Reconstructing stellar DEM and metallicity using high ...hea- · 15/06/2010 · Gibbs in a nutshell • Initiating parameters ... Abundance trace 0 2000 6000 2.3 2.5 2.7 2.9 Element

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


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.


Previously estimated Capella DEM


log10(DEM) used for simulations

●●●●●

●●

●●●●

●

●●

●

●

●

●●

●●●●●

●●

●

●

●

●

●

●

●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

6.5 7.0 7.5 8.0

16.0

17.0

18.0

log10(DEM) used for simulations (16 nodes)

log10(T)

log1

0(D

EM

)

●

●

●

●

●

●

●

●

●

●●

● ● ● ● ●

6.5 7.0 7.5 8.0

16.0

17.0

18.0

log10(DEM) used for simulations (16 nodes)

log10(T)

log1

0(D

EM

)


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


Estimated DEM

●●

●

●

●

●

●

●● ● ● ● ● ● ● ●

6.5 7.0 7.5 8.0

0e+

001e

+18

2e+

183e

+18

4e+

18

Stronger smoothing

log10(T)

DE

M

●●

●

●

●

●

●

●

● ● ● ● ● ● ● ●● ●

●

●

●

●

●

●

● ● ● ● ● ● ● ●● ●

●

●

●

●

●

●

● ● ● ● ● ● ● ●

●●

●

●

●

●

●

●● ● ● ● ● ● ● ●

6.5 7.0 7.5 8.0

0e+

001e

+18

2e+

183e

+18

4e+

18

No smoothing

log10(T)

DE

M

● ●●

●

●

●

●

●

● ● ● ● ● ● ● ●

●●

●

●

●

●

●

●● ● ● ● ● ● ● ●

●●

●

●

●

●

●

●● ● ● ● ● ● ● ●


Estimated log10(DEM)

●

●

●

●

●

●

●

●

●

●

●● ●

●● ●

6.5 7.0 7.5 8.0

16.5

17.0

17.5

18.0

18.5

Stronger smoothing

log10(T)

log1

0(D

EM

)●

●

●

●

●

●

●

●

● ● ● ●● ● ● ●

● ●

●

●

●

●

●

●

● ●● ●

● ●● ●

●

●

●

●

●

●

●

●

● ● ● ● ● ● ● ●

●

●

●

●

●

●

●

●

●

●

●● ●

●● ●

6.5 7.0 7.5 8.0

16.5

17.0

17.5

18.0

18.5

No smoothing

log10(T)

log1

0(D

EM

)

● ●

●

●

●

●

●

●

●

●● ●

●● ● ●

●

●

●

●

●

●

●

●

●

●

●● ● ●

●●

●

●

●

●

●

●

●

●

●

●

●● ● ●

●●


Estimated abundance

01

23

4

Stronger smoothing

Elements

Abu

ndan

ce

C N O Ne Mg Al Si S Ar Ca Fe

●

●

●

●

●

●●

●

●

●

●

01

23

4

No smoothing

Elements

Abu

ndan

ce

C N O Ne Mg Al Si S Ar Ca Fe

●

●

●

●

●

●●

●

●

●

●


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

●

●

●

●

●

●●

●

●

●

●

●

● ● ●

0.0

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

●

●

●

●

●

●●

●

●

●

●

●

● ● ●


Estimated rho

●

● ●

●

●

●

●

●

●●

● ● ● ● ● ●

6.0 6.5 7.0 7.5 8.0

0e+

002e

+18

4e+

18

Real data

log10(T)

DE

M

●

●

●

●

●

●

●

●

●

●●

●

0.0

0.4

0.8

Real data

Elements

Abu

ndan

ce

C N O Ne Mg Al Si S Ar Ca Fe Ni

●

●

●

●

●

●

●

●

●

●●

●


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


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


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


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


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


Mu total trace

0 2000 4000 6000 8000

2820

0028

6000

2900

00

Mu total, R = 1

Index

DE

M


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


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)




i

p(Yobsi |µ,γ),

where


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)




i

p(Yobsi |µ,γ),

where


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)


Parametrization of the multiscale smoothing


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

!





p(µ0,ρ|γ,Y)∝ p(µ0)∏

r,np(ρr,n)

∏

i

p(Yi|ρ,µ0,γ).



T∑

t=1

hi,tµt

!

= Pois

µ0

T∑

t=1

hi,tqt,R

!

= Pois�

siµ0�



αµ+∑

i

Yi

!

/

βµ+∑

i

si

!





p(µ0,ρ|γ,Y)∝ p(µ0)∏

r,np(ρr,n)

∏

i

p(Yi|ρ,µ0,γ).



T∑

t=1

hi,tµt

!

= Pois

µ0

T∑

t=1

hi,tqt,R

!

= Pois�

siµ0�



αµ+∑

i

Yi

!

/

βµ+∑

i

si

!





p(µ0,ρ|γ,Y)∝ p(µ0)∏

r,np(ρr,n)

∏

i

p(Yi|ρ,µ0,γ).



T∑

t=1

hi,tµt

!

= Pois

µ0

T∑

t=1

hi,tqt,R

!

= Pois�

siµ0�



αµ+∑

i

Yi

!

/

βµ+∑

i

si

!


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))


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))


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?


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







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






Documents

Reconstructing stellar DEM and metallicity using high ...hea- · 15/06/2010 · Gibbs in a nutshell • Initiating parameters ... Abundance trace 0 2000 6000 2.3 2.5 2.7 2.9 Element