Analysis of Fermi GRB T90 distribution · T 90 distributions of Fermi GRBs ˜2 tting ableT 2:Parameters of a three-Gaussian t w i i ˙i A i ˜2 p -val 1 -0.030 0.603 102.0 0.27 2

Analysis of Fermi GRB T90 distribution

Mariusz Tarnopolski

Astronomical Observatory

Jagiellonian University

23 July 2015

Cosmology School, Kielce

Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 1 / 41

Presentation plan

1 Introduction and overview

2 T90 distributions of Fermi GRBsχ2 �ttingMaximum log-likelihood �tting

3 Hurst Exponents (HEs) & Machine Learning (ML)

4 On the limit between short and long GRBs

5 Conclusions


Presentation plan


2 T90 distributions of Fermi GRBs

χ2 �ttingMaximum log-likelihood �tting



5 Conclusions


Presentation plan


2 T90 distributions of Fermi GRBsχ2 �tting

Maximum log-likelihood �tting



5 Conclusions


Presentation plan





5 Conclusions


Presentation plan





5 Conclusions


Presentation plan





5 Conclusions


Presentation plan





5 Conclusions


Introduction and overview

Satellites

CGRO/BATSE

Swift/BATÐ→RHESSI

BeppoSAX/GRBM

Fermi/GBM

HETE-2/FREGATE

INTEGRAL/SPI-ACS

SUZAKU



KONUS (Mazets et al. 1981)



BATSE 1B

Kouveliotou et al. (1993) �tted a quadratic function between the twopeaks of 222 GRBs and determined its minimum to be at (1.2 ± 0.4) s,which rounded of to the next integer bin edge, is 2.0 s.Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July 2015 5 / 41


BATSE 3B (Horváth 1998)

-2 -1 0 1 2 3

0

10

20

30

40

50

logT90

Counts



BATSE 4B (current) (Horváth 2002)

-2 -1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

logT90

PDF

Horvath

chi2

MLE



Swift (Horváth et al. 2008)



Swift (Huja et al. 2009)



Swift (Huja & �ípa 2009)



RHESSI (�ípa et al. 2009)



BeppoSAX (Horváth 2009)


T90 distributions of Fermi GRBs χ2 �tting

A mixture of Gaussians:

fk =k

∑i=1

AiNi(µi , σ2i )

fk =k

∑i=1

Ai√2πσi

exp(− (x−µi )2

2σ2i

)

is �tted to a histogram of log (T90).A signi�cance level of α = 0.05 isadopted; 25 binnings are applied,de�ned by the bin widths w from0.30 to 0.06 with a step of 0.01.The corresponding number of binsrange from 15 to 69.

w=0.27

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.26

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.25

-1 0 1 2 3

0

50

100

150

200

250

w=0.2

-1 0 1 2 3

0

50

100

150

200

w=0.13

-1 0 1 2 3

0

50

100

150

logT90



A mixture of Gaussians:

fk =k

∑i=1

AiNi(µi , σ2i )

fk =k

∑i=1

Ai√2πσi

exp(− (x−µi )2

2σ2i

)

is �tted to a histogram of log (T90).A signi�cance level of α = 0.05 isadopted; 25 binnings are applied,de�ned by the bin widths w from0.30 to 0.06 with a step of 0.01.The corresponding number of binsrange from 15 to 69.

