Reconstructing stellar DEM and metallicity using high ...hea- · 15/06/2010 · Gibbs in a...
Transcript of Reconstructing stellar DEM and metallicity using high ...hea- · 15/06/2010 · Gibbs in a...
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Reconstructing stellar DEM and metallicityusing high-resolution X-ray Spectra
Victoria LiublinskaJoint work with CHASC: X.-L. Meng, V. Kashyap, D. van Dyk
Harvard University
June 15, 2010
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
The model
Let Yobsi ∼ Pois(ξi), where ξi is photon intensity in wavelength
channel i, i= 1 . . . I and
ξ= ξsource+ ξbkg =MD
λC +∑
k
λLk
!
+ ξbkg
=MD
GC +∑
k
γkGLk
!
µ+ ξbkg,
Parameters are γ and µ.
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Previously estimated Capella DEM
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
log10(DEM) used for simulations
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log10(DEM) used for simulations (16 nodes)
log10(T)
log1
0(D
EM
)
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log10(DEM) used for simulations (16 nodes)
log10(T)
log1
0(D
EM
)
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Other properties of current simulations
• Includes all censoring, adds informative prior to abundances
• 16 nodes and some smoothing (αρ=20): Depending on thestarting point EM converges in 200-500 iterations
• 16 nodes and some smoothing (αρ=2): EM converges in800-1100 iterations
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Estimated DEM
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6.5 7.0 7.5 8.0
0e+
001e
+18
2e+
183e
+18
4e+
18
Stronger smoothing
log10(T)
DE
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6.5 7.0 7.5 8.0
0e+
001e
+18
2e+
183e
+18
4e+
18
No smoothing
log10(T)
DE
M
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EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Estimated log10(DEM)
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Stronger smoothing
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log1
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EM
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No smoothing
log10(T)
log1
0(D
EM
)
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EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Estimated abundance
01
23
4
Stronger smoothing
Elements
Abu
ndan
ce
C N O Ne Mg Al Si S Ar Ca Fe
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01
23
4
No smoothing
Elements
Abu
ndan
ce
C N O Ne Mg Al Si S Ar Ca Fe
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●
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Estimated rho
0.0
0.2
0.4
0.6
0.8
1.0
Stronger smoothing
Multiscale levels
Rho
(sp
littin
g fa
ctor
s)
1 2 2 3 3 3 3 4 4 4 4 4 4 4 4
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0.2
0.4
0.6
0.8
1.0
No smoothing
Multiscale levels
Rho
(sp
littin
g fa
ctor
s)
1 2 2 3 3 3 3 4 4 4 4 4 4 4 4
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EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Estimated rho
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6.0 6.5 7.0 7.5 8.0
0e+
002e
+18
4e+
18
Real data
log10(T)
DE
M
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0.0
0.4
0.8
Real data
Elements
Abu
ndan
ce
C N O Ne Mg Al Si S Ar Ca Fe Ni
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EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Gibbs in a nutshell
• Initiating parameters γ,µ
• 5 steps of sampling different missing data
• Updating abundance γ(t+1)|µ(t),γ(t),Ymis
• 3 more steps of sampling missing data
• Updating (scaled) DEM µ̃(t+1)|Ymis, where µ̃= f(γ)µ• Rescale DEM µ(t+1) = µ̃(t+1)/f
�
γ(t+1)�
• Properties of a current simulation: minimal smoothing(αρ=2), 16 nodes for log10(T), 10000 simulations including1000 of burn-in
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Gibbs in a nutshell
• Initiating parameters γ,µ
• 5 steps of sampling different missing data
• Updating abundance γ(t+1)|µ(t),γ(t),Ymis
• 3 more steps of sampling missing data
• Updating (scaled) DEM µ̃(t+1)|Ymis, where µ̃= f(γ)µ• Rescale DEM µ(t+1) = µ̃(t+1)/f
�
γ(t+1)�
• Properties of a current simulation: minimal smoothing(αρ=2), 16 nodes for log10(T), 10000 simulations including1000 of burn-in
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Rho trace
0 2000 4000 6000 8000
0.94
0.96
0.98
1.00
Scale 1, R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 4000 6000 8000
0.12
0.18
Scale 2, R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 4000 6000 80000.
20.
