The principal-component analysis is a statistical method, which lowers the number of important variables in a data set. The use of this method for the bursts’ spectra and afterglows is discussed in this paper. The analysis indicates that three principal components are enough among the eight ones to describe the variablity of the data. The correlation between the spectral index α and the redshift

suggests that the thermal emission component becomes more dominant at larger redshifts

Bagoly, Z.

Balázs, L. G.

Borgonovo, L.

Horváth, I.

Larsson, S.

Mészáros, A.

Ryde, F.

English

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2005-10045-1IL NUOVO CIMENTO Vol. 28 C, N. 3 Maggio-Giugno 2005Principal-Component Analysis of Gamma-Ray Bursts’ Spectra(∗)Z. Bagoly(1), I. Horva´th(2), L. G. Bala´zs(3), L. Borgonovo(4), S. Larsson(4),A. Me´sza´ros(5) and F. Ryde(4)(1) Laboratory for Information Technology, Eo¨tvo¨s University - H-1117 BudapestPa´zma´ny P. s. 1./A, Hungary(2) Department of Physics, Bolyai Military University - H-1456 Budapest, POB 12, Hungary(3) Konkoly Observatory - H-1525 Budapest, POB 67, Hungary(4) Stockholm Observatory, AlbaNova - SE-106 91 Stockholm, Sweden(5) Astronomical Institute of the Charles University - V Holesˇovicˇka´ch 2CZ-180 00 Prague 8, Czech Republic(ricevuto il 23 Maggio 2005; pubblicato online il 20 Settembre 2005)Summary. — The principal-component analysis is a statistical method, whichlowers the number of important variables in a data set. The use of this method forthe bursts’ spectra and afterglows is discussed in this paper. The analysis indicatesthat three principal components are enough among the eight ones to describe thevariablity of the data. The correlation between the spectral index α and the redshiftsuggests that the thermal emission component becomes more dominant at largerredshifts.PACS 98.70.-f – Unidentified sources of radiation outside the Solar System.PACS 98.70.Rz – γ-ray sources; γ-ray bursts.PACS 98.80.-k – Cosmology.PACS 01.30.Cc – Conference proceedings.1. – IntroductionThe principal-component analysis (PCA) is a powerful statistical method, if the ob-servational data contain several variables [1]. If there are some correlations among thevariables, then PCA lowers the number of variables needed for the description of thedataset without any essential loss of generality. It introduces the quantities called prin-cipal components (PCs). Then one can omit some PCs not leading to essential loss ofinformation about the dataset (for more details about PCA see [1]).In the topic of gamma-ray bursts (GRBs) the paper [2] used PCA on the dataset of theBATSE Catalog [3]. This Catalog contains nine measured variables in the gamma-bandalone (three different peak fluxes, the four fluences in the four different energy channels,and the two durations) for any GRB. Bagoly et al. [2] have shown that of these nine(∗) Paper presented at the “4th Workshop on Gamma-Ray Burst in the Afterglow Era”, Rome,October 18-22, 2004.c© Societa` Italiana di Fisica 295296 Z. BAGOLY, I. HORVA´TH, L. G. BALA´ZS, L. BORGONOVO ETC.variables only three are also enough (one of the durations, a combination of the fluencesand the peak fluxes, and the fluence on the highest energy channel alone).As far as it is known, since 1998 no further PCA was applied on any GRBs’ datasets.This is remarkable, because for the dozens of GRBs further observational quantitiesare also known from the analyses of spectra and from the afterglow (AG) data. Thus,the usefulness of such generalized PCA is a triviality. This contribution presents thepreliminary results of a new PCA, in which more variables than in [2] are included.In Jochen Greiner’s list [4], up to September 30, 2004, there are 38 GRBs with knownredshift z: for GRB980329 and GRB011030X there are upper limits only, and in onecase z is queried (GRB980326). These 41 GRBs define the primary