w=0.27

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.26

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.25

-1 0 1 2 3

0

50

100

150

200

250

w=0.2

-1 0 1 2 3

0

50

100

150

200

w=0.13

-1 0 1 2 3

0

50

100

150

logT90



Table 1: Parameters of a two-Gaussian �t

w i µi σi Ai χ2 p-val

0.271 -0.042 0.595 100.2

6.467 0.6922 1.477 0.465 325.7

0.261 -0.063 0.569 92.30

12.23 0.2702 1.475 0.473 318.8

0.251 -0.125 0.510 79.55

22.00 0.0242 1.453 0.494 316.1

0.201 -0.049 0.611 73.34

22.09 0.0772 1.473 0.476 243.6

0.131 -0.030 0.607 48.56

33.74 0.1422 1.480 0.468 157.2



w=0.27

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.26

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.25

-1 0 1 2 3

0

50

100

150

200

250

w=0.2

-1 0 1 2 3

0

50

100

150

200

w=0.13

-1 0 1 2 3

0

50

100

150

logT90

w=0.27

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.26

-1 0 1 2 3

0

50

100

150

200

250

300

w=0.25

-1 0 1 2 3

0

50

100

150

200

250

w=0.2

-1 0 1 2 3

0

50

100

150

200

w=0.13

-1 0 1 2 3

0

50

100

150

logT90



Table 2: Parameters of a three-Gaussian �t

w i µi σi Ai χ2 p-val

1 -0.030 0.603 102.0

0.27 2 1.466 0.455 317.1 5.333 0.502

3 2.027 0.201 6.014

1 -0.210 0.461 71.48

0.26 2 1.119 0.450 128.6 6.819 0.448

3 1.598 0.421 208.4

1 -0.137 0.492 77.73

0.25 2 1.414 0.480 300.4 13.52 0.095

3 1.939 0.128 14.49

1 -0.204 0.493 57.30

0.20 2 1.221 0.488 144.0 17.71 0.087

3 1.665 0.396 113.1

1 -0.058 0.581 46.61

0.13 2 1.453 0.464 153.7 29.42 0.167

3 1.903 0.092 4.328



∆χ2 = χ21 − χ22.= χ2(∆ν)

Table 3: Improvements of a three-Gaussian over a two-Gaussian �t

w ∆χ2 p-value

0.27 1.134 0.7670.26 5.411 0.1440.25 8.480 0.0370.20 4.380 0.2230.13 4.320 0.229



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Horvá

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Horvá

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Huja&

ípaH200

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al.H200

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Horvá

thH200

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Thiswor

k-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5BATSE

3B

BATSE4B

SwiftSwift

SwfitRHESSI

Beppo

SAX

Fermi

logT

90



Results 1

T90 distribution of Fermi GRBs is bimodal � no evidence fora (phenomenological) third (intermediate) class


T90 distributions of Fermi GRBs Maximum log-likelihood �tting

It feels like a waste of data to bin ∼ 1600 events into a few dozens of bins.

Having a distribution with a PDF given by f = f (x ; θ) (possibly a mixture),where θ = {θi}pi=1 is a set of parameters, the log-likelihood function isde�ned as

L =N

∑i=1

log f (xi ; θ),

where {xi}Ni=1 are the datapoints from the sample to which a distribution is�tted. The �tting is performed by searching a set of parameters θ forwhich the log-likelihood L is maximized.



It feels like a waste of data to bin ∼ 1600 events into a few dozens of bins.

Having a distribution with a PDF given by f = f (x ; θ) (possibly a mixture),where θ = {θi}pi=1 is a set of parameters, the log-likelihood function isde�ned as

L =N

∑i=1

log f (xi ; θ),

where {xi}Ni=1 are the datapoints from the sample to which a distribution is�tted. The �tting is performed by searching a set of parameters θ forwhich the log-likelihood L is maximized.



For nested as well as non-nested models, the Akaike information criterion(AIC ) may be applied. The AIC is de�ned as

AIC = 2p − 2Lp.

A preferred model is the one that minimizes AIC . The formulation of AICpenalizes the use of an overly excessive number of parameters, hencediscourages over�tting. Among candidate models with AICi , let AICmin

denote the smallest. Then,

Pri = exp(∆i

2) ,

where ∆i = AICmin −AICi , can be interpreted as the relative (compared toAICmin) probability that the i-th model minimizes the AIC .




AIC = 2p − 2Lp.


denote the smallest.

Then,

Pri = exp(∆i

2) ,





AIC = 2p − 2Lp.


denote the smallest. Then,

Pri = exp(∆i

2) ,




A mixture of k standard normal (Gaussian) N (µ,σ2) distributions:

f(N )

k(x) =

k

∑i=1

Aiϕ(x − µiσi

) =k

∑i=1

Ai√2πσi

exp(−(x − µi)2

2σ2i

)

A mixture of k skew normal (SN) distributions:

f(SN)

k(x) =

k

∑i=1

2Aiϕ(x − µiσi

)Φ(αix − µiσi

)

A mixture of k sinh-arcsinh (SAS) distributions:

f(SAS)

k(x) =

k

∑i=1

Aiσi

[1 + ( x−µiσi)2]− 12

βi cosh [βi sinh−1 ( x−µiσi) − δi]×

F(SAS)

k(x) =

k

∑i=1

× exp [−12sinh [βi sinh−1 ( x−µiσi

) − δi]2]

