Power Spectrum Estimation and Likelihood Function

Power Spectrum Estimation and Likelihood Function
Luca Pagano
May 12, 2015
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Power Spectrum Estimation
A scalar field ∆T (n) defined over the full sky can be decomposed in spherical harmonic
coefficients
Z
∗
a`m =
dn∆T (n)Y`m
(n),
(1)
with,
∆T (n) =
`
X X
a`m Y`m (n).
(2)
`>0 m=−`
Assuming ∆T Gaussian distributed,
and
where hC` i ≡
C`th
ha`m i = 0,
(3)
ha`m a`∗0 m0 i = δ``0 δmm0 hC` i,
(4)
is specified by the theory of primordial perturbations.
(5)
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Power Spectrum Estimation
An unbiased estimator of C`th is given by
C` =
`
X
1
|a`m |2 .
2` + 1 m=−`
(6)
C` -s are χ2ν -distributed with the mean equal to C`th , ν = 2` + 1 degrees of freedom (dof), and a
2
variance of 2C`th /ν.
In the case of non-fullsky CMB measurements a position dependent weighting W (n) must be
applied to the measured data
Z
1
dnW i (n)
fsky wi =
4π 4π
(7)
is the i-th moment of the arbitrary weighting scheme.
The window
function can also be expanded in spherical harmonics with the coefficients
R
∗ (n), and with a power spectrum
w`m = dnW (n)Y`m
2 w 2 and
for which W(` = 0) = 4πfsky
1
L UCA PAGANO
W` =
1 X
|w`m |2 ,
2` + 1 m
P
W(`)(2` + 1) = 4πfsky w2 .
`≥0
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(8)
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Power Spectrum Estimation
A sky temperature fluctuation map ∆T (n) on which a window W (n) is applied can be
decomposed in spherical harmonics coefficients
Z
∗
ã`m =
dn∆T (n)W (n)Y`m
(n)
X
∗
≈ Ωp
∆T (p)W (p)Y`m
(p),
(9)
(10)
p
where the integral over the sky is approximated by a discrete sum over the pixels that make the
map, with an individual surface area Ωp .
e can be defined as
The pseudo power spectrum C
e` =
C
L UCA PAGANO
`
X
1
|ã`m |2 .
2` + 1 m=−`
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Power Spectrum Estimation
e ` is clearly different from the full sky angular spectrum C` , but their
The pseudo power spectrum C
ensemble averages can be related by
X
e` i =
M``0 hC`0 i,
(12)
hC
`0
where M``0 describes the mode-mode coupling resulting from the cut sky. This kernel depends
only on the geometry of the cut sky and can be expressed simply in terms of the power spectrum
W` of the spatial window applied to the survey.
Starting from:
Z
ã`m
=
∗
dn∆T (n)W (n)Y`m
(n) =
X
Z
a`0 m0
∗
dnY`0 m0 (n)W (n)Y`m
(n) =
(13)
`0 m 0
=
X
a`0 m0 K`m`0 m0 [W ],
(14)
`0 m0
where the kernel K describes the mode-mode coupling resulting from the sky weighting.
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Power Spectrum Estimation
Note that ã is a linear combination of Gaussian variables and is therefore Gaussian as well, but
the ã`m -s are not independent.
P
If we use the series representaion of the window function, W (n) = `m w`m Y`m (n), the coupling
kernel reads
Z
K`1 m1 `2 m2 ≡
dnY`1 m1 (n)W (n)Y`∗2 m2 (n)
(15)
Z
X
=
w`3 m3
dnY`1 m1 (n)Y`3 m3 (n)Y`∗2 m2 (n)
`3 m3
=
X
w`3 m3 (−1)m2
`3 m3
×
`1
0
`2
0
(2`1 + 1)(2`2 + 1)(2`3 + 1)
4π
`3
`1
`2
`3
,
0
m1 −m2 m3
with
M`1 `2 =
2`2 + 1 X
`1
(2`3 + 1)W`3
0
4π
`
`2
0
`3
0
1/2
(16)
2
(17)
.
3
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Power Spectrum Estimation
The effect of the instrumental beam, experimental noise, and filtering of the TOD stream can be
included as follows
X
e` i =
e` i,
hC
M``0 F`0 B`20 hC`0 i + hN
(18)
`0
where B` is a window function describing the combined smoothing effects of the beam and finite
e` i is the average noise power spectrum, We seek the solution such that
pixel size, hN
hC` i = `(` + 1)hC` i/2π is a piece-wise constant. If we replace hC` i by Q`b Pbl 0 hC`0 i = Q`b hCb i
then
−1
e
e
hCb i = Kbb
(19)
0 Pb 0 ` hC` i − hN` i ,
where
Kbb0
L UCA PAGANO
=
Pb` K``0 Q`0 b0 , = Pb` M``0 F`0 B`20 Q`0 b0 .
