How to Time a Black Hole

2020-02-27 228浏览

  • 1.! Tiana_Athriel How to Time " dhuppenkothen a Black Hole Time series analysis for the multi-wavelength future Daniela Huppenkothen NYU Center for Cosmology and Particle Physics NYU Center for Data Science
  • 2.How do we convert data into knowledge?
  • 3.There are three major ways to look at the sky (in EM)
  • 4.Images
  • 5.Spectra Mortlock et al 2011
  • 6.Time
  • 7.300 1.0 200 Time 100 300 50200 50300 50400 50500 50600 50700 50800 50900 51000 51100 51200 51300 51400 51500 51600 51700 51800 51900 52000 52100 52200 52300 52400 52500 52600 52700 52800 52900 53000 53100 53200 53300 53400 53500 53600 53700 53800 53900 54000 54100 54200 54300 54400 54500 54600 54700 54800 54900 55000 55100 200 100 300 0.8 200 100 300 200 100 300 Count rate [counts/s] 200 0.6 100 300 200 100 300 200 0.4 100 300 200 100 300 200 100 0.2 300 200 100 300 200 100 0.0 0.0 55200 0.2 55300 0.4 55400 Time in MJD 0.6 55500 0.8 55600 1.0
  • 8.jet physics and particle acceleration stellar winds radiative processes relativistic plasmas + magnetic fields general relativity
  • 9.How do we convert data into knowledge?
  • 10.time series How do we convert data into knowledge? black hole physics
  • 11.1) How can we detect variability? 2) How can we classify variable sources? 3) How can we characterize variability?
  • 12.machine learning (maybe) 1) How can we detect variability? 2) How can we classify variable sources? 3) How can we characterize variability?
  • 13.machine learning (maybe) 1) How can we detect variability? 2) How can we classify variable sources? 3) How can we characterize variability?
  • 14.L118 WATTS & STROHMAYER Vol. 637 Transients 15.2 15.0 14.8 Fig. 1). Although the flare was not directly in the RHESSI field of view, most photons in the front segments would have been direct. Given RHESSI’s native time resolution of 1 binary ms (2!20 s), these events are clearly suitable for high-frequency timing analysis. The rear segment flux, by contrast, comprises scattered photons from the front segments, direct photons entering through the walls of the spacecraft, and albedo flux. The latter, which could be as much as 40%–50% of the direct flux in the energy range of interest (McConnell et al. 2004), has a severe impact on timing analysis. At the time of the flare, ● ● RHESSI was passing the limbs of the Earth (as viewed from ● ● ● ● ● the SGR). Albedo flux is limb-brightened, particularly ●● if the ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● incoming flux is polarized (Willis et al. 2005). This means that ● ● ●● ● ● ● ● ● a large fraction of the detected photons could have incurred ● additional delays of up to ≈0.02 s, smearing out signals above ≈50 Hz. Note that although count rates in the rear segments exceed those recorded by RXTE, count rates in the front segments are slightly lower. It should also be noted that scattering from the spacecraft walls and the Earth will cause the photon −2 0 2 energies recorded by RHESSI, particularly in the rear segments, to deviate from the true energies of the incident photons. Quantifying this effect precisely is extremely difficult. For this reason we use broad energy bands in our analysis and urge some care in interpreting the recorded photon energies. We started by extracting event lists from the RHESSI data, ● excluding only events occurring in a 2 s period ≈270 s after ● ● ● ● ● ● ● ● ● ● ● the peak of the flare when an attenuator is removed●●(the as● ● ● ● ● ● ● ● ● ● ● ● sociated spike introduces spurious variability, particularly in ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● the front segments). Timing analysis was carried out using ●●the ● 2 Z n statistic (Buccheri et al. 1983; Strohmayer & Markwardt 1999). Israel et al. (2005) showed that the presence of the high● ●● ● ● ● frequency signals was dependent on the●● phase of ● ● ● ● ● ● ● ●● ● the 7.6 s ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● rotational pulse; the signals appeared most ●● strongly ● ● at phases ● V mag Fig. 2.