A mixture of k alpha-skew-normal (ASN) distributions:

f(ASN)

k(x) =

k

∑i=1

Ai

(1 − αi x−µiσi)2+ 1

2 + α2i

ϕ(x − µiσi

)




f(N )

k(x) =

k

∑i=1

Aiϕ(x − µiσi

) =k

∑i=1

Ai√2πσi

exp(−(x − µi)2

2σ2i

)


f(SN)

k(x) =

k

∑i=1

2Aiϕ(x − µiσi

)Φ(αix − µiσi

)


f(SAS)

k(x) =

k

∑i=1

Aiσi

[1 + ( x−µiσi)2]− 12


F(SAS)

k(x) =

k

∑i=1


) − δi]2]


f(ASN)

k(x) =

k

∑i=1

Ai

(1 − αi x−µiσi)2+ 1

2 + α2i

ϕ(x − µiσi

)




f(N )

k(x) =

k

∑i=1

Aiϕ(x − µiσi

) =k

∑i=1

Ai√2πσi

exp(−(x − µi)2

2σ2i

)


f(SN)

k(x) =

k

∑i=1

2Aiϕ(x − µiσi

)Φ(αix − µiσi

)


f(SAS)

k(x) =

k

∑i=1

Aiσi

[1 + ( x−µiσi)2]− 12


F(SAS)

k(x) =

k

∑i=1


) − δi]2]


f(ASN)

k(x) =

k

∑i=1

Ai

(1 − αi x−µiσi)2+ 1

2 + α2i

ϕ(x − µiσi

)




f(N )

k(x) =

k

∑i=1

Aiϕ(x − µiσi

) =k

∑i=1

Ai√2πσi

exp(−(x − µi)2

2σ2i

)


f(SN)

k(x) =

k

∑i=1

2Aiϕ(x − µiσi

)Φ(αix − µiσi

)


f(SAS)

k(x) =

k

∑i=1

Aiσi

[1 + ( x−µiσi)2]− 12


F(SAS)

k(x) =

k

∑i=1


) − δi]2]


f(ASN)

k(x) =

k

∑i=1

Ai

(1 − αi x−µiσi)2+ 1

2 + α2i

ϕ(x − µiσi

)



●

●

●

●

●

2 3 4 5 6

3432

3434

3436

3438

3440

3442

number of components

AIC

AIC vs. number of components in a mixture of standard normaldistributions. The minimal value corresponds to a three-Gaussian.



PD

F

HaL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 HbL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

HcL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 HdL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

HeL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 HfL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

HgL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 HhL

-1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

logT90

Distributions �tted tologT90 data. Colordashed curves are thecomponents of the (blacksolid) mixturedistribution. The panelsshow a mixture of (a) twostandard Gaussians, (b)three standard Gaussians,(c) two skew-normal, (d)three skew-normal, (e)two sinh-arcsinh, (f) threesinh-arcsinh, (g) onealpha-skew-normal, and(h) twoalpha-skew-normaldistributions.



æ

æ

æ

æ

æ

æ

æ

æ

HaL HbL HcL HdL HeL HfL HgL HhL

3430

3435

3440

3445

3450

3455

2-

G3-

G2-

SN

3-

SN

2-

SA

S

3-

SA

S

1-

ASN

2-

ASN

Distribution

AIC

æ AIC

ç

ç

ç

ç

ç

ç çç

0.0

0.2

0.4

0.6

0.8

1.0

Pr

ç Pr

AIC and relative probability (Pr) for the models examined.



Results 2

Log-likelihood method supported the non-existence of a third(intermediate) component in the T90 distribution of Fermi.

A two-component mixture of skewed distributions (2-SN and 2-SAS)describes the data better than a three-Gaussian.


Hurst Exponents (HEs) & Machine Learning (ML)

Methods � HE � de�nition

HE is a measure of persistency/long-term memory/self-similarity of aprocess.

Two ways of de�ning:

1 a process Y (t) (non-stationary) is self-similar with self-similarityparameter H, if

Y (λt) .= λHY (t)2 a process X (t) (stationary) is self-similar if ∃α ∈ (0,2):

limτ→∞

ρ(t)∝ τ−α, α = 2 − 2H



Methods � HE � properties

0 < H ≤ 1

H = 0.5 for a random walk (Brownion motion)

H < 0.5 for anti-persistent (anti-correlated, short memory) process

H > 0.5 for persistent (correlated, long memory) process

H = 1 for periodic time series

fractal dimension D = 2 −H



-0.2 0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

HE

Counts



Methods � MVTS

Light curves are binned in to narrow time bins. Optimum bin-width atwhich the non-statistical variability in the light curve becomes signi�cant.Prompt duration emission and equal duration of background region.Ratio of the variances of the GRB and the background divided by thebin-width as a function of bin-width. For binnings beyond the minimum thesignal is indistinguishable from Poissonian �uctuations. Left fromminimum, signi�cant variability in the light curve may vanish (coarsebinning). Optimum bin-width is at the minimum.