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Likelihood function
It’s possible to evaluate the likelihood of the data for a given theoretical model exactly from the
temperature and polarization maps. The standard likelihood is given by
h
i
~ t (S + N)−1 m
~
exp − 12 m
~
dm
~ |S)d m
~ =
L(m
,
(20)
1/2
3n
|S + N|
(2π) p /2
~ is the data vector containing the data, S and N are the signal and noise covariance
where m
matrix respectively.
Computational prohibitive for high resolution maps In case of full sky observation and uncorrelated
noise we can re-write it as:
n
o
~ |C` ) = (2` + 1) Tr [Ĉ` C`−1 ] + ln |C` |
L(m
(21)
Issue: likelihood in case of cut sky and inomogeneus noise
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Likelihood function
To approximate the likelihood on the cut-sky, the usual approach when analysing the CMB
temperature is to develop a form for the log likelihood that is quadratic in some function of the C` ,
and hence can easily be generalized to the cut-sky using an estimate of the C` covariance matrix.
At large `, Eq. (21) is approximated by a symmetric Gaussian distribution where the variance is
determined by the estimators themselves:
2` + 1
2
"
− 2 ln LS (C` |Ĉ` ) =
2` + 1
2
"
−2 ln LQ (C` |Ĉ` ) =
Ĉ` − C`
#2
Ĉ`
.
(22)
.
(23)
Or by the theory:
Ĉ` − C`
C`
#2
This distribution is closer to the true likelihood. It is often somewhat misleadingly referred to as the
‘Gaussian approximation’, even though it does not have the determinant term required for
P(Ĉl |C` ) to be a normalized Gaussian distribution. Another possibility is
− 2 ln Lf (C` |Ĉ` ) =
2` + 1
2
"
Ĉ` − C`
Cf `
#2
= −2 ln LPlanck (C` |Ĉ` ),
(24)
where Cf ` is some fixed fiducial model assumed to be smooth and close to the model C` under
consideration. This is more interesting as although the shape of the likelihood is wrong at any `,
gives results consistent with the exact likelihood function.
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Likelihood function
Adding a C` -dependent determinant term to the quadratic approximation can also produce valid
results.
"
#2
2` + 1 Ĉ` − C`
−2 ln LD (C` |Ĉ` ) =
+ ln |C` |.
(25)
2
C`
Beyond these quadratic/Gaussian approximations, other approximations that have been used
include the log-normal distribution where the log-likelihood is quadratic in the log of the power
−2 ln LLN (C` |Ĉ` ) =
"
2` + 1
ln
2
Ĉ`
C`
!#2
(26)
.
This distribution is also somewhat biased: it only matches the exact full-sky result to second order
in Ĉl /C` − 1.
A weighted combination of the quadratic and the log-normal distributions can be a more accurate
approximation to the exact likelihood, being correct to third order in Ĉl /C` − 1. This approximation
was adopted in the analysis of WMAP data at high `:
ln LWMAP (C` |Ĉ` ) =
L UCA PAGANO
2
1
ln LQ (C` |Ĉ` ) + ln LLN (C` |Ĉ` ).
3
3
PSE AND L IKELIHOOD
(27)
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WMAP Likelihood function - high-`
Why is this likelihood third order correct?
Writing Ĉ` as C` (1 + ). We can expand the exact expression for the likelihood around its
maximum. Then, for a single multipole `
"
#
2
3
−2 ln LExact = (2` + 1) [ − ln(1 + )] ' (2` + 1)
−
+ O(4 )
2
3
We note that the Gaussian likelihood approximation is equivalent to the above expression
truncated at 2
" #
2
−2 ln LGauss = (2` + 1)
2
The expression for the log-normal likelihood approximation is:
"
#
2
3
−2 ln LLN = (2` + 1)
−
2
2
(28)
(29)
(30)
Thus, the WMAP approximation of the likelihood function is given by:
ln LWMAP (C` |Ĉ` ) =
L UCA PAGANO
1
2
ln LGauss (C` |Ĉ` ) + ln LLN (C` |Ĉ` ).
3
3
PSE AND L IKELIHOOD
(31)
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WMAP Likelihood function - low-`
Page,L. et al. (2007). Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Polarization Analysis. The Astrophysical Journal
Supplement Series, 170(2), 335–376.
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Panck Likelihood function - high-`
Again factorized in low-` and high-` parts:
where
Different masking for each channel. More aggressive for high frequencies.
Planck Collaboration. (2013). Planck 2013 results. XV. CMB power spectra and likelihood. Astronomy & Astrophysics, 571(0), 62.
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Panck Likelihood function - high-`
Masks. Point Sources and galactic
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Panck Likelihood function - high-`
Residual Foregrounds fitted at power spectrum level.
Taking into account:
Unresolved point sources
Clustered galaxies
Thermal Sunyaev-Zeodovich
Kinetic Sunyaev-Zeodovich
Cross-correlation between tSZ and Cosmic Infrared Background
Inter calibration parameters
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Panck Likelihood function - high-`
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Panck Likelihood function - high-`
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Panck Likelihood function - high-`
The Poisson point source contribution to the 143?217 spectrum
Kinetic SZ: the kSZ template used
Cross-spectra between frequencies νi and νj , the tSZ template
Cosmic infrared background
Thermal-SZ/CIB cross-correlation
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Panck Likelihood function - high-`
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Panck Likelihood function - high-`
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Panck Likelihood function - low-`
At low-` real space likelihood:
M = C + N where C is:
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Panck Spectrum
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Theoretical Power Spectrum
We are able now to assign to a given C` a likelihood.
