Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

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1 Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1

2 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton coeffcent: r ( x x)( y y) ( x x) 2 ( y y) 2 r = 1: perfect correlaton r = 0: no correlaton r = 1: perfect antcorrelaton 2

3 Correlaton Between Tme-Varyng Parameters In fact, the data shown n the example are contnuum and H fluxes n a varable Seyfert 1 galaxy, Mrk 335. x = C(t) y = L(t) The contnuum and emsson-lne fluxes are hghly correlated. Mrk 335 data conssts of 24 ponts average spacng of 7.9 days. 3

4 Instead of lettng x = C(t) and y = L(t), mprove the correlaton by lettng x = C(t) and y = L(t + ), where s the tme-shft or lag 4

5 Cross-correlatng evenly spaced data s trval Shft Shft Shft = = unts unts Goal: fnd the value of the shft that maxmzes the correlaton coeffcent. 5

6 Frst practcal problem: n general, data are not evenly spaced. One soluton s to nterpolate between real data ponts. 6

7 Each real datum C(t) n one tme seres s matched wth an nterpolated value L(t + ) n the other tme seres and the lnear correlaton coeffcent s computed for all possble values of the lag. Interpolated lne ponts lag behnd correspondng contnuum ponts by 16 days. 7

8 Cross-Correlaton Functon Lnear correlaton coeffcent as a functon of tme lag s the cross-correlaton functon (CCF). The formal defnton of the CCF as a contnuous functon s the convoluton ntegral: CCF( ) L ( t) C( t ) dt 8

9 The Tme-Shft Improves the Lnear Correlaton 9

10 Dscrete Correlaton Functon Potental problem: what f you cannot reasonably nterpolate between actual observatons? Alternatve method s the DCF. For each contnuum pont, search for lne ponts wthn a certan bn wdth of the value of beng computed. (DCF) Only contrbutng ponts n the current computaton 10

11 Dscrete Correlaton Functon (DCF) Same example shown. Bn wdth s somewhat arbtrary, but medan samplng (lower panel) seems to gve good results. DCF s a very conservatve approach. Valuable check when gaps n data. In general performs much worse than nterpolated CCF. Bn =4 d Bn =8 d 11

12 Computatonal Subtletes Can the results be mproved wth hgher-order (non-lnear) nterpolaton? Doesn t seem to be any advantage, and non-lnear functons are hard to control. Can accuracy greater than average tme nterval be obtaned? Yes, provded that that lght curves are reasonably well sampled. 12

13 Computatonal Subtletes If you have tme seres wth N ponts, you are usng all N ponts n the calculaton only at = 0. Otherwse, ponts at the end of the seres drop out of the calculaton. Ths has two consequences: Tme seres are almost always non-statonary. The sgnfcance of any lag must be assessed n terms of the actual number of pont contrbutng to the correlaton coeffcent at the measured lag. 13

14 computatonally actve regon Whle the orgnal tme seres has N = 24 ponts, at the shft = 15.6 days, the number of actual pont contrbutng are N = 22 n the contnuum and N = 21 n the lne. Ths effect can be mportant for larger values of the lag. 14

15 Statstcally speakng, a seres s statonary f the mean and standard devaton do not change when ndvdual ponts at the ends of the seres drop out. Snce AGN tme seres are short, ths s never true. For each lag, means and standard devatons must be recomputed to correctly normalze the CCF. Non-Statonary Seres r ( x ( x x) 2 x)( y y) ( y 15 y) 2

16 Uncertantes n Cross-Correlaton Lags Determnng errors n cross-correlaton lags has been a vexng problem for over a decade. At present, the best method s a modelndependent Monte-Carlo method called FR/RSS : FR: Flux redstrbuton accounts for the effects of uncertantes n flux measurement RSS: Random subset selecton accounts for effects of samplng n tme 16

17 RSS s based on a computatonally ntensve method for evaluatng sgnfcance of lnear correlaton known as the bootstrap method. Bootstrap method: for N real data ponts, select at random N ponts wthout regard to whether or not they have been prevously selected. Determne r for ths subset Repeat many tmes to obtan a dstrbuton n the value of r. From ths dstrbuton, compute the mean and standard devaton for r. Bootstrap Method Orgnal data set A bootstrap realzaton Another realzaton 17

18 Random Subset Selecton How do you deal wth redundant selectons n a tme seres, where order matters? Ether: Ignore redundant selectons Each realzaton has typcally 1/e fewer ponts than the orgnal (orgn of the name RSS). Numercal experments show that ths then gves a conservatve error on the lag (the real uncertanty may be somewhat smaller). Weght each datum accordng to number of tmes selected Phlosophcally closer to orgnal bootstrap. 18

19 Assume that flux uncertantes are Gaussan dstrbuted about measured value, wth uncertanty. Take each measured flux value and alter t by a random Gaussan devate. Decrease by factor n 1/2, where n s the number of tmes the pont s selected. Flux Redstrbuton 19

20 A sngle FR/RSS realzaton. Red ponts are selected at random from among the real (black) ponts, redundant ponts are dscarded, and survvng ponts redstrbuted n flux usng random Gaussan devates scaled by the quoted uncertanty for each pont. The realzaton shown here gves cent = 17.9 days (value for orgnal data s 15.6 days) 20

21 Cross-Correlaton Centrod Dstrbuton Many FR/RSS realzatons are used to buld up the cross-correlaton centrod dstrbuton (CCCD). The rms wdth of ths dstrbuton (whch can be non- Gaussan) can be used as an estmate of the lag. Mrk 335 FR/RSS result: cent days 21

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