—Average power spectra from 2.27 s intervals (0.3 cycles) centered on different rotational phases, computed using photons from the front segments with recorded energies in the range 25–100 keV. The top curve was computed using 15 successive 2.27 s intervals, ≈150–260 s after the main flare, at a rotational phase that includes the secondary peak and part of the DC phase. The frequency resolution is 1 Hz. The middle curve shows the same spectrum with 2 Hz frequency resolution. The QPO at 92.5 Hz is clearly visible. The bottom curve is for the same time period but is an average of rotational phases !2.27 s away from the 92.5 Hz signalphase:no QPOs are detected. Characteristic error bars are shown for each spectrum. Vaughan et al, 2016 searched over a range of DF, Np, and energy bands for any signals with significance 13 j. ● We● started by searching for signals in the range 50–1000 Hz, ● ● ● ● ●● ●● ● using only data from the front●segments. In this range the noise ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●two signals that meet our profile is Poissonian. We find only ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● criterion. ● search ● ● ● ● ● ● ● ● ● ● ● ● ● ● The first, for photons with ● ● ● recorded energies in the range ●● ● ● ● ● ● ● ● ● ● ● ● 25–100 keV, is● the QPO at ● 92.5 Hz previously reported by ● ● ● ● ● ● Israel et al. (2005), shown in Figure 2. This signal, which we ● ● ● ● detect only at a ● rotational phase away from the main peak, is ● ● ● ●●● ● ● ● ● strongest ≈150–260 s after the initial flare. As noted by Israel ● ● ● ● ● ● ● ● ● ● occurs in conjunction with an increase in et al. (2005), this unpulsed emission. At Dn p 1 Hz the QPO is resolved; at Dn p 2 Hz it is not. We estimate the significance of the Dn p 2 Hz power using a x2 distribution with 68 degrees of 4 6 8 10 12 freedom, which is the distribution expected based on the number of independent frequency bins and pulses averaged. The time peak at 93(yr) Hz has a single-trial probability of 2 # 10!7. Applying a correction for the number of frequency bins, independent time periods, and rotational phases searched, we arrive at a significance of ≈1 # 10!3. That this is lower than the significance reported by Israel et al. (2005) is to be expected, ● given that the RHESSI front segment count rate is lower than ● ● ● ● ● ● ● that of RXTE. A search for the signal in the RHESSI rear ● ● ● ●● ● ● ● ● ● ● ● ● segments●●indicates the signal has indeed been smeared out ● ● ● ● that ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● due to albedo flux. Fitting the QPO with a Lorentzian profile, ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● centroid frequency of 92.7 ! 0.1 Hz, with a coher● we find a ● ● ● ● ● ●● ● ● ● ● ●● ● ence value Q of 40. The integrated ● ● ● ● ● rms fractional amplitude is ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● with ●● ●● ● ● ● 10% ! 0.3% , in good agreement Israel et al. (2005). ●●● ● ●● ● ● ● ● (a) 14.8 variable sources 15.0 re have been many reports of periodic or s from AGN, spanning the range of AGN Maselli et al, 2013 from minutes mma-rays, and on timescales field has a chequered history. Many reports e based on very few observed cycles of the ailure to properly account for the random hich can produce intervals of seemingly peess (1978) for a general discussion of this Uttley (2006) for some specific examples wn just from X-ray observations of nearby ons of the same targets usually fail to show oherent oscillations expected from a truly enter the era of massive time-domain surg 105 107 targets, it is becoming more imss detection procedures in order to under- Watts & Strohmayer, 2005 V mag t al. Fig. 1.—Light curves.Top:Front segments, 25–100 keV band.Middle:Rear segments, 25–100 keV band.Bottom:Front segments, 100–200 keV band. The plots show the main peak and decaying tail with the 7.6 s double-peaked pulse profile. The spike in the front segments at 270 s is due to the removal of an attenuator. Zero time corresponds to 21:30 UTC on 2004 December 27. (b) ●● ● ● ● ●
  • 15.Stationarity vs Non-Stationarity “The joint probability distribution does not change over time.”