Methods � SVM

Not probabilistic, but methods exist (probability calibration, e.g. distanceto the hyperplane) to make it probabilistic.



Methods � Monte Carlo & SVM

(2220) = 231 subsamples from short GRBs; for each, 435 subsamples of 42

(out of 46) long GRBs → training set. Remaining � validation set.

≈ 105 realisations.

Success ratio r : rshort and rlong; rtot =2rshort+4rlong

6.









Results 3

1 H and MVTS alone give unsatisfactory classi�cations

2 T90 works as expected

3 (H, logMVTS) � unsatisfactory

4 (H, logT90) � better than H and logT90 alone

5 (logMVTS, logT90) � �←� worse, �→� better6 complementing (logMVTS, logT90) with HEs � accuracy increased by

7%; comparable to T90 alone.


On the limit between short and long GRBs

As in (Kouveliotou et al. 1993), the limitting T90 value is found as alocal minimum.

ML �t of a two-Gaussian instead of a parabola.

Datasets: BATSE 1B (for comparison; 226 events), BATSE current,Swift, BeppoSAX and Fermi (∼ 1000 − 2000 events).

Parameter errors: parametric bootstrap.



PDF

(a)

-2 -1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5 (b)

-2 -1 0 1 2 30.00.10.20.30.40.50.6

(c)

-2 -1 0 1 2 30.00.10.20.30.40.50.6 (d)

-2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

(e)

-2 -1 0 1 2 30.00.10.20.30.40.50.60.7

logT90



Table 4: Parameters of the �ts. Errors are estimated using the bootstrap method.

Label Dataset N i µi δµi σi δσi Ai δAi min. δmin.

(a) BATSE 1B 2261 −0.393 0.099 0.465 0.069 0.272

0.040 2.158 0.0492 1.460 0.056 0.532 0.044 0.728

(b)BATSE

20411 −0.095 0.051 0.627 0.033 0.336

0.018 3.378 0.272current 2 1.544 0.018 0.429 0.013 0.664

(c) Swift 9141 −0.026 0.255 0.740 0.120 0.139

0.042 � �2 1.638 0.031 0.528 0.023 0.861

(d) BeppoSAX 10031 0.626 0.186 0.669 0.075 0.355

0.084 � �2 1.449 0.035 0.393 0.027 0.645

(e) Fermi 15961 −0.072 0.073 0.525 0.044 0.215

0.021 2.049 0.2482 1.451 0.021 0.463 0.014 0.785



Results 4

Datasets from Swift and BeepoSAX are unimodal, hence no new limitmay be inferred.

BATSE 1B and Fermi are consistent with the conventional 2 s value.

A limit of 3.38 ± 0.27 s was obtained for BATSE current.

This leads to diminishing the fraction of long GRBs in the sampe by4%.

T90 is an ambiguous GRB type indicator.


Conclusions

Conclusions

Both χ2 and ML �tting lead to a bimodal T90 distribution

This is a hint that there are only two GRB classes

Two types of two-component skewed distributions are a better �t thana three-Gaussian

It is unlikely that the third, intermediate-duration, GRB class is a realphysical phenomenon

It was suggested (Zitouni 2015) that an assymetric T90 distributionmay be due to an assymetric distribution of envelope masses of theprogenitors

HE might serve as a GRB class indicator � including it in the SVMscheme increased accuracy by 7%

A division between short and long GRBs at T90 of 3.38 s is moreappropriate for the BATSE current dataset than the the conventionalvalue of 2 s.


Conclusions

Conclusions









Conclusions

Conclusions









References

[1] Tarnopolski M., 2015, A&A, in press (arXiv:1506.07324)

[2] Tarnopolski M., 2015 (arXiv:1506.07801)



[5] www.oa.uj.edu.pl/M.Tarnopolski

Thank you for your attention.


www.oa.uj.edu.pl/M.Tarnopolski

References

[1] Tarnopolski M., 2015, A&A, in press (arXiv:1506.07324)




[5] www.oa.uj.edu.pl/M.Tarnopolski

Thank you for your attention.


www.oa.uj.edu.pl/M.Tarnopolski

Documents

Analysis of Fermi GRB T90 distribution · T 90 distributions of Fermi GRBs ˜2 tting ableT 2:Parameters of a three-Gaussian t w i i ˙i A i ˜2 p -val 1 -0.030 0.603 102.0 0.27 2