6
Scale 2, R = 1.04
Index
Mul
tisca
le s
moo
thin
g le
vels
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Rho trace
0 2000 4000 6000 8000
0.20
0.40
Scale 3, R = 1.07
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 4000 6000 8000
0.66
0.72
Scale 3, R = 1.02
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 4000 6000 8000
0.2
0.6
Scale 3, R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 4000 6000 8000
0.0
0.4
0.8
Scale 3, R = 1.08
Index
Mul
tisca
le s
moo
thin
g le
vels
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Rho trace
0 2000 6000
0.0
0.4
0.8
Scale 4, R = 1.28
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.2
0.3
0.4
0.5
0.6
Scale 4, R = 1.09
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.15
0.25
Scale 4, R = 1.04
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.80
0.85
0.90
0.95
Scale 4, R = 1.04
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.2
0.4
0.6
0.8
1.0
Scale 4, R = 1.05
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.2
0.4
0.6
0.8
1.0
Scale 4, R = 1.06
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.2
0.4
0.6
0.8
1.0
Scale 4, R = 1.01
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.0
0.4
0.8
Scale 4, R = 1.08
Index
Mul
tisca
le s
moo
thin
g le
vels
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Mu total trace
0 2000 4000 6000 8000
2820
0028
6000
2900
00
Mu total, R = 1
Index
DE
M
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Abundance trace
0 2000 6000
2.3
2.5
2.7
2.9
Element N , R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
0.44
0.50
0.56
Element O , R = 1.01
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
1.95
2.10
2.25
Element Ne , R = 1.01
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
1.4
1.6
Element Mg , R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
1.0
2.0
Element Al , R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
2.5
3.0
3.5
4.0
Element S , R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
1.4
1.8
2.2
Element Ar , R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
0 2000 6000
1.00
1.10
Element Fe , R = 1
Index
Mul
tisca
le s
moo
thin
g le
vels
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Posterior distribution of µ|γ, Yobs
p(µ|γ, Yobs)∝ p(µ)∏
i
p(Yobsi |µ,γ),
where
Yobsi |µ,γ∼ Pois
J∑
j=1
Mi,jdj
T∑
t=1
GCj,tµt+
T∑
t=1
∑
k
γkGLk,j,tµt
!
+ ξbkgi
= Pois
T∑
t=1
hi,tµt+ ξbkgi
!
,
with µ1 = µ0∏R
r=1ρr,0, µT = µ0∏R
r=1(1−ρr,2r−1) and each µt is aproduct of µ0 and R splitting factors, either ρr,n or (1−ρr,n). Thefollowing example clarifies the representation (R=4)
µ12 = µ0q12,R = µ0(1−ρ0,0)q12,R−1 = µ0(1−ρ0,0)ρ1,1q12,R−2
= · · ·= µ0(1−ρ0,0)ρ1,1(1−ρ2,3)(1−ρ2,6)
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Posterior distribution of µ|γ, Yobs
p(µ|γ, Yobs)∝ p(µ)∏
i
p(Yobsi |µ,γ),
where
Yobsi |µ,γ∼ Pois
J∑
j=1
Mi,jdj
T∑
t=1
GCj,tµt+
T∑
t=1
∑
k
γkGLk,j,tµt
!
+ ξbkgi
= Pois
T∑
t=1
hi,tµt+ ξbkgi
!
,
with µ1 = µ0∏R
r=1ρr,0, µT = µ0∏R
r=1(1−ρr,2r−1) and each µt is aproduct of µ0 and R splitting factors, either ρr,n or (1−ρr,n).
Thefollowing example clarifies the representation (R=4)
µ12 = µ0q12,R = µ0(1−ρ0,0)q12,R−1 = µ0(1−ρ0,0)ρ1,1q12,R−2
= · · ·= µ0(1−ρ0,0)ρ1,1(1−ρ2,3)(1−ρ2,6)
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Posterior distribution of µ|γ, Yobs
p(µ|γ, Yobs)∝ p(µ)∏
i
p(Yobsi |µ,γ),
where
Yobsi |µ,γ∼ Pois
J∑
j=1
Mi,jdj
T∑
t=1
GCj,tµt+
T∑
t=1
∑
k
γkGLk,j,tµt
!
+ ξbkgi
= Pois
T∑
t=1
hi,tµt+ ξbkgi
!
