Resolving GRB Light Curves

Transcription

1 Resolving GRB Light Curves Robert L. Wolpert Duke University w/mary E Broadbent (Duke) & Tom Loredo (Cornell) 2014 August 03 17:15{17:45 Robert L. Wolpert Resolving GRB Light Curves 1 / 60

2 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 2 / 60

3 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 3 / 60

4 Overview of Levy Adaptive Regression Kernels I Goal: inference on unknown function f I Kernel regression approximates unknown function with weighted sum of functions I Adaptive kernel regression infers the parameters of the kernel instead of using xed dictionary f (x) JX j=1 u j K (x j s j ; j ) where s j is the location of the kernel on the domain of f. I LARK models the number, weights, and locations of the kernels as the largest jump discontinuities in a Levy process I Main advantage: stochastic process gives coherent way to incorporate threshold for approximation Skip ahead Robert L. Wolpert Resolving GRB Light Curves 4 / 60

5 Levy (Innitely Divisible) Processes I For a one-dimensional stochastic process X (t) to be a Levy Process the increments X (t 2 ) X (t 1 ) for t 1 < t 2 must satisfy: Stationarity The distribution of an increment depends only on the length of the interval (t 2 t 1 ). Independence Disjoint increments are independent I The Levy-Khinchine theorem states that for an innitely-divisible random variable X, the natural log of the characteristic function log E expfi! 0 X g = i! 0 m 1 2!0!+ Z R e i!0 u 1 i! 0 h(u) (du) I The (du) corresponding to an innitely divisible process is called its Levy measure Robert L. Wolpert Resolving GRB Light Curves 5 / 60

6 Levy processes as a Poisson integral (W+Ickstadt, 1998) I For a measure space (S; F ; (du)), the Poisson random measure H(du) is a random function from F to N such that for disjoint A i 2 F, H(A i ) ind Po (A i ). I When (du) is the Levy measure for an innitely-divisible random variable X, then H[u] = Z uh(du) = R for (random) support fu j g of H. 1X j=1 u j d = X Robert L. Wolpert Resolving GRB Light Curves 6 / 60

7 Levy Processes as Poisson Random Fields I Previous slide: ID RV can be represented as the countable sum of the support of H Po((du)). I For stochastic process case, we can write H 1 f0<stg = Z X uh(du ds) = fuj j 0 < s j tg = d X (t) R (0;t] where fs j g are distributed uniformly on S and u j are distributed proportionally to (du) I In simulation, only a nite number of pairs f(s j ; u j )g are drawn, creating an approximation to the true process; J Po (; 1) ; fu j j u j > g iid (du)1 fu>g =(; 1) Robert L. Wolpert Resolving GRB Light Curves 7 / 60

8 Levy Adaptive Regression Kernels Z f (t) := = X s j t K (t j s; )1 fstgh(du ds d) u j K (t j s j ; j ) where (s j ; u j ) represent the countable innovations resulting from the random integral representation of a Levy process. For tractable simulation, we use a truncation parameter > 0, and simulate from X f (t) := u j K (t j s j ; j )1 fsj tg1 fuj >g We use functions of this form to do Bayesian inference on unknown functions, such as light curves. Robert L. Wolpert Resolving GRB Light Curves 8 / 60

9 Example of a LARK model Robert L. Wolpert Resolving GRB Light Curves 9 / 60

10 LARK is coherent under aggregation Robert L. Wolpert Resolving GRB Light Curves 10 / 60

11 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 11 / 60

12 Anatomy of a gamma-ray burst Photo credit: NASA Goddard Space Center Robert L. Wolpert Resolving GRB Light Curves 12 / 60

13 Scientic Questions I Number and shape of pulses I Interaction between time and energy spectra I Inverse problem from transformation of data by telescope Robert L. Wolpert Resolving GRB Light Curves 13 / 60

14 Pulse Number and Shape I Figure: A variety of GRB photon rate time series (known as light curves) I Timescale: 0:5s to 100s I Number of pulses: 1 to 5 or more Robert L. Wolpert Resolving GRB Light Curves 14 / 60

15 Inverse Problem Induced by Detector I Figure: Photons are sorted into 4 energy channels, based on the energy deposited (not the incident energy) I Channel 1 is lowest energy; Channel 4 is highest I Energy deposited is less than incident energy; scientic interest is in incident space Robert L. Wolpert Resolving GRB Light Curves 15 / 60

