A Maximum Likelihood Method to Improve Faint-Source Flux and Color Estimates

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1 Publicatins f the Astnmical Sciety f the Pacific, 110:77 731, 1998 June The Astnmical Sciety f the Pacific. All ights eseved. Pinted in U.S.A. A Maximum Likelihd Methd t Impve Faint-Suce Flux and Cl Estimates David W. Hgg 1, and Edwin L. Tune,3 Received 1997 Nvembe 1; accepted 1998 Febuay ABSTRACT. Flux estimates f faint suces tansients ae systematically biased high because thee ae fa me tuly faint suces than bight. Cectins that accunt f this effect ae pesented as a functin f signalt-nise ati and the (tue) slpe f the faint-suce numbe-flux elatin. The cectins depend n the suce being iginally identified in the image in which it is being phtmeteed. If a suce has been identified in the data, the cectins ae diffeent; a pesciptin f calculating the cectins is pesented. Implicatins f these cectins f analyses f suveys ae discussed; the mst imptant is that suces identified at signal-t-nise atis f 4 less ae pactically useless. 1. INTRODUCTION Given a nisy phtmetic measuement f a vey faint suce, what is the best estimate f its tue flux? The best answe t this questin is I dn t knw integate lnge t educe the nise! Hweve, in sme cases, this is nt pssible. F example, the ultadeep images f the Hubble Deep Field (HDF) (Williams et al. 1996) epesent s much Hubble Space Telescpe (HST) bseving time that in pactice they cannt be much impved. Fm the gund they cannt even in pinciple be impved because any gund-based images significantly deepe than existing nes (see, e.g., Djgvski et al. 1995; Metcalfe et al. 1995; Smail et al. 1995; Hgg et al. 1997a, 1997b) wuld be ttally cnfusin-limited (see, e.g., Cndn 1974). As anthe example, bsevatins f tansients, such as gamma-ay busts supenvae, cannt be impved because they cannt be epeated, even in pinciple. Given that in sme cases deepe imaging is nt an ptin, the easn the questin des nt have a tivial answe is that the numbe cunts f faint suces tend t ise with deceasing flux, s me suces ae available f upscatteing t a given measuement than ae available f dwnscatteing. A familia analgy is with tignmetic paallaxes, whee lw signal-t-nise ati measuements ae biased lage, since given any bseved paallax p and assciated e, thee is a finite ange f tue paallaxes p cnsistent with it, but thee ae fa me suces in the sky with small paallaxes p! p than lage p 1 p. The Lutz-Kelke cectins that accunt f this ae easy t cmpute and apply (Lutz & Kelke 1973; Hansen 1979); 1 Theetical Astphysics, Califnia Institute f Technlgy. Institute f Advanced Study, Olden Lane, Pincetn, NJ 08540; hgg@ias.edu. 3 Pincetn Univesity Obsevaty, Pincetn Univesity, Peytn Hall, Pincetn, NJ they have been essential in pviding unbiased distances in astnmy. Thee is a cnceptually simila set f cectins f lw signal-t-nise ati measuements f faint-suce fluxes. In this aticle, these cectins ae cmputed and discussed. As in the case f paallaxes, the cectins depend n hw the suces ae selected and n the intinsic distibutin f the measued quantity, in this case the tue numbe-flux elatin. Unftunately, the numbe-flux elatin is nt exactly knwn in mst cases f inteest, since the faint-suce phtmety is usually being pefmed in de t detemine this vey elatin! Futheme, in many cases f inteest, the cectin pesented hee des nt epesent the lagest suce f systematic e. Hweve, unlike the the suces f e, this cectin applies t all flux-selected suces, independent f instumentatin analysis technique. One nte f teminlgy: the systematic bias discussed hee is ften imppely efeed t as the Malmquist bias. The Malmquist bias is the effect that in a flux-limited sample, thee is a lage than epesentative factin f high-luminsity suces because they can be seen t geate distances and hence ve a lage vlume (Malmquist 194; Mihalas & Binney 1981). It is because f the intinsic scatte in suce luminsities. Malmquist bias is emved, e.g., when