resolving grb light curves - statistical sciencerlw/talks/jsm14.pdflark for joint time and energy...

IntroductionGamma-ray bursts

LARK for Time SeriesLARK for Joint Time and Energy

Discussion

Resolving GRB Light Curves

Robert L. Wolpert

Duke University

[email protected]

w/Mary E Broadbent (Duke) & Tom Loredo (Cornell)

2014 August 03 17:15{17:45

Robert L. Wolpert Resolving GRB Light Curves 1 / 60



Discussion

Outline

Introduction

Gamma-ray bursts

LARK for Time SeriesModelSample of Results

LARK for Joint Time and EnergyModelGRB 501GRB 493

Discussion




Discussion

Outline

Introduction

Gamma-ray bursts



Discussion




Discussion

Overview of L�evy Adaptive Regression Kernels

I Goal: inference on unknown function f

I Kernel regression approximates unknown function withweighted sum of functions

I Adaptive kernel regression infers the parameters of the kernelinstead of using �xed dictionary

f (x) �JX

j=1

ujK (x j sj ; �j)

where sj 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 L�evy processI Main advantage: stochastic process gives coherent way to

incorporate threshold for approximation Skip ahead




Discussion

L�evy (In�nitely Divisible) Processes

I For a one-dimensional stochastic process X (t) to be a L�evyProcess the increments X (t2)�X (t1) for t1 < t2 must satisfy:

Stationarity The distribution of an increment dependsonly on the length of the interval (t2 � t1).

Independence Disjoint increments are independent

I The L�evy-Khinchine theorem states that for anin�nitely-divisible random variable X , the natural log of thecharacteristic function

log E expfi!0Xg = i!0m�12!

0�!+

ZR

�e i!

0u � 1� i!0h(u)��(du)

I The �(du) corresponding to an in�nitely divisible process iscalled its L�evy measure




Discussion

L�evy processes as a Poisson integral (W+Ickstadt, 1998)

I For a measure space (S ;F ; �(du)), the Poisson randommeasure H(du) is a random function from F to N such that

for disjoint Ai 2 F , H(Ai )ind� Po

��(Ai )

�.

I When �(du) is the L�evy measure for an in�nitely-divisiblerandom variable X , then

H[u] =

ZR

uH(du) =1Xj=1

ujd= X

for (random) support fujg of H.




Discussion

L�evy Processes as Poisson Random Fields

I Previous slide: ID RV can be represented as the countablesum of the support of H � Po(�(du)).

I For stochastic process case, we can write

H�1f0<s�tg

�=

ZR�(0;t]

uH(du ds) =X

fuj j 0 < sj � tg d= X (t)

where fsjg are distributed uniformly on S and uj aredistributed proportionally to �(du)

I In simulation, only a �nite number of pairs f(sj ; uj)g aredrawn, creating an approximation to the true process;

J� � Po��(�;1)

�; fuj j uj > �g iid� �(du)1fu>�g=�(�;1)




Discussion

L�evy Adaptive Regression Kernels

f (t) :=

ZK (t j s; �)1fs�tgH(du ds d�)

=Xsj�t

ujK (t j sj ; �j)

where (sj ; uj) represent the countable innovations resulting fromthe random integral representation of a L�evy process. For tractablesimulation, we use a truncation parameter � > 0, and simulate from

f�(t) :=X

ujK (t j sj ; �j)1fsj�tg1fuj>�g

We use functions of this form to do Bayesian inference onunknown functions, such as light curves.




