X-ray Dark Sources Detection

Transcription

1 X-ray Dark Sources Detection Lazhi Wang Department of Statistics, Harvard University Nov. 5th, / 19

2 Data Y i, background contaminated photon counts in a source exposure over T = seconds (13.6 hours), X, photon counts in the exposure of pure background over T seconds. 2 / 19

3 Goals of the Project 1 To develop a fully Bayesian model to infer the distribution of the intensities of all the sources in a population. 2 To identify the existence of dark sources in the population. 3 / 19

4 Outline 1 The basic hierarchical Bayesian model 2 Extensions of the basic model 3 Extensive simulation studies: Robustness of the model Non-informativeness of the prior 4 Identifying the existence of dark sources via hypothesis testing: Calculation of test-statistic and posterior predictive p-value Simulation study 5 Real Data Application 6 One Difficult Problem and Discussion 4 / 19

5 Basic Hierarchical Bayesian Model Level I: Y i = S i + B i S i λ i Poisson(r i e i T λ i ) B i ξ Poisson(a i T ξ) X ξ Poisson(AT ξ) S i (counts): number of photons from source i in the source region, B i (counts): number of photons from the background in the source region, λ i (counts/s/cm 2 ): the intensity of source i, ξ (counts/s/pixels): the intensity of background, T (seconds): exposure time, T = , e i (cm 2 ): the telescope effective area, r i :proportion of photons from source i expected to fall in source region, a i (pixels): the size of source region i, A (pixels): the size of background region. S i, B i, λ i, ξ are all unobserved/latent, T, e i, r i, a i, A are all known constant. Y i, X are observed data. 5 / 19

6 Basic Hierarchical Bayesian Model Level II: ξ Gamma(α 0, β 0 ) { λ i α, = 0 with probability π d, β, πd Gamma(α, β) with probability 1 π d. Level III: Prior on the hyper-parameters π d, µ = α β, θ = α β 2 P(µ, θ) π d Unif (0, 1) 1 1 c1 2 + (µ c 2) 2 c3 2 + (θ c 4) 2 I µ>0,θ>0, 5 / 19

7 Model Extension I: Overlapping Sources Notation: O = {i 1,, i k } indicates the region formed by the overlap of source i 1,, i k. For example, O 1 = {1, 2, 4}, O 2 = {1}. O: the collection of all such regions. Level I model: Y o = S o + B o = j O S oj + B o, S oj λ j Poisson(r oj e o T λ j ) B o ξ Poisson(a o T ξ) 6 / 19

8 Model Extension II: Different Background Intensities In our data, the background intensity has an increasing trend as the projected angle (in arcmin) on the sky from the center of the field of view increases from 0 to 16. Projected Angle Counts (counts) Region (pixels) Intensity (counts/pixels) overall / 19

9 Model Extension II: Different Background Intensities Notation: Model: X k (counts): number of photons collected in background region k over T seconds ξ k (counts/s/pixels): the background intensity in regions k A k (pixels): the size of background region k O k : the collection of source regions in the background region k For counts from the pure background: X k ξk Poisson(A k T ξ k ) For counts from the source region O O k : B o ξ k Poisson(a o T ξ k ) 7 / 19

10 Simulation Study: The Robustness of the Model Y i Poisson(r i e i T λ i + 5), for i = 1,, 1000, X = , { = 0 with probability π d, r i e i T λ i Gamma[µ = 15, θ ] with probability 1 π d. 8 / 19

11 Simulation Study: The Robustness of the Model Y i Poisson(r i e i T λ i + 10), for i = 1,, 1000, X = , { = 0 with probability π d, r i e i T λ i Gamma[µ = 15, θ ] with probability 1 π d. 8 / 19

12 Simulation Study: Non-informativeness of the Prior B i Poisson(5), π d = 0.4, µ = 15, θ = 100 Histogram of Y i 's frequency coverage of HPD intervals π d µ θ simulation setting Photon counts in the source regions norminal percentage 9 / 19

13 Simulation Study: Non-informativeness of the Prior B i Poisson(5), π d = 0.4, µ = 15, θ = 500 Histogram of Y i 's frequency coverage of HPD intervals π d µ θ simulation setting Photon counts in the source regions norminal percentage 9 / 19

14 Simulation Study: Non-informativeness of the Prior B i Poisson(5), π d = 0.4, µ = 15, θ = 1000 Histogram of Y i 's frequency coverage of HPD intervals π d µ θ simulation setting Photon counts in the source regions norminal percentage 9 / 19

15 Hypothesis Testing for Existence of Dark Sources Hypothesis Testing: H 0 : π d = 0, H a : π d > 0. Reject H 0 if the p-value is low, p-value = P(T (D) T obs H 0 ), where D H 0 and T (D) is a test statistic. 10 / 19

16 Hypothesis Testing for Existence of Dark Sources Hypothesis Testing: H 0 : π d = 0, H a : π d > 0. Reject H 0 if the p-value is low, p-value = P(T (D) T obs H 0 ), where D H 0 and T (D) is a test statistic. However, D H 0 is unknown because α and β are unknown: λ i α, β Gamma(α, β) Posterior predictive p-value (ppp): ppp = P 0 (T (D) T obs D obs ), where D D H 0 with (α, β) α, β D obs, H / 19

17 Hypothesis Testing for Existence of Dark Sources Estimation of ppp: 1 Draw (α (t), β (t) ) from (α, β) D obs for t = 1, 2,, m, 2 For each pair (α (t), β (t) ), simulate D (t) from the null model and calculate T (t) = T (D (t) ), 3 Estimate ppp by ppp 1 m m I t=1 ( T (t) T obs). 11 / 19