ne cmputes a luminsity functin fm sta cunts. It des nt invlve any kind f measuement e; it des nt g away if ne btains me pecise phtmety! The bias cected f hee esults fm the bsevatinal scatte in fluxes; the measuement es. It des indeed g away when the fluxes ae emeasued at much highe pecisin; it nly needs t be cnsideed when lw signal-t-nise data ae being used. What is discussed in this pape is clsely elated t Eddingtn bias, the effect f lw signal-t-nise flux measuements n faint-suce numbemagnitude elatins. Statistical cectins t bseved numbemagnitude elatins ae cmputed by Eddingtn (1913); flux cectins f individual suvey suces ae cmputed hee. 77

2 78 HOGG AND TURNER Fig. 1. Likelihd cuves f seveal numbe-magnitude expnents q and signal-t-nise atis 3 (wst), 5 (middle), and 10 (best). COMPUTATION OF CORRECTIONS Cnside the simplest case, in which a suce is being phtmeteed in the image in which it was fist detected. That is, it is being measued in the data in which it was selected. The likelihd p(sfs ) (pbability pe unit flux) that a suce has tue flux S given that it is bseved t have flux S is elated t the likelihd p(sfs) that it is bseved t have S when it has S by Bayes s theem: p(sfs ) p(sfs)p(s), (1) whee a pptinality is used because the nmalizatin is being igned (f nw) and p(s) (pbability pe unit flux) is the tue undelying distibutin f fluxes, given by the (tue, nt bseved) numbe-flux elatin. If the numbe f suces N (!m) bighte than magnitude m as a functin f m is a pwe law, d lg N d lg N.5 q, () dm d lg S then the cnditinal pbability becmes (q 1) S (S S ) ) [ S j ] p(sfs ) exp, (3) ( whee it is assumed that the bsevatinal e is Gaussiandistibuted and j is the uncetainty in the bseved flux S, S /j is the signal-t-nise ati. Figue 1 shws these likelihd cuves f numbe-flux expnent q.0, 1.5, 1.0, and 0.5 and signal-t-nise atis 3, 5, and 10. This figue demnstates that measuements at a signal-t-nise ati f 3 d nt stngly cnstain the tue flux, whateve the slpe f the numbe cunts, but paticulaly if the cunts have the Eu PASP, 110:77 731

3 MAXIMUM LIKELIHOOD METHOD 79 clidean 4 slpe f q 1.5 ( geate). It is wth emphasizing that the abve equatin and the cuves pltted in Figue 1 ae essentially identical t thse cmputed f paallax cectins (Lutz & Kelke 1973; Hansen 1979), except that the paallax cectins ae cmputed f nly ne paticula expnent value. If the flux measuement was unbiased, the peak in the likelihd functin p(sfs ) wuld be at S/S 1. Hweve, taking the deivative dp/ds, it is fund that the maximum likelihd tue flux S is in fact S 1 1 4q 4 1, (4) S whee q is the numbe-magnitude expnent defined abve and is the signal-t-nise ati. Thee is n finite maximum likelihd value at all if! 4q 4; an example is the q 1.5, 3 cuve in Figue 1. The abve equatin specifies a c- ectin that shuld in pinciple be applied t all flux measuements in a flux-limited sample. When the signal-t-nise is gd enugh ( k 4q 4) the cectin can be appximated as S q 1 1 when k 4q 4, (5) S in tems f the magnitude cectin Dm { m m, 1.086q Dm when k 4q 4. (6) Things change slightly if the likelihd is cmputed in the magnitude (i.e., lg flux athe than flux) dmain; afte all, maximum likelihd techniques ae sensitive t the metic f the space in which the likelihd is cmputed. When cmputed puely in the magnitude lg flux dmain, the cectin is 1.086q.171 Dm when k 4q 8. (7) The fact that the slutin depends n the type f data space (lg linea) demnstates that the specific value f the cectin is nt cmpletely specified, because it nly pvides a best guess (a subjective estimate) f the tue flux. It is wthy f nte that the Lutz-Kelke paallax cectins ae similaly subjective, as ae essentially all statistical estimats. 