sample here.For every GRB in our sample we will consider 8 variables: z, α, β, AGbreak, Epeak,AGα, T90 and fluence. The last two variables are the usual T90 durations and fluencesused already in the BATSE Catalog. The indices α and β are the slopes in Band’sspectrum, Epeak defines the photon energy, where the Band spectrum EF (E) peaks(F (E)dE is the fluence from the infinitesimal photon energy interval E, (E+dE)). Thesethree variables define the spectrum of GRBs. AGbreak is the observed time (in days),when the power law decay of AG changes into a faster decay; the slope of the decaybefore this AGbreak is denoted as AGα. (This slope is usually denoted also as α, butusing this notation there would be a confusion with the spectrum’s slope α.)All this means that we have taken only two variables from BATSE Catalog (theT90 duration and fluence), which in essence defined the two PCs in [2]. The third PC ofBATSE Catalog, namely the fluence at the highest energy channel, will not be consideredhere, because for this channel the data are often vanishing and/or noisy. Our main aimhere is to obtain information and conclusions about the six remaining new quantities, notabout the BATSE values themselves; hence, it is better not to consider here the fluenceat the highest energy channel.There are 16 GRBs from the mentioned 41 GRBs that have all these variables inGreiner’s list. These 16 GRBs define the net sample for our PCA. For fluence, T90,AGbreak (days), 1 + z and Epeak we will take the logarithms of the measured values.2. – Principal-Component AnalysisPCA needs, first, the calculation of correlations among the variables. The followingpairs (with the corresponding probabilities p) give strong correlations: log T90-log(1+ z)(p = 66.49%), log(1 + z)-α (p = 99.75%), logFluence-log T90 (p = 95.49%), logEpeak-log T90 (p = 97.98%) and logEpeak-log Fluence (p = 99.72%). All other correlations areweaker.The second step in PCA is to calculate the eigenvalues and eigenvectors. PCA givesus the eigenvalues and the eigenvectors of the “best” projection (table I).Then the third step in PCA is to define the important PCs. In accordance with [1] theimportant PCs are that ones, which have eigenvalues above 1.0. From table I we obtainthe result that only 3 PCs are important. This is a highly remarkable conclusion: wehave 8 variables, and their reduction to 3 is an essential lowering. In addition, two PCswere expected already from the earlier paper [2], because these two PCs were given by theBATSE data alone. In other words, the 6 new quantities—not used already in [2]—defineonly one further new important PC.As is seen from table I, the first three eigenvectors (Comp1-3) are the linear combina-tion of logEpeak, log T90, logFluence, log(1 + z), α, log AGbreak and AGα. No variablealone is dominating in these PCs. All this means that, beyond the two PCs comingPRINCIPAL-COMPONENT ANALYSIS OF GAMMA-RAY BURSTS’ SPECTRA 297Table I. – PCA result.Comp1 Comp2 Comp3 Comp4 Comp5 Comp6 Comp7log(1 + z) 0.345 −0.359 −0.399 0.301 0.278 0.348 −0.538α 0.355 −0.394 −0.283 0.336 −0.392 −0.188 0.497β −0.342 0.016 0.394 0.798 0.226 0.128 0.098logAGbreak −0.225 0.473 −0.560 0.023 0.132 0.135 −0.013AGα 0.381 −0.142 0.484 −0.285 0.217 0.377 0.051logEpeak −0.427 −0.285 −0.145 −0.173 −0.218 0.716 0.313log T90 −0.316 −0.498 −0.154 −0.212 0.640 −0.349 0.213logFluence −0.400 −0.378 0.120 −0.055 −0.442 −0.181 −0.555Eigenvalue 4.314 1.761 1.224 0.464 0.189 0.038 0.009from the BATSE Catalog, there is only one further important PC, and this one is acomplicated linear combination of all variables. In other words: Although the correla-tions among the variables are generally weak, these correlations are enough to ensure acoupling among the variables.The logEpeak + log T90 + log Fluence correlation is well known. This correlation inessence ensures that logEpeak alone is not independent. The correlation log(1+z) and αalso ensures that the redshift alone is also not independent—a quite remarkable behavior.