40% galactic masking) in the case where 2 external bins ([10,40],[400,1000]) are added.
var | no lens | 143x217GHz | 143GHz | 217GHz |
H0 | 67.05+-1.24 | 68.34+-0.98 | 68.56+-1.06 | 68.28+-1.03 |
omega_b | 0.0220+-0.0003 | 0.0221+-0.0002 | 0.0222+-0.0002 | 0.0221+-0.0002 |
omega_cdm | 0.1216+-0.0029 | 0.1186+-0.0021 | 0.1186+-0.0023 | 0.1188+-0.0024 |
log(10^10A_s) | 3.22+-0.03 | 3.20+-0.03 | 3.19+-0.03 | 3.20+-0.03 |
n_s | 0.957+-0.007 | 0.962+-0.006 | 0.964+-0.006 | 0.963+-0.006 |
z_reio | 10.76+-0.98 | 10.64+-0.93 | 10.50+-0.96 | 10.53+-0.98 |
- same effect
- comparison wrt to previous plots in next section
LCDM 6 parameters behaviour w/o external bins for 143+217 combination
- We show here the results for the cosmological parameters obtained respectively with default binning on adding external bins for the combination 143GHzX217GHz.
var | default | ext bins |
H0 | 68.07+-1.00 | 68.34+-0.98 |
omega_b | 0.0221+-0.0003 | 0.0221+-0.0002 |
omega_cdm | 0.1193+-0.0023 | 0.1186+-0.0022 |
log(10^10A_s) | 3.21+-0.03 | 3.20+-0.03 |
n_s | 0.961+-0.006 | 0.962+-0.006 |
z_reio | 10.71+-0.99 | 10.64+-0.93 |
- slightly stronger shift when adding exterbal bins
- marginal error improvement
LCDM 6 parameters behaviour with respect to galactic masking
- We show here the results for the cosmological parameters obtained for the 143GHz, cleaned with 857Ghz map and with default binning but with different masking: 30% and 40% respectively .
var | gal30 | gal40 |
H0 | 68.17+-1.00 | 68.08+-1.05 |
omega_b | 0.0221+-0.0002 | 0.0221+-0.0003 |
omega_cdm | 0.1190+-0.0023 | 0.1192+-0.0024 |
log(10^10A_s) | 3.20+-0.02 | 3.21+-0.03 |
n_s | 0.962+-0.006 | 0.962+-0.006 |
z_reio | 10.52+-0.98 | 10.66+-0.97 |
217GHz
- We show here the results for the cosmological parameters obtained respectively with default binning or adding external bins for the 217GHz.
var | default | err default | larger bins | err larger bins |
H0 | 68.12 | 1.03 | 68.28 | 1.03 |
omega_b | 0.0221 | 0.0025 | 0.0221 | 0.0002 |
omega_cdm | 0.1190 | 0.0023 | 0.1188 | 0.0024 |
log(10^10A_s) | 3.20 | 0.03 | 3.20 | 0.03 |
n_s | 0.962 | 0.006 | 0.963 | 0.006 |
z_reio | 10.64 | 1.00 | 10.53 | 0.98 |
-
2011-2012
Chi-square distribution from gaussian variables
u=/sum_{i=1,k}x_i^2 ~ /chi^2_k where k=dof and x~N(0,1)
Metropolis: simmetrical proposal q(X|Y)=q(Y|X)
1D CASE
- target distribution: Chi-Square with 3 dof
- proposal distribution: N( last parameter value in the chain, sigma=0.7 ) and N( last parameter value in the chain, sigma=1.5 )
- Parallel evolution of the chains in the 2 cases of sigma=0.7 and sigma=1.5. We can see the different burn in periods.
- Gelman-Rubin R parameter with respect of the length of the chains in the 2 cases of sigma=0.7 and sigma=1.5.
We considered a chain converged when R is always less then 1.03
We can see that the first step proposal is too small and we need longer chains to see convergence.
- As final example we show here the histogram of the values of the parameter for one of the chain. We recognize the form of the chi-square with 3 dof? (to check)
2D CASE
- target distribution: Bivariate distribution with mean=(0,0), sigma=(1,2) and correlation=0.8
- proposal distribution: N( last parameter value in the chain, sigma=1 ) for both parameters.
- We first show the evolution of the 4 chain to see if they at least seem to converge.
- We monitor convergence using Gelman Rubin R. In our 2D case we will have a value for each of the two parameter with respect of the length of the chains.