18 Hypothesis Testing for Existence of Dark Sources Estimation of ppp: 1 Draw (α (t), β (t) ) from (α, β) D obs for t = 1, 2,, m, 2 For each pair (α (t), β (t) ), simulate D (t) from the null model and calculate T (t) = T (D (t) ), 3 Estimate ppp by ppp 1 m Likelihood Ratio Test Statistics: m I t=1 ( T (t) T obs). R(D) = sup α,β,π d L a (α, β, π d D) sup α,β L 0 (α, β, D) We use T (D) = log(r(d)) as the test statistic. 11 / 19

19 Calculation of Test Statistics One simplification: ξ = X L 0 (α, β Y): P 0 (Y α, β) = i=1 = C βα Γ(α) P(Y λ)p 0 (λ α, β)dλ N Y i ( ) c j Yi Γ(Yi j + α) i j (β + r i e i T ) Y. i j+α i=1 j=1 L a (α, β, π d Y): P a (Y α, β, πd ) = P(Y λ)pa (λ α, β, πd )dλ N = C π d c Y i i + (1 π d ) βα Y i ( ) c j Yi Γ(Yi j + α) i Γ(α) j (β + r i e i T ) Y. i j+α j=1 12 / 19

20 Simulation Study Y i Poisson(r i e i T λ i + 5), for i = 1,, 1000, X = , { = 0 with probability π d, r i e i T λ i Gamma[µ = 15, θ ] with probability 1 π d. 13 / 19

21 Simulation Study Y i Poisson(r i e i T λ i + 10), for i = 1,, 1000, X = , { = 0 with probability π d, r i e i T λ i Gamma[µ = 15, θ ] with probability 1 π d. 13 / 19

22 Simulation Study: Distribution of ppp B i Poisson(5), π d = 0.4, µ = 15, θ = 100 All the ppp s are / 19

23 Simulation Study: Distribution of ppp B i Poisson(5), π d = 0.4, µ = 15, θ = 500 Histogram of ppp's ppp's ppp samples from Uniform(0,1) 14 / 19

24 Simulation Study: Distribution of ppp B i Poisson(5), π d = 0.4, µ = 15, θ = 1000 Histogram of ppp's ppp's ppp samples from Uniform(0,1) 14 / 19

25 Real Data Analysis: No overlap sources, arcmin 6 Posterior distribution of the hyper-parameters µ θ π d / 19

26 Real Data Analysis: No overlap sources, arcmin 6 Histogram of the test statistics: ppp probability test statistics 15 / 19

27 Real Data Analysis: all the overlap sources, arcmin 6 Posterior distribution of the hyper-parameters µ θ π d / 19

28 Real Data Analysis: all the overlap sources, arcmin 8 Posterior distribution of the hyper-parameters (two background intensities). µ θ π d same background intensity / 19

29 Difficulty Calculation of ppp in the presence of overlapping sources. We need to calculate the likelihood ratio test statistic: R(Y) = sup α,β,π d L a (α, β, π d Y) sup α,β L 0 (α, β, Y) 18 / 19

30 Difficulty Calculation of ppp in the presence of overlapping sources. We need to calculate the likelihood ratio test statistic: R(Y) = sup α,β,π d L a (α, β, π d Y) sup α,β L 0 (α, β, Y) For simplicity: N = 2, the two sources overlap. O = {O 1 = {1}, O 2 = {2}, O 3 = {1, 2}} 18 / 19

31 Difficulty Calculation of ppp in the presence of overlapping sources. We need to calculate the likelihood ratio test statistic: R(Y) = sup α,β,π d L a (α, β, π d Y) sup α,β L 0 (α, β, Y) For simplicity: N = 2, the two sources overlap. O = {O 1 = {1}, O 2 = {2}, O 3 = {1, 2}} The complete data likelihood under the null hypothesis is P 0 (Y, λ α, β) = P(Y 1 λ 1 )P(Y 2 λ 2 )P(Y 3 λ 1, λ 2 )P(λ 1, λ 2 α, β) e c 1λ 1 c 2 λ 2 λ α 1 1 λ α 1 2 (1 + c 3 λ 1 ) Y 1 (1 + c 3 λ 2 ) Y 2 (1 + c 5 λ 1 + c 6 λ 2 ) Y 3, where c i s are some constants. 18 / 19

32 Difficulty Calculation of ppp in the presence of overlapping sources. We need to calculate the likelihood ratio test statistic: R(Y) = sup α,β,π d L a (α, β, π d Y) sup α,β L 0 (α, β, Y) For simplicity: N = 2, the two sources overlap. O = {O 1 = {1}, O 2 = {2}, O 3 = {1, 2}} The complete data likelihood under the null hypothesis is P 0 (Y, λ α, β) = P(Y 1 λ 1 )P(Y 2 λ 2 )P(Y 3 λ 1, λ 2 )P(λ 1, λ 2 α, β) e c 1λ 1 c 2 λ 2 λ α 1 1 λ α 1 2 (1 + c 3 λ 1 ) Y 1 (1 + c 3 λ 2 ) Y 2 (1 + c 5 λ 1 + c 6 λ 2 ) Y 3, where c i s are some constants. We need to integrate out λ 1 and λ 2 to get the likelihood L 0 (α, β Y). The calculation is feasible but very complicated when we have more overlaps and when N is large. 18 / 19

33 Real Data Analysis: all the overlap sources, arcmin 8 Posterior distribution of the hyper-parameters (same background intensities). µ θ π d real data analysis / 19