,
with µ1 = µ0∏R
r=1ρr,0, µT = µ0∏R
r=1(1−ρr,2r−1) and each µt is aproduct of µ0 and R splitting factors, either ρr,n or (1−ρr,n). Thefollowing example clarifies the representation (R=4)
µ12 = µ0q12,R = µ0(1−ρ0,0)q12,R−1 = µ0(1−ρ0,0)ρ1,1q12,R−2
= · · ·= µ0(1−ρ0,0)ρ1,1(1−ρ2,3)(1−ρ2,6)
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Parametrization of the multiscale smoothing
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Posterior distribution of (µ0,ρ)|γ,Yobs
• µ0 ∼ Gamma(αµ)/βµ, where flat prior would correspond toαµ = 1 and βµ = 0• ρr,n ∼ Beta(αρ,αρ), r= 0, . . . , R− 1 and n= 0, . . . , 2r− 1• Instead of working with Yobs we can also use Y (counts free
from the background contamination)
p(µ0,ρ|γ,Y)∝ p(µ0)∏
r,np(ρr,n)
∏
i
p(Yi|ρ,µ0,γ).
As it was noted previously
Yi|µ0,ρ,γ∼ Pois
T∑
t=1
hi,tµt
!
= Pois
µ0
T∑
t=1
hi,tqt,R
!
= Pois�
siµ0�
Therefore, µ0 can be updated in closed form without any missingdata imputation
µ0|ρ,Y,γ∼ Gamma
αµ+∑
i
Yi
!
/
βµ+∑
i
si
!
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Posterior distribution of (µ0,ρ)|γ,Yobs
• µ0 ∼ Gamma(αµ)/βµ, where flat prior would correspond toαµ = 1 and βµ = 0• ρr,n ∼ Beta(αρ,αρ), r= 0, . . . , R− 1 and n= 0, . . . , 2r− 1• Instead of working with Yobs we can also use Y (counts free
from the background contamination)
p(µ0,ρ|γ,Y)∝ p(µ0)∏
r,np(ρr,n)
∏
i
p(Yi|ρ,µ0,γ).
As it was noted previously
Yi|µ0,ρ,γ∼ Pois
T∑
t=1
hi,tµt
!
= Pois
µ0
T∑
t=1
hi,tqt,R
!
= Pois�
siµ0�
Therefore, µ0 can be updated in closed form without any missingdata imputation
µ0|ρ,Y,γ∼ Gamma
αµ+∑
i
Yi
!
/
βµ+∑
i
si
!
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Posterior distribution of (µ0,ρ)|γ,Yobs
• µ0 ∼ Gamma(αµ)/βµ, where flat prior would correspond toαµ = 1 and βµ = 0• ρr,n ∼ Beta(αρ,αρ), r= 0, . . . , R− 1 and n= 0, . . . , 2r− 1• Instead of working with Yobs we can also use Y (counts free
from the background contamination)
p(µ0,ρ|γ,Y)∝ p(µ0)∏
r,np(ρr,n)
∏
i
p(Yi|ρ,µ0,γ).
As it was noted previously
Yi|µ0,ρ,γ∼ Pois
T∑
t=1
hi,tµt
!
= Pois
µ0
T∑
t=1
hi,tqt,R
!
= Pois�
siµ0�
Therefore, µ0 can be updated in closed form without any missingdata imputation
µ0|ρ,Y,γ∼ Gamma
αµ+∑
i
Yi
!
/
βµ+∑
i
si
!
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Posterior distribution of (µ0,ρ)|γ,Yobs
• µ0 ∼ Gamma(αµ)/βµ, where flat prior would correspond toαµ = 1 and βµ = 0• ρr,n ∼ Beta(αρ,αρ), r= 0, . . . , R− 1 and n= 0, . . . , 2r− 1• Instead of working with Yobs we can also use Y (counts free
from the background contamination)
p(µ0,ρ|γ,Y)∝ p(µ0)∏
r,np(ρr,n)
∏
i
p(Yi|ρ,µ0,γ).
As it was noted previously
Yi|µ0,ρ,γ∼ Pois
T∑
t=1
hi,tµt
!
= Pois
µ0
T∑
t=1
hi,tqt,R
!
= Pois�
siµ0�
Therefore, µ0 can be updated in closed form without any missingdata imputation
µ0|ρ,Y,γ∼ Gamma
αµ+∑
i
Yi
!
/
βµ+∑
i
si
!
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Updating splitting factorsSince ρr,n ∼ Beta(αρ,αρ) and
p(ρ0,0|ρ−,µ0,γ,Y)∝ p(ρ0,0)∏
i
p(Yi|ρ,µ0,γ).