Discussion

Example of a LARK model




Discussion

LARK is coherent under aggregation




Discussion

Outline

Introduction

Gamma-ray bursts



Discussion




Discussion

Anatomy of a gamma-ray burst

Photo credit: NASA Goddard Space CenterRobert L. Wolpert Resolving GRB Light Curves 12 / 60



Discussion

Scienti�c Questions

I Number and shape of pulses

I Interaction between time and energy spectra

I Inverse problem from transformation of data by telescope




Discussion

Pulse Number and Shape

I Figure: A variety of GRBphoton rate time series(known as light curves)

I Timescale: 0:5s to 100s

I Number of pulses: 1 to 5 ormore




Discussion

Inverse Problem Induced by Detector

I Figure: Photons are sortedinto 4 energy channels,based on the energydeposited

(not the incident energy)

I Channel 1 is lowest energy;Channel 4 is highest

I Energy deposited is less thanincident energy; scienti�cinterest is in incident space




Discussion

Interaction of Time and Energy

BATSE GRB Trigger 501




Discussion

ModelSample of Results

Outline

Introduction

Gamma-ray bursts



Discussion




Discussion


Outline

Introduction

Gamma-ray bursts



Discussion




Discussion


Modeling a Single Light Curve




Discussion


Model

I Observe counts (Y1; : : : ;YN) in intervals (t0; t1] : : : (tN�1; tN ].

I Model: Ykind� Po(�k) where

�k =

Z tk

tk�1

f (t)dt

for some uncertain function f .

I We model f in LARK form: f (t) = B +PJ

j=1 AjK (t j Tj ; �j)

I The number of kernels, J, is unknown,as are f(Aj ;Tj ; �j)g for 1 � j � J

I We �x the baseline B and treat it as a known quantity




Discussion


Kernel Selection

I Fast rise, exponential decay (FRED) shapeI Norris+ (2005):

KN(t j T ; �1; �2) / exp

�� 1t � T

� t � T

�2

�1ft>Tg

I Considered broader \GiG" class but data were inconclusive:

KGiG (t j T ; p; �1; �2) / (t�T )p exp

�� 1t � T

� t � T

�2

�1ft>Tg




Discussion


Summary of Poisson Random Field Representation

I Model pulse start times Tj 's, and their maximum amplitudesAj 's as the jumps in a L�evy (e.g., Gamma or �-Stable) process

I The Poisson random �eld representation allows us to see theGamma process as the sum of countably many jumps

I whose times (Tj) are uniformly distributed on an intervalI whose heights (Aj) larger than any � > 0 are distributed

proportionally to

��(du) = �u�1 expf��ug1fu>�g du:I In�nitely 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�LR1� �(du)

�distribution.




Discussion


Computation

I RJ-MCMC is a Metropolis-Hastings algorithm, but with atrans-dimensional proposal distribution, needed to doinference on our parameter space, the dimension of which israndom and variable.

I Our proposal distribution is a mixture of �ve di�erentproposals: birth, death, walk, split, and merge.

I Parallel thinning, a new variation on parallel tempering whichexploits the in�nite divisibility of LARK distributions to createinterpretable auxiliary chains, also aids in mode-�nding andmixing between modes




Discussion


Outline

Introduction

Gamma-ray bursts



Discussion




Discussion


GRB 501{ Channel 1




Discussion


GRB 501{ Channel 1

(a) (b)




Discussion


GRB 501{ Channel 2




Discussion


GRB 501{ Channel 2

(a) (b)




Discussion


GRB 501{ Channel 3




Discussion


GRB 501{ Channel 3

(a) (b)




Discussion


GRB 501{ Channel 4




Discussion


GRB 501{ Channel 4

(a) (b)




Discussion


Models in di�erent channels are incoherent

Number of pulses in each channel

(a): Channel 1 (b): Channel 2

(c): Channel 3 (d): Channel 4




Discussion

ModelGRB 501GRB 493

Outline

Introduction

Gamma-ray bursts



Discussion




Discussion

ModelGRB 501GRB 493

Model

i : time indexk : channel indexj : pulse index

E(Yik) =�ik

�ik =

Z ti+1

ti

Bkdt

+ Rk

�Z ti+1

ti

JXj=1

Vj �(t j E ; �j)� �(E j �j)dt dE�

Rk is the response function for channel kBk is the empirically-determined baseline photon rate