  • 16.Stationarity vs Non-Stationarity “The joint probability distribution does not change over time.” NGC 5548: Peterson et al (1999)
  • 17.Figure 1 The parameter space of SMBH binary pairs. The expected orbital periods for SMBH close binary pairs at the specified separations as a function of total black-hole mass. The solid upper line for each separation indicates a z = 5 track and the solid lower line a z = 0.05 track whilst the two internal dotted lines show z = 1.0 (lower) and z = 2.0 (upper) tracks respectively. The hatched region indicates the range over which CRTS has temporal coverage of 1.5 cycles or more of a periodic signal. The pink shaded region shows the region of detection for the best CRTS candidate given the range of virial blackhole masses reported in the literature. Also shown (solid black star) is the location of the best known SMBH binary candidate, OJ 2876 . Periodicity Detection 14.4 14.5 14.6 TIMING FEATURES OF ACCRETING X-RAY BINARIES 399 0DJnLtuGe 14.7 14.8 14.9 15.0 15.1 0 GDrcLD et Dl. 1999 0L6 A6A6 LI1EA5 C66 EJJers et Dl. 2000 1000 2000 3000 4000 0-D - 49100 5000 6000 7000 Graham et al, 2015 Figure 2 The composite light curve for PG 1302-102 over a period of 7,338 days (⇠ 20 years). The light curve combines data from two CRTS telescopes (CSS and MLS) with n !P! form for SLX 1735"269 (observation historical data from the LINEAR and ASAS surveys, and the literature (see Methods for del and its components. details). The error bars represent one standard deviation errors on the photometry values. The dashed line indicates a sinusoid with period 1,884 days and amplitude 0.14 mag. Z sourceThe GXuncertainty 17+2 frominthe thesample measured period is 88 days. Note that this does not reflect the t al. (2002), corresponding to SZ waveform which will depend on the physical properties of expected shape of the periodic iddle ofthe thesystem. horizontal branch; see MJD, modified Julian day. tice that since the two kHz peaks in uite well separated from the other 8 elow 200 Hz do not affect the kHz se high-frequency peaks we could rted in Homan et al. (2002). We h a model consisting of six Lorentg. 8): a broad (Q ¼ 0:6) component , a narrow one with a subharmonic Fig. 8) identified with LLF , and two Fig. 6.—Power spectra in !P! form for GS 1826"24 (observations N and Q). Lines mark the best-fit model and its components. 1) components with characteristic which we indicate as L0 and L00 . Belloni et al, 2002 Kjeldsen et al, 2009
  • 18.Periodicity Detection I 1) Fourier Analysis 2) Lomb-Scargle Periodogram 3) Z statistic, Rayleigh statistic, … 4) Gregory & Loredo 1992 5) Wavelets 6) … 2
  • 19.Almost all algorithms used for periodicity detection assume stationary processes and/or white noise and even sampling
  • 20.credit:Richard Freeman, flickr.com/photos/freebird710/, CC licensed
  • 21.constant backgroundcredit:Richard Freeman, flickr.com/photos/freebird710/, CC licensed
  • 22.Wijnands + van der Klis 1998 constant backgroundcredit:Richard Freeman, flickr.com/photos/freebird710/, CC licensed
  • 23.Willem van de Velde the Younger, “The Gust”
  • 24.variable background! Willem van de Velde the Younger, “The Gust”
  • 25.Lachowicz+Done, 2010 variable background! Willem van de Velde the Younger, “The Gust”
  • 26.Lachowicz+Done, 2010 variable background! Willem van de Velde the Younger, “The Gust”
  • 27.The human brain is awesome at pattern recognition
  • 28.*see:pareidoliaCorollary:the human brain is prone to overfitting* + see spurious patterns (awesome for survival, less awesome for science)
  • 29.*see:pareidoliaCorollary:the human brain is prone to overfitting* + see spurious patterns By Viking 1, NASA [Public domain], via Wikimedia Commons (awesome for survival, less awesome for science)
  • 30.15.2 ● ● ● ● ● ● ● ● ● ● −2 0 2 4 ●● ●● ● 6 8 10 12 14.8 time (yr) ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 15.0 ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● (b) ● ● ● ● ● ● ●● ● ●● ●● ● −2 0 2 ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ● ● ●● ● 15.2 V mag ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● 4 6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 8 10 12 time (yr) .0 ag 14.8 Vaughan et al, 2016 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● (c)
  • 31.Hypothesis Testing
  • 32.Hypothesis Testing “What is the probability that I observe a frequency with power Pobs or higher, if my data is pure noise*?”