16 Interaction of Time and Energy BATSE GRB Trigger 501 Robert L. Wolpert Resolving GRB Light Curves 16 / 60

17 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 17 / 60

18 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 18 / 60

19 ing a Single Light Curve Robert L. Wolpert Resolving GRB Light Curves 19 / 60

20 I Observe counts (Y 1 ; : : : ; Y N ) in intervals (t 0 ; t 1 ] : : : (t N 1; t N ]. ind I : Y k Po( k ) where k = Z tk t k 1 f (t)dt for some uncertain function f. I We model f in LARK form: f (t) = B + P J j=1 A j K (t j T j ; j ) I The number of kernels, J, is unknown, as are f(a j ; T j ; j )g for 1 j J I We x the baseline B and treat it as a known quantity Robert L. Wolpert Resolving GRB Light Curves 20 / 60

21 Kernel Selection I Fast rise, exponential decay (FRED) shape I Norris + (2005): K N (t j T ; 1 ; 2 ) / exp 1 t T 1 ft>tg t T 2 I Considered broader \GiG" class but data were inconclusive: K GiG (t j T ; p; 1 ; 2 ) / (t T ) p exp 1 t T 1 ft>tg t T 2 Robert L. Wolpert Resolving GRB Light Curves 21 / 60

22 Summary of Poisson Random Field Representation I pulse start times T j 's, and their maximum amplitudes A j 's as the jumps in a Levy (e.g., Gamma or -Stable) process I The Poisson random eld representation allows us to see the Gamma process as the sum of countably many jumps I whose times (T j ) are uniformly distributed on an interval I whose heights (A j ) larger than any > 0 are distributed proportionally to (du) = u 1 expf ug1 fu>g du: I Innitely many jumps lie below any threshold, but we only simulate the nitely many jumps larger than. I The number J of jumps larger than in a time interval of length L has Po LR 1 (du) distribution. Robert L. Wolpert Resolving GRB Light Curves 22 / 60

23 Computation I RJ-MCMC is a Metropolis-Hastings algorithm, but with a trans-dimensional proposal distribution, needed to do inference on our parameter space, the dimension of which is random and variable. I Our proposal distribution is a mixture of ve dierent proposals: birth, death, walk, split, and merge. I Parallel thinning, a new variation on parallel tempering which exploits the innite divisibility of LARK distributions to create interpretable auxiliary chains, also aids in mode-nding and mixing between modes Robert L. Wolpert Resolving GRB Light Curves 23 / 60

24 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 24 / 60

25 { Channel 1 Robert L. Wolpert Resolving GRB Light Curves 25 / 60

26 { Channel 1 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 26 / 60

27 { Channel 1 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 27 / 60

28 { Channel 2 Robert L. Wolpert Resolving GRB Light Curves 28 / 60

29 { Channel 2 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 29 / 60

30 { Channel 2 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 30 / 60

31 { Channel 3 Robert L. Wolpert Resolving GRB Light Curves 31 / 60

32 { Channel 3 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 32 / 60

33 { Channel 3 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 33 / 60

34 { Channel 4 Robert L. Wolpert Resolving GRB Light Curves 34 / 60

35 { Channel 4 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 35 / 60

36 { Channel 4 (a) (b) Robert L. Wolpert Resolving GRB Light Curves 36 / 60

37 s in dierent channels are incoherent Number of pulses in each channel (a): Channel 1 (b): Channel 2 (c): Channel 3 (d): Channel 4 Robert L. Wolpert Resolving GRB Light Curves 37 / 60

38 GRB 493 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 38 / 60

39 GRB 493 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 39 / 60

40 GRB 493 i: time index k: channel index j: pulse index E(Y ik ) = ik ik = Z ti+1 t i B k dt + R k Z ti+1 t i JX j=1 V j (t j E ; j ) (E j j )dt de R k B k is the response function for channel k is the empirically-determined baseline photon rate Robert L. Wolpert Resolving GRB Light Curves 40 / 60

41 GRB 493 Kernel Function { Time Spectrum for a Single Pulse Time uence conditioned on energy: (t j E ; T ; ; ) = t T + ln(e =100) =K (x) = exp( x)1 fx>0g : I This model has the pulse arrive instantly (time T, for photons of energy E = 100keV) I : time decay parameter I : controls the fact that pulses arrive at dierent times at dierent energies (typically, more energetic ones arrive earlier) I K is a normalizing constant needed for the interpretation of as pulse uence V j Robert L. Wolpert Resolving GRB Light Curves 41 / 60