4 Anthe nte n teminlgy: what is called the Euclidean slpe eally ught t be called the n-evlutin, nnexpanding slpe, because even in a Euclidean space, the numbe cunts have q ( 1.5 at lage distance if eithe the univese is expanding the suces ae evlving. Me bust than maximum likelihd estimates ae cnfidence intevals, because these d nt depend n the chice f metic. Cnfidence intevals ae fund by integating the likelihd cuves. Unftunately, the aeas unde the cuves shwn in Figue 1 d nt cnvege; the likelihd distibutins ae nt nmalizable! This nnnmalizability cmes fm the divegence f p(sfs ) as S 0 (nt visible in sme f the cuves in Fig. 1 simply because at high the divegence happens at vey small S/S ). Thee ae tw espects in which this divegence nnnmalizability is unphysical: fist, thee cannt be an infinite numbe f suces in the visible univese; thee ae nt even an infinite numbe f atms in the univese! Secnd, mst ultadeep images f the sky, including the HDF, ae clse t thei cnfusin limits, beynd which the bseved numbe cunts have t cut ff n matte hw much integatin time is emplyed. Neithe f these effects can be simply taken int accunt in geneal; they depend n the data quality and the suces unde study. The equatins in this sectin have assumed that bsevatinal es ae Gaussian-distibuted, which is nt tue f all phtmetic measuements. The equatins ae easily genealized (althugh they d nt necessaily emain analytic) with the Gaussian in equatin (3) eplaced by whateve e distibutin is apppiate f the measuement in questin. 3. AN EMPIRICAL TEST The cectin can be tested with any imaging data in which the numbe-flux elatin is knwn. Hee, the HST HDF data in the F606W (0.6 mm) bandpass ae used. Nise was added t the 104 # 104 Vesin msaics f the HST images f the HDF (Williams et al. 1996) t make the pixel-t-pixel sky nise 10 times as bad as in the iginal msaics. The highe nise msaics will be efeed t as the bad images and the iginals as the gd images. A catalg f suces was chsen in the bad images dwn t vey faint levels using the SExtact suce detectin package (Betin & Anuts 1996) in essentially its default mde: smth with a pixel FWHM tiangula filte and select suces whse cental pixel in the smthed image is abve a given theshld. These suces wee then phtmeteed with the NOAO IRAF sftwae in matched 0.16 ( pixel) diamete apetues in bth the bad and gd images. The bad/gd flux atis ae pltted against signal-t-nise ati in Figue alng with the expected cectin cmputed with equatin (4) and the (knwn) cunt slpe q 0.5 (Williams et al. 1996). The cectin des vey well dwn t signal-t-nise atis 3. At! 4, a significant numbe f spuius (ze flux in the gd image) suces stat t appea. Figue shws, as with the Lutz-Kelke cectins, that the cectins ae n the same de as the intinsic scatte due t measuement e, s sme suces have undeestimated athe than veestimated fluxes. Hweve, the cectin is 1998 PASP, 110:77 731

4 730 HOGG AND TURNER f the distibutin f suce fluxes is imptant. The likelihd functin (pbability pe unit flux) f the tue I-band flux S given the bseved flux S and the knwn V-band flux (V) S is then (Q 1) S (S S ) (V) p(s FS, S ) ( ) exp [ ], (9) S j which leads t the maximum likelihd cectin S Q 1 S 1 when k 4Q 4, (10) in tems f the magnitude cectin Dm { m m, 1.086Q Dm when k 4Q 4. (11) Fig.. Ratis f flux measuements in the bad image, in which suces in the HDF wee detected, t the flux measuements in the gd image. The bad image is simply the gd image plus additinal nise (see text). The slid line is the expected ati S /S f the maximum likelihd flux t the bseved flux given by eq. (4) f the numbe-magnitude expnent q 0.5 (see text). still necessay if an unbiased estimat f the tue flux is desied. 