Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hea-www.harvard.edu/AstroStat/Stat310_1112/jx_20120207.pdf Äàòà èçìåíåíèÿ: Tue Feb 7 22:59:48 2012 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 04:18:55 2012 Êîäèðîâêà:

Outline Background Problem description Methodology Research New Results Two concerns

New Results of Fully Bayesian
JIN XU
UCI

February 7, 2012

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Background Problem description Calibration Samples Methodology Research Principle Component Analysis Model Building Three source parameter sampling schemes New Results Simulation Quasar data sets Two concerns New data sets Applying wavelets to replace PCA
JIN XU New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Background

High-Energy Astrophysics Spectral Analysis Calibration Products Scientific Goals

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

High-Energy Astrophysics

Provide understanding into high-energy regions of the Universe. Chandra X-ray Observatory is designed to observe X-rays from high-energy regions of the Universe. X-ray detectors typically count a small number of photons in each of a large number of pixels. Spectral Analysis aims to explore the parameterized pattern between the photon counts and energy.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

An Example of One Dataset
TITLE = EXTENDED EMISSION AROUND A GIGAHERTZ PEAKED RADIO SOURCE DATE = 2006-12-29 T 16:10:48

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Calibration Uncertainty
Effective area records sensitivity as a function of energy. Energy redistribution matrix can vary with energy/location. Point Spread Functions can vary with energy and location.
800 0 ACIS-S effective area (cm2) 200 400 600

0.2

1 E [keV]

10

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Incorporate Calibration Uncertainty

Calibration Uncertainty in astronomical analysis have been generally ignored. No robust principled method is available. Our goal is to incorporate the uncertainty by Bayesian Methods. In this talk, we focus on uncertainty in the effective area.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Calibration Samples

Two Main Problems

The true effective area curve can't be observed, when we try to incorporate calibration uncertainty in estimating source parameters. We don't have parameterized form for effective area curve. It makes sampling hard to approach.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Calibration Samples

Generating Calibration Samples

Drake et al. (2006), suggests to generate calibration samples of effective area curves to represent the uncertainty. Calibration Samples: {A1 , A2 , A3 , ..., AL }

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Principle Component Analysis Mo del Building Three source parameter sampling schemes

Three Main Steps

Use Principle Component Analysis to parameterize effective area curve. Model Building, that it combining source model with calibration uncertainty. Three source parameter sampling schemes.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Principle Component Analysis Mo del Building Three source parameter sampling schemes

Use PCA to represent effective area curve

¯ A = A0 + + A0 : default effective area, ¯ : mean deviation from A0 ,

m j =1 ej rj vj

rj and vj : first m principle component eigenvalues & vectors, ej : independent standard normal deviations. Capture 95% of uncertainty with m = 6 - 9.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Principle Component Analysis Mo del Building Three source parameter sampling schemes

Use PCA to represent effective area curve
PCA method has nicely parameterized effective area curve.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Principle Component Analysis Mo del Building Three source parameter sampling schemes

A simplified model of telescope response, only concerning effective area uncertainty
M (E ; ) = S (E ; ) A(E ) (E ; ): Observed Photon Distribution, S (E ; ): True Source Model, we set it as poisson distribution with expectation equal to exp (-nH sigma(E )) Amp E (-gamma) + bkg A(E ): Effective Area Curve. : source parameter, = {nH , Amp , gamma, bkg }
JIN XU New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Principle Component Analysis Mo del Building Three source parameter sampling schemes

Scheme One: Fixed Effective Area Curved

We assume A = A0 , where A0 is the default affective area curve, and may not be the true one, This scheme doesn't incorporate any calibration uncertainty, The estimation may be biased and error bars may be underestimated. Only one sampling step involved: p (|M , A0 ) L(M |, A0 )p (A0 )

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Principle Component Analysis Mo del Building Three source parameter sampling schemes

Scheme Two: Pragmatic Bayesian, Lee et al(2011, Apj)

Main purpose is to reduce complexity of sampling. This scheme "completely" incorporates the calibration uncertainty, Step One: sample A from p (A) Step Two: sample from p (|M , A) L(M |, A)p ()

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Principle Component Analysis Mo del Building Three source parameter sampling schemes

Scheme Three: Fully Bayesian

Use correct Bayesian Approach, This scheme concerns about letting the current data influence calibration products, Step One: sample A from p (A|M , ) L(M |, A)p (A) Step Two: sample from p (|M , A) L(M |, A)p () Most difficult approach to sample.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Eight simulated data sets
The first four data sets were all simulated without background contamination using the XSPEC model wabs*powerlaw, nominal default effective area A0 from the calibration sample of Drake et al. (2006), and a default RMF for ACIS-S. Simulation 1: = 2,NH = 223 cm Simulation 2: = 1,NH = 221 cm Simulation 3: = 2,NH = 223 cm Simulation 4: = 1,NH = 221 cm
-2 -2 -2 -2

, and 105 counts; , and 105 counts; , and 104 counts; , and 104 counts;

The other four data sets (Simulation 5-8) were generated using an extreme instance of an effective area.
JIN XU New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 1

