2012-2013

HiLLiPOP 20 pars (ACib, Asz, Adust)

TT vs TT+pol

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TT only

Results (bias corrected)

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\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

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TT+pol

Results (bias corrected)

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\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

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\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

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Attach:LALmask_cosmo.png Δ Δ

astro nuicance

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calib

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Correlation matrix

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Convergence test

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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

Attach:fft_ada_no_lens_cosmo_po.png Δ Δ

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.

Metropolis algorithm

Adaptive Metropolis algorithm

A graphical confront between the two algorithms