4. CHANGING THE SELECTION TECHNIQUE The next case t cnside is phtmety f a suce in ne image (say the I band) afte it is detected (and its psitin is knwn) in anthe image (say the V band). In this case, Bayes s theem is still used, but f p(s) the tue distibutin f V I cls is used athe than the I-band numbe cunts. Actually, athe than the cl distibutin, it is bette t think (V) f the cnditinal I-band flux distibutin p(s FS ) (pba- (V) bility pe unit flux), given that the V-band flux S is knwn (in what fllws, it is assumed that the V-band detectin is at vey high signal-t-nise ati s the V-band flux is well knwn). Because, unlike the numbe cunts, these cnditinal distibutins ae nt geneally pwe laws, the flux cectin depends nt nly n the shape f the distibutin but als n whee in the distibutin the bseved flux S lies. Ftunately, when the signal-t-nise ati is lage enugh, it is pssible t lineaize Bayes s fmula aund the bseved flux s nly the lcal pwe-law slpe S F (V) d lg p(s FS ) Q { (8) d lg S F S Nte that the cectin can be psitive negative, depending n the sign f the lcal slpe Q. Cectins applicable when me cmplicated selectin pcedues have been used can be cmputed in analgus ways. 5. SUMMARY AND DISCUSSION Maximum likelihd cectins f faint-suce flux measuements have been cmputed f the case in which the suces ae measued at lw signal-t-nise in the data in which they wee iginally selected. It is fund that since the numbeflux elatin tends t be ising at the faint end, the lw signalt-nise flux measuements ae usually veestimates f the tue flux. At signal-t-nise atis! 4, flux measuements (f this type i.e., in the data in which the suces wee selected) ae almst meaningless because they ae cnsistent with almst any tue flux between ze and the measued value. The bias cnsideed hee tends t steepen measued numbe-flux elatins at the faint end; i.e., the measued d lg N/d lg S is me negative than the tue value because the vey numeus faint suces ae scatteed up t bighte levels. This effect is nly significant at vey faint levels, whee it is usually mitigated in fact canceled ut by incmpleteness. The best way t cect measued numbe-flux elatins f bth the flux bias and incmpleteness is t pefm full cmpleteness simulatins, which, if dne cectly, will accunt f bth effects simultaneusly (see, e.g., Smail et al. 1995; Hgg et al. 1997b), and f cuse a full accunting f all systematic es equies detailed mdeling f evey stage in the bseving and analysis pcedues. Althugh the cectins pesented hee d nt cmpehensively accunt f mst f these systematic biases, they ae vey geneal, impving flux estimates f individual suces independent f bsevatinal technique PASP, 110:77 731

5 MAXIMUM LIKELIHOOD METHOD 731 These cectins ught t be applied t the suce fluxes at the faint end f the catalgs fm all huge (and theefe difficult t impve upn) suveys, such as the Palma Obsevaty Sky Suveys, the Infaed Astnmical Satellite (IRAS) suvey, and fm futue huge suveys such as the Micn All Sky Suvey and the Slan Digital Sky Suvey. In fact, the IRAS catalgs wee cected at the faint end f sme elated biases but nt this bias pe se (IRAS Explanaty Supplement 1988). Als, all tansients discveed at lw signalt-nise atis in tansient seaches, such as faint gamma-ay busts, f which n additinal measuements can be made afte the fact, shuld have these cectins applied. We thank Rge Blandfd, Jhn Gizis, Gey Neugebaue, Neill Reid, and Yun Wang f useful discussins and the HDF team f planning, taking, educing, and calibating the HDF data. Sme financial suppt was pvided by NSF gant AST REFERENCES Betin, E., & Anuts, S. 1996, A&AS, 117, 393 Cndn, J. J. 1974, ApJ, 188, 79 Djgvski, S., et al. 1995, ApJ, 438, L13 Eddingtn, A. S. 1913, MNRAS, 73, 359 Hansen, R. B. 1979, MNRAS, 186, 875 Hgg, D. W., Neugebaue, G., Amus, L., Matthews, K., Pahe, M. A., Sife, B. T., & Weinbege, A. J. 1997a, AJ, 113, 474 Hgg, D. W., Pahe, M. A., McCathy, J. K., Chen, J. G., Blandfd, R. D., Smail, I., & Sife, B. T. 1997b, MNRAS, 88, 404 IRAS Catalgs and Atlases: Explanaty Supplement. 1988, ed. C. A. Beichman, G. Neugebaue, H. J. Habing, P. E. Clegg, & T. J. Cheste (Washingtn: GPO) Lutz, T. E., & Kelke, D. H. 1973, PASP, 85, 573 Malmquist, K. G. 194, Medd. Lund. Astn. Obs. Se. II, 3, 64 Metcalfe, N., Shanks, T., Fng, R., & Rche, N. 1995, MNRAS, 73, 57 Mihalas, D., & Binney, J. 1981, Galactic Astnmy (d ed.; New Yk: Feeman) Smail, I., Hgg, D. W., Yan, L., & Chen, J. G. 1995, ApJ, 449, L105 Williams, R. E., et al. 1996, AJ, 11, PASP, 110:77 731