How can we connect the theoretical C` to the cosmological parameters?
Using a Boltzmann code: Code for Anisotropies in the Microwave Background (CAMB)
http://camb.info/readme.html
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Esercizio
Stimare l’ampiezza dello spettro di potenza AS
Dati:
http://irsa.ipac.caltech.edu/data/Planck/release_1/ancillary-data/
previews/COM_PowerSpect_CMB_R1.10/COM_PowerSpect_CMB_R1.10.txt
Generare uno spettro di potenza:
http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm
In realtà già tutto fatto, trovate dati e spettro qui:
https://dl.dropboxusercontent.com/u/17494978/
Stimare il parametro in approssimazione di likelihood uncorrelated Gaussian
Media e intervalli di confidenza
Ampiezza parametro lineare
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Metodi per il best-fit e Stima delle incertezze
Diversi metodi
Local Minimization: Partendo da un modello si utilizza un algoritmo di minimizzazione
iterativa (Levenberg-Marquardt).
Simulated Annealing: Migliora l’algoritmo di minimizzazione introducendo perturbazioni
random che vengono gradualmente raffreddate
Genetic Algoritms: Ottimizzazione di algoritmi di evoluzione biologica
Dobbiamo descrivere le incertezze sulla stima dei parametri del modello
Discuteremo nel dettaglio due metodi
Fit dei dati simulati
MCMC, algoritmo Metropolis-Hastings
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Teorema di Bayes e applicazione
Il teorema di Bayes ci dice che
p(~
d|~x )p(~x )
p(~x |~
d) = R
~
p(d|~x )p(~x )d ~x
(32)
Scegliendo una prior piatta p(~x ) ∼ 1 si ottiene che la posterior coincide con la likelihood
p(~
d|~x ) ∼ exp(−χ2 (~x )/2)
(33)
Si utlizza il Monte Carlo Markov Chain
MCMC è un metodo efficiente per analisi Bayessiana
E’ particolarmente indicato nel caso di problemi a molti parametri, in quanto è più
performante del semplice campionamento a step
E’ veloce e affidabile
Si basa sulla costruzione di una catena di stati randomizzati che per un grande numero di
step converge alla distribuzione di probabilità
L’algoritmo più utilizzato è il Metropolis-Hastings
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L’algoritmo Metropolis-Hastings
L’algoritmo Metropolis-Hastings puù essere implementato nel seguente modo:
1
2
3
4
5
Inizializzare la catena in ~
x0 a n = 0
Proporre un nuovo stato ~
x 0 che avrà come probabilità di transizione q(~
x 0 |~
xn )
Calcolare f (~
x 0 )/f (~
xn ) ∼ exp(−[χ2 (~
x 0 ) − χ2 (~
xn )]/2)
Generare un numero random,
distribuzione
uniforme tra 0 e 1
n 0u, da una
o
f (~
x )q(~
x |~
x0)
Se u ≤ α(~
x 0 |~
xn )) ≡ min f (~x )q(~xn0 |~x ) , 1 allora ~
xn+1 = ~
x 0 altrimenti ~
xn+1 = ~
xn
n
6
7
n
n =n+1
Ricominciare dal passo 2
La principale complicazione pratica di questo metodo è la scelta di q(~x 0 |~x )
Una scelta sbagliata può portare ad una lenta convergenza del metodo
La scelta più efficiente possibile sarebbe la posterior p(~x 0 |~
d) stessa
La scelta più comune per q(~x 0 |~x ) è una Gaussiana centrata in ~x
Per incrementare l’efficienza e l’esplorazione dello spazio dei parametri, si può modificare il
codice in modo da cambiare solo uno o due parametri più correlati in uno step
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Parameter Estimation - Real World
Explore the parameter space
Public code Cosmological MonteCarlo (CosmoMC)
CosmoMC is a Fortran 2008 Markov-Chain Monte-Carlo (MCMC) engine for exploring
cosmological parameter space http://cosmologist.info/cosmomc/
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Parameter Estimation - Real World
Parametrization
Convergence diagnostics using getdist
Data:
CMB
SN
BAO
MPK
HST
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External Datasets
CMB:
Ground: BICEP, ACT, SPT...
Balloon: Boomerang, MAXIMA, Spider, EBEX...
All small scales observation
Astrophysical:
SN Ia: Several data collection
Observable: Luminosity distance
BAO: SDSS, BOSS, 2dF
Observable: periodic fluctuations in the density of the visible baryonic matter of the universe
as "standard ruler" for length scale in cosmology. Angular diameter distance
MPK: SDSS, 2dF
Observable: Matter power spectrum
HST: Cepheids measurements
Observable: Luminosity distance
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