(As is seen on fig. 1, the positive correlation between log(1+z) and α is obvious, becausethe trend—increasing of z with increasing α—is well seen. Contrary to this, no suchtrend is seen between z and β.) Similarly, the remaining correlations are enough to lowerthe number of important variables.Note that, interestingly, β stands practically alone in the fourth PC (Comp4). Hence,the only variable, which can define alone a PC, is the quantity β. But, in accordancewith [1], this PC has already an eigenvalue smaller than 1.0.-3-2.5-2-1.5-1-0.5 5 4 3 2 11+zNet sample n=16alphabetaFig. 1. – 1 + z distribution of the α and β values of the net data.298 Z. BAGOLY, I. HORVA´TH, L. G. BALA´ZS, L. BORGONOVO ETC.3. – DiscussionThe importance of the PCA is given also by the fact that it uses exclusively theobservational data, and any model of GRBs must respect the reduction of importantvariables. Here we briefly discuss the consequences of PCA on the models of GRBs.The prompt emission in GRBs can in most cases be described by various models ofoptically thin synchrotron emission. Theoretical considerations suggest that, apart fromthis non-thermal emission, a thermal emission component, from the photosphere of theoutflow, could be equally strong in the gamma-ray band [5, 6]. In [7] it was found thatthe observed spectra can indeed be described by a thermal black-body superimposed ona non-thermal spectrum. The relative strengths of these two components vary. A largeα is measured if the thermal component is dominant, while a lower value (more like asynchrotron value) is found if the non-thermal component dominates. The variations inα therefore could suggest a variation in importance of the photosphere.Note that α values are measured on the integrated spectra. These are necessarily softerthan the instantaneous spectra [8]. It is the instantaneous spectrum that reflects the phys-ical radiation process shaping the spectrum. We therefore have that αrad.proc. > αint..The strong correlation between α and 1 + z therefore suggests that the thermal compo-nent becomes more dominant at larger redshifts. In [5] it was showed that a combinationof the variation time scale at the central engine and the dimensionless entropy of the out-flow determines this relation. Low variability at the central engine combined with largeLorentz factors should therefore have been dominant at large redshifts. [9] also foundthis α-vs.-1 + z correlation. They explained it in terms of the Epeak ∝ E0.5rad correlation,where Erad is the total emitted energy of GRBs.4. – ConclusionsThe PCA method indicates that three PCs are enough to describe the variability ofthe data. Hence, a reduction of the eight variables to three is strongly suggested. Thestrong correlation between α and 1 + z suggests that the thermal emission componentbecomes more dominant at larger redshifts.∗ ∗ ∗Thanks are due to the valuable discussions with C.-I. Bjo¨rnsson, I. Csabai, C.Fransson, J. P. Uldall Fynbo, P. Me´sza´ros and G. Tusna´dy. This researchwas supported through OTKA grants T034549 and T48870, and by a grant from theWenner-Gren Foundations (A.M.).REFERENCES[1] Jolliffe I.T., Principal Component Analysis (Springer) 1986.[2] Bagoly Z. et al., ApJ, 498 (1998) 342.[3] Meegan C.A. et al., http://www.gammaray.msfc.nasa.gov/batse.[4] Greiner J., http://www.mpe.mpg.de/∼jcg/grbgen.html.[5] Me´sza´ros P. and Rees M., ApJ, 541 (2000) L5.[6] Daigne F. and Moshkovich R., MNRAS, 342 (2000) 587.[7] Ryde F., ApJ, 614 (2004) 827.[8] Ryde F. and Svensson R., ApJ, 512 (1999) 639.[9] Amati L. et al., A&A, 390 (2002) 81.

Principal-Component Analysis of Gamma-Ray Bursts’ Spectra

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Principal-Component Analysis of Gamma-Ray Bursts’ Spectra

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