We notice that, from a certain point on, R_1 and R_2 show very similar behaviours. This is certainly due to the high correlation of the target distribution.
We can check this dependence analysing the case of no correlation.
Here the Rs behave very differently, the slower convergence of R_2 due to a poor choice of the jump.
- Finally we show the 2D histogram of the parameters of the chains, once rejected the burn in period.
1D CASE: autocorrelation
- target distribution: Chi-Square with 3 dof
- proposal distribution: N( last parameter value in the chain, sigma ) with sigma respectively equal to 1, 6, 10
- Histograms of the numbers of consecutive rejection moves in the three cases
- Correlation with respect to the lag in the three cases, the correlation length is obtained fitting with an exponential function.
- Which is the optimal acceptance rate for minimising the correlation length? Confirmation of the Rule of thumb
sigma | burn_in: p/pmax<0.1 | acceptance rate | correlation length | Gelman-Rubin R |
0.5 | 1200 | 0.93 | 64.12 | 0.10 |
1 | 800 | 0.88 | 16.07 | 0.05 |
3 | 700 | 0.70 | 4.44 | 0.02 |
6 | 700 | 0.50 | 3.50 | 0.01 |
10 | 650 | 0.36 | 3.60 | 0.01 |
20 | 600 | 4.45 | 4.45 | 0.03 |
- Test for convergence using FFT: Joanna Dunkley et al Mon.Not. R. Astron. Soc. 356,925-936 (2005)
Description of the parameter:
alpha= in case of random walk alpha is equal to 2. We see that increasing the trial steps the probability of rejection of a jump increase consequently departing from 2.
correlation length= we have with this parameter the occasion of a double check with the previous method. Once again is possible to see that an acceptance rate of approximately 0.5 corresponds to a minimum in correlation length.
P0=the criterion of convergence here is r=P0/N<0.01We can with this establish for how long we should run the chain before saying we are actually sampling from a fair approximation of the target distribution.
N | r |
500 | 0.06 |
2000 | 0.025 |
3000 | 0.013 |
4500 | 0.008 |
8500 | 0.006 |
ND case
Acceptance rate and autocorrelation
Target distribution: 4D and 10D gaussian respectively with identity as covariance matrix. Proposal distribution: 4D and 10D gaussian respectively with diagonal covariance matrix. sigma is then the ratio between the proposal and the target jump.
- For N=5000, burn_in=1000 and mediating on 1000 repetition we obtain the following plots.
From the first plot we clearly see how choosing a too big jump lowers the acceptance rate.
The second plot give as a confirmation of the rule for choosing the sigma of the proposal: sigma proposal/ sigma target = 2.38^2/sqrt(D).
The comparison between the two plots give as instead a confirmation of the rule of thumb.
Metropolis vs Multiparticles Algorithm (stretch move)
We show below the comparison between the two methods first in 2D case and then in Nd case with N=6.
Using FFT analysis we investigate autocorrelation and convergence starting of the final sample forcing computational effort to be the same. L=l*W=100*100 where L is the length of the chain for MH, l the number of steps in the multiparticle method and W is the number of walkers. Acceptance rate is instead chose as advised by the authors of the papers we refer to. For MH for D>1 the ratio between the sigma of the proposal and the target should equal to 2.44/sqrt(D) so 1.7 for the 2d case (ar=0.35) and 1 for the 6d case (ar=0.25). For the SM we use the parameter a=2 the gives an ar=0.5
- 2D case TARGET: bivariate gaussian with identity as covariance matrix
- Metropolis Algorithm
- Stretch Move Algorithm
- 6D case TARGET: 6d gaussian with identity as covariance matrix
- Metropolis Algorithm
- Stretch Move Algorithm
SM for Rosenbrock distribution
Metropolis vs Adaptive Metropolis
- Autocorreation of the chains
The first case has as target distribution a multivariate with a generic covariance matrix. We can see clearly the faster decrease of autocorrelation (meaning a better mixing) in the case of adaptive metropolis.
The second case shows instead that in case of a symmetrical target distribution (the multivariate gaussian has diagonal covariance matrix) the performance of the two algorithms is almost the same.
- Checking convergence
In the case of a generic covariance matrix for the target distribution, is presented here the evolution of the Gelman-Rubin parameter R as we use longer chains for the case of a 4D distribution and 6D.
The adaptive algorithm is found to converge in only 300 steps in both cases.
- Adaptive Metropolis algorithm
- A graphical confront between the two algorithms