The distribution of background-free counts can be represented as
Yi|µ0,ρ,γ= Pois
T/2∑
t=1
hi,tµ0ρ0,0qt,R−1+T∑
t=T/2+1
hi,tµ0(1−ρ0,0)qt,R−1
= Pois�
si,1ρ0,0+ si,2(1−ρ0,0)�
,
then the conditional distribution of ρ0,0|ρ−,µ0,Y,γ has thefollowing form
ραρ−10,0 (1−ρ0,0)
αρ−1∏
i
(si,1ρ0,0+ si,2(1−ρ0,0))Yie−(si,1ρ0,0+si,2(1−ρ0,0))
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Updating splitting factorsSince ρr,n ∼ Beta(αρ,αρ) and
p(ρ0,0|ρ−,µ0,γ,Y)∝ p(ρ0,0)∏
i
p(Yi|ρ,µ0,γ).
The distribution of background-free counts can be represented as
Yi|µ0,ρ,γ= Pois
T/2∑
t=1
hi,tµ0ρ0,0qt,R−1+T∑
t=T/2+1
hi,tµ0(1−ρ0,0)qt,R−1
= Pois�
si,1ρ0,0+ si,2(1−ρ0,0)�
,
then the conditional distribution of ρ0,0|ρ−,µ0,Y,γ has thefollowing form
ραρ−10,0 (1−ρ0,0)
αρ−1∏
i
(si,1ρ0,0+ si,2(1−ρ0,0))Yie−(si,1ρ0,0+si,2(1−ρ0,0))
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Updating splitting factorsPossible idea: update splitting factors by scale level. The numberof simultaneously updated factors would be 1, 2, 4, 8, 16,etc.
For example letηi = si,2,1ρ1,0+ si,2,2(1−ρ1,0) + si,2,3ρ1,1+ si,2,4(1−ρ1,1),then p(ρ1,0,ρ1,1|ρ−,µ0,γ,Y)∝ραρ−11,0 (1−ρ1,0)
αρ−1ραρ−11,1 (1−ρ1,1)
αρ−1∏
iηYii e−ηi
Implementation questions:
• Which proposal is the best? Truncated MVN (or MVt), others?OR simply evaluate the posterior and sample from the grid(since each ρr,n < 1)?
• Other updating scheme? One-by-one, in pairs etc.
• Update µ directly? (the posterior is not that simple since eachprior on ρr,n contains all µ1, . . . ,µT)
• Alternate this scheme with full augmentation?
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Updating splitting factorsPossible idea: update splitting factors by scale level. The numberof simultaneously updated factors would be 1, 2, 4, 8, 16,etc.For example letηi = si,2,1ρ1,0+ si,2,2(1−ρ1,0) + si,2,3ρ1,1+ si,2,4(1−ρ1,1),then p(ρ1,0,ρ1,1|ρ−,µ0,γ,Y)∝ραρ−11,0 (1−ρ1,0)
αρ−1ραρ−11,1 (1−ρ1,1)
αρ−1∏
iηYii e−ηi
Implementation questions:
• Which proposal is the best? Truncated MVN (or MVt), others?OR simply evaluate the posterior and sample from the grid(since each ρr,n < 1)?
• Other updating scheme? One-by-one, in pairs etc.
• Update µ directly? (the posterior is not that simple since eachprior on ρr,n contains all µ1, . . . ,µT)
• Alternate this scheme with full augmentation?
EM results from the simulation EM results on a real data Gibbs results from the simulation Alternative sampling of the DEM
Updating splitting factorsPossible idea: update splitting factors by scale level. The numberof simultaneously updated factors would be 1, 2, 4, 8, 16,etc.For example letηi = si,2,1ρ1,0+ si,2,2(1−ρ1,0) + si,2,3ρ1,1+ si,2,4(1−ρ1,1),then p(ρ1,0,ρ1,1|ρ−,µ0,γ,Y)∝ραρ−11,0 (1−ρ1,0)
αρ−1ραρ−11,1 (1−ρ1,1)
αρ−1∏
iηYii e−ηi
Implementation questions:
• Which proposal is the best? Truncated MVN (or MVt), others?OR simply evaluate the posterior and sample from the grid(since each ρr,n < 1)?
• Other updating scheme? One-by-one, in pairs etc.
• Update µ directly? (the posterior is not that simple since eachprior on ρr,n contains all µ1, . . . ,µT)
• Alternate this scheme with full augmentation?