[SIM 1] NH=1023; =2; N=10

5

5

10

15

fix ARF fully bayesian pragmatic bayesian

0

1.6

1.8

2.0

2.2

2.4

2.6

2.8

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 2

[SIM 2] NH=1021; =1; N=10

5

20

30

40

10

fix ARF fully bayesian pragmatic bayesian

0
0.8

0.9

1.0

1.1

1.2

1.3

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 3

[SIM 3] NH=1023; =2; N=10

4

2

3

4

fix ARF

1

fully bayesian pragmatic bayesian

0

1.6

1.8

2.0

2.2

2.4

2.6

2.8

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 4

[SIM 4] NH=1021; =1; N=10

4

5

10

15

fix ARF fully bayesian pragmatic bayesian

0
0.8

0.9

1.0

1.1

1.2

1.3

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 5

[SIM 5] NH=1023; =2; N=10

5

5

10

15

fix ARF fully bayesian pragmatic bayesian

0

1.6

1.8

2.0

2.2

2.4

2.6

2.8

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 6

30

[SIM 6] NH=1021; =1; N=10

5

10

15

20

25

fix ARF

5

fully bayesian pragmatic bayesian

0
0.8

0.9

1.0

1.1

1.2

1.3

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 7

2

3

4

5

[SIM 7] NH=1023; =2; N=10

4

fix ARF

1

fully bayesian pragmatic bayesian

0

1.6

1.8

2.0

2.2

2.4

2.6

2.8

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Results for Simulation 8

[SIM 8] NH=1021; =1; N=10

4

5

10

15

fix ARF fully bayesian pragmatic bayesian

0
0.8

0.9

1.0

1.1

1.2

1.3

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

Quasar results

16 Quasar data sets were fit by these three models: 377, 836, 866, 1602, 3055, 3056, 3097, 3098, 3100, 3101, 3102, 3103, 3104, 3105, 3106, 3107. Most interesting founding for fully bayesian model is shift of parameter fitting, besides the change of standard errors. Both comparisons of mean and standard errors among three models are shown below.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

mean: fix-prag

2.5

q

q

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

q

2.0

q q q

µp

rag

()

0.0
0.0

0.5

1.0

1.5

0.5

1.0

1.5

2.0

2.5

JIN XU

µfix() Results of Fully Bayesian New


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

mean: fix-full

2.5

q

µfull(

q

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

q

2.0

q q q

)
0.0 0.5 1.0

1.5
0.0

0.5

1.0

1.5

2.0

2.5

JIN XU

µfix() Results of Fully Bayesian New


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

mean: prag-full

2.5

q

µfull(

q

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

q

2.0

q q q

)
0.0 0.5 1.0

1.5
0.0

0.5

1.0

1.5 µp
rag

2.0

2.5

JIN XU

() New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

sd: fix-prag

1.00

q

q

0.05

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

0.50

q q

()

prag

0.10

0.20

q q

0.01
0.01

0.02

0.02

0.05

0.10

0.20

0.50

1.00

JIN XU

fix() Results of Fully Bayesian New


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

sd: fix-full

1.00

q

q

0.05

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

0.50

q q

full(

)
0.10

0.20

q q

0.01
0.01

0.02

0.02

0.05

0.10

0.20

0.50

1.00

JIN XU

fix() Results of Fully Bayesian New


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

sd: prag-full

1.00

q

q

0.05

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

0.50

q q

full(

)
0.10

0.20

q q

0.01
0.01

0.02

0.02

0.05

0.10
prag

0.20

0.50

1.00

JIN XU

() New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

more plots
µ ^
prag

() =

µprag ()-µfix () fix ()
0.50

, these lines cover 2 sd.

q

fix(

q

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

q

q

0.05

0.10

)

0.20

q

q

0.02

-4

-2

0 ^ µp
rag

2

4

JIN XU

New Results of Fully Bayesian

()


Outline Background Problem description Methodology Research New Results Two concerns

Simulation Quasar data sets

more plots
µ ^
full

() =

µfull ()-µfix () fix ()
0.50

, these lines cover 2 sd.

q

fix(

q

q

q

377 836 866 1602 3055 3056 3097 3098 3100 3101 3102 3103 3104 3105 3106 3107

q

q

0.05

0.10

)

0.20

q

q

0.02

-4

-2

0 ^ µfull(

2

4

JIN XU

) New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

New data sets Applying wavelets to replace PCA

Data set 1878
model: xsphabs.abs1*(xsapec.kT1+xsapec.kT2)
617138 0.250 a1878[, 1] a1878[, 2] a1878[, 3]
0 2000 Index 4000

617134

0.240

617130

0.230

0

2000 Index

4000

0.074
0

0.078

0.082

0.086

2000 Index

4000

0.320

0.50

a1878[, 4]

a1878[, 5]

0.315

a1878[, 6]
0 2000 Index 4000

0.40

0.310

0

2000 Index

4000

0.305

0.0165
0

0.30

0.0175

0.0185

2000 Index

4000

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

New data sets Applying wavelets to replace PCA

Data set 1878

model: xsphabs.abs1*(xsapec.kT1+xsapec.kT2) even for fixed arf model, the results are not good; try to add one proportion parameter, and add data augmentation sampler to the code; till now, only one naive simulation has been done so far.

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

New data sets Applying wavelets to replace PCA

Discrete wavelet transformation (DWT) for quiet.arf

T-2V6

T-3W6 T-3 T-3 T-3 T-2 T-2 W5 W4 W3 W W2
1

y 0 200

500

X 0 200 400

t

600

800

1000

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

New data sets Applying wavelets to replace PCA

Discrete wavelet transformation (DWT) for quiet0934.arf

T-2V6

T-3W6 T-3 T-3 T-3 T-2 T-2 W5 W4 W3 W W2
1

400

y

800

X 0 200 400

0

t

600

800

1000

JIN XU

New Results of Fully Bayesian


Outline Background Problem description Methodology Research New Results Two concerns

New data sets Applying wavelets to replace PCA

DWT

how to make summary of those parameters are the key point. future work is to sample these parameters and transform back to arf.

JIN XU

New Results of Fully Bayesian