42 GRB 493 Kernel Function { Energy Spectrum for a Single Pulse Comptonized Energy Spectrum: E (E j ; E c ) = expf E =E 100 c g1 fe >0g I Energy spectrum decays as a power law with index < 0 I After cuto E c, spectrum decays exponentially I Inference is challenging due to having only 4 channels of DISCLA (Discriminator Large Area) data Robert L. Wolpert Resolving GRB Light Curves 42 / 60

43 GRB 493 How and interact to create pulse shape Robert L. Wolpert Resolving GRB Light Curves 43 / 60

44 GRB 493 -Stable process I Volumes of pulses > are modeled by the largest jumps of a fully-skewed -Stable process with = 3=2 f (v) = 3p V 8 5=2 1 fv >g I = 3=2, reects the observation that photon uence decays as a power law with index 5=2 I = 1 photon/cm 2 I = 0:1074 so that E(J ) = 3 Robert L. Wolpert Resolving GRB Light Curves 44 / 60

45 GRB 493 Overdispersion I To compensate for model misspecication, an overdispersion parameter, r, is introduced. This parameter does not depend on the time interval or the energy channel. I Instead of modeling Y ij as Poisson, Y ij NB n = r ij ; p = r 1 + r I E(Y ij ) = ij I V(Y ij ) = ij 1+r r Robert L. Wolpert Resolving GRB Light Curves 45 / 60

46 GRB 493 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 46 / 60

47 GRB 493 Robert L. Wolpert Resolving GRB Light Curves 47 / 60

48 GRB 493 Robert L. Wolpert Resolving GRB Light Curves 48 / 60

49 GRB 493 { Results Figure: 95% Credible Interval for Mean, Robert L. Wolpert Resolving GRB Light Curves 49 / 60

50 GRB 493 { Results Figure: 95% posterior predictive intervals for Robert L. Wolpert Resolving GRB Light Curves 50 / 60

51 GRB 493 { Results (a) (b) Figure: Number of Pulses, Robert L. Wolpert Resolving GRB Light Curves 51 / 60

52 GRB 493 { Overdispersion Figure: Variance ination Robert L. Wolpert Resolving GRB Light Curves 52 / 60

53 GRB 493 { Parameter Estimates Parameter Pulse 1 Pulse 2 Pulse 3 T (s) ( 0:542; 0:344) (1:58; 2:44) ( 0:602; 0:287) V (/s) (10:2; 19:3) (1:02; 6:46) (1:00; 1:45) (s) (0:5923; 0:875) (0:0119; 1:370) ( 0:665; 0:154) (s 1 ) (0:344; 0:511) (0:415; 1:19) (1:01; 1:59) ( 0:229; 0:0) ( 2:09; 0:001) ( 1:42; 0:053) E c (kev) (53:1; 71:5) (28:3; 386:0 (79:3; 362:0) Table: Highest posterior density intervals (95%) for the parameters of the three-pulse models,. Robert L. Wolpert Resolving GRB Light Curves 53 / 60

54 GRB 493 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 54 / 60

55 GRB 493 GRB 493 { Doesn't always work as desired! Figure: 95% Credible Interval for Mean, GRB 493 Robert L. Wolpert Resolving GRB Light Curves 55 / 60

56 GRB 493 GRB 493 { Doesn't always work as desired! (a) (b) Figure: Number of Pulses, GRB 493 Robert L. Wolpert Resolving GRB Light Curves 56 / 60

57 Outline Introduction GRB 493 Robert L. Wolpert Resolving GRB Light Curves 57 / 60

58 I Our model successfully models the number and shapes of pulses for some GRBs, but not others, due to strict modeling of the rises of pulses as \instantaneous" in incident space. I This modeling approach does not depend on xed-length time or energy bins I Our approach successfully incorporates knowledge of the photon detector and corrects for systematic bias in photon energy assignment. Robert L. Wolpert Resolving GRB Light Curves 58 / 60

59 Future work I Flexible model for better rise-time modeling I Improve sampling for better eective sample sizes I Incorporate 16-channel data (MER) instead of 4 (DISCLA) for improved recovery of spectral parameters I Share information among bursts to infer population parameters Robert L. Wolpert Resolving GRB Light Curves 59 / 60

60 Thanks for your attention! Thanks too to NSF (DMS & PHY) and NASA AISR Program, and to Jon Hakkila (College of Charleston, Dept Physics & Astro). Robert L. Wolpert Resolving GRB Light Curves 60 / 60