Discussion

ModelGRB 501GRB 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)1fx>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 di�erent times atdi�erent energies (typically, more energetic ones arrive earlier)

I K is a normalizing constant needed for the interpretation ofVj as pulse uence




Discussion

ModelGRB 501GRB 493

Kernel Function { Energy Spectrum for a Single Pulse

Comptonized Energy Spectrum:

�(E j �;Ec) =

�E

100

��

expf�E=Ecg1fE>0g

I Energy spectrum decays as a power law with index � < 0

I After cuto� Ec , spectrum decays exponentially

I Inference is challenging due to having only 4 channels ofDISCLA (Discriminator Large Area) data




Discussion

ModelGRB 501GRB 493

How � and interact to create pulse shape




Discussion

ModelGRB 501GRB 493

�-Stable process

I Volumes of pulses > � are modeled by the largest jumps of afully-skewed �-Stable process with � = 3=2

f (v) = 3�p8�

V�5=21fV>�g

I � = 3=2, re ects the observation that photon uence decaysas a power law with index �5=2

I � = 1 photon/cm2

I � = 0:1074 so that E(J�) = 3




Discussion

ModelGRB 501GRB 493

Overdispersion

I To compensate for model misspeci�cation, an overdispersionparameter, r , is introduced. This parameter does not dependon the time interval or the energy channel.

I Instead of modeling Yij as Poisson,

Yij � NB

�n = r�ij ; p =

r

1 + r

�

I E(Yij) = �ijI V(Yij) = �ij

1+rr




Discussion

ModelGRB 501GRB 493

Outline

Introduction

Gamma-ray bursts



Discussion




Discussion

ModelGRB 501GRB 493

GRB 501




Discussion

ModelGRB 501GRB 493

GRB 501 { Results

Figure: 95% Credible Interval for Mean, GRB 501




Discussion

ModelGRB 501GRB 493

GRB 501 { Results

Figure: 95% posterior predictive intervals for GRB 501




Discussion

ModelGRB 501GRB 493

GRB 501 { Results

(a) (b)

Figure: Number of Pulses, GRB 501




Discussion

ModelGRB 501GRB 493

GRB 501 { Overdispersion

Figure: Variance in ation




Discussion

ModelGRB 501GRB 493

GRB 501 { Parameter Estimates

Parameter Pulse 1 Pulse 2 Pulse 3T (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)Ec (keV) (53:1; 71:5) (28:3; 386:0 (79:3; 362:0)

Table: Highest posterior density intervals (95%) for the parameters of thethree-pulse models, GRB 501.




Discussion

ModelGRB 501GRB 493

Outline

Introduction

Gamma-ray bursts



Discussion




Discussion

ModelGRB 501GRB 493

GRB 493 { Doesn't always work as desired!

Figure: 95% Credible Interval for Mean, GRB 493




Discussion

ModelGRB 501GRB 493

GRB 493 { Doesn't always work as desired!

(a) (b)

Figure: Number of Pulses, GRB 493




Discussion

Outline

Introduction

Gamma-ray bursts



Discussion




Discussion

Discussion

I Our model successfully models the number and shapes ofpulses for some GRBs, but not others, due to strict modelingof the rises of pulses as \instantaneous" in incident space.

I This modeling approach does not depend on �xed-length timeor energy bins

I Our approach successfully incorporates knowledge of thephoton detector and corrects for systematic bias in photonenergy assignment.




Discussion

Future work

I Flexible model for better rise-time modeling

I Improve sampling for better e�ective sample sizes

I Incorporate 16-channel data (MER) instead of 4 (DISCLA) forimproved recovery of spectral parameters

I Share information among bursts to infer populationparameters




Discussion

Thanks for your attention!

Thanks too to NSF (DMS & PHY) and NASA AISR Program,and to Jon Hakkila (College of Charleston, Dept Physics & Astro).


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