  • 33.Hypothesis Testing “What is the probability that I observe a frequency with power Pobs or higher, if my data is pure noise*?” *Gaussian or Poisson
  • 34.Hypothesis Testing periodic signal SAXJ1808.4-3658:Wijnands + van der Klis (1998) Noise Detection threshold *unevenly sampleddata:Lomb-Scargle periodogram
  • 35.Parameter Mean 5 per cent 95 per cent α β γ δ 2.7 1.6 × 10−2 0.10 2.1 × 10−4 2.4 0.95 × 10−2 0.084 0.97 × 10−4 3.1 2.7 × 10−2 0.12 3.4 × 10−4 Hypothesis Testing II Vaughan Huppenkothen 2013 Figure 5. 2010, Mrk 766 data and modelet(Hal, 1 ) computed at the posterior mode. propagate model uncertainty! The panels are the same as in Fig. 3. partial analysis of the data – it is in effect the application of a datadependent ‘stopping rule’ – and it is extremely difficult to see how densities on the words, for eac σ 2i ), where the width of the pr hyperparamete ies of nearby, Markowitz et as outlined bel intervals, pairw dictive p-value inferences are Previous stu in the range α a prior centred sion (standard of the power s sources, with β p(log β H 1 ) = of β ∼ 10−2 . rate, which ca ray observation bend/break fre parameters suc line width (e.g assuming RE J (Middleton et
  • 36.Fit a Model to the Power Spectrum "269 (observation om the sample onding to SZ al branch; see o kHz peaks in om the other affect the kHz aks we could l. (2002). We of sixLorent:6)component a subharmonic h LLF , and two characteristic 0 00 Fig. 6.—Power spectra in !P! Belloni et al, 2002 form for GS 1826"24 (observations N and Q). Lines mark the best-fit model and its components.
  • 37.… but all my data is unevenly sampled!
  • 38.Go back to the time domain!
  • 39.Go back to the time domain!
  • 40.Go back to the time domain! (Gaussian Processes)
  • 41.Fit a Model to the Covariance Foreman-Mackey et al, 2017, Rasmussen + Williams
  • 42.Fit a Model to the Covariance rdinates ⇣ coordinates X = ⇣ x1 X= x1 1 T ln p(y X, ✓, ↵) = r ✓ K↵ 1 ln p(y X, ✓, ↵) = 2 r ✓ T K↵ 2 r✓ = ⇣ n m ere ⇣ ··· ··· 1 1 r✓ r✓ ⌘T xN ⌘T xN 1 N 12 ln det K↵ N 2 ln (2⇡) ln det K↵ ln (2⇡) 2 2 ⌘T ⌘T y µ (x ) · · · y µ (x ) 1 ✓ 1 N ✓ N r ✓ = y1 µ✓ (x1 ) ··· yN µ✓ (xN ) vector of residuals and the elements of the covariance matrix K a he vector of residuals and the elements of the covariance matrix K are k (x , x ). The maximum likelihood values for the paramete m] = ↵ n m Foreman-Mackey Rasmussen + Williams = k (x ,etxal, 2017, ). The maximum likelihood values for the parameters ↵ nm ↵
  • 43.Fit a Model to the Covariance Rasmussen + Williams
  • 44.C. E. Rasmussen & C. K. I. Williams, G ISBN 026218253X. c 2006 Massachuset Fit a Model to the Covariance 1.1 A Pictorial Introduction to Bay 2 f(x) 1 0 −1 −2 0 Rasmussen + Williams 0.5 input, x 1 (a), prior Figure 1.1: Panel (a) shows four samples d (b) shows the situation after two datapoints h is shown as the solid line and four samples lines. In both plots the shaded region deno input value x.
  • 45.C. E. Rasmussen & C. K. I. Williams, G ISBN 026218253X. c 2006 Massachuset Fit a Model to the Covariance 1.1 A Pictorial Introduction to Bay 2 f(x) 1 0 −1 −2 0 Rasmussen + Williams 0.5 input, x 1 (a), prior Figure 1.1: Panel (a) shows four samples d (b) shows the situation after two datapoints h is shown as the solid line and four samples lines. In both plots the shaded region deno input value x.
  • 46.C. E. Rasmussen & C. K. I. Williams, G ISBN 026218253X. c 2006 Massachuset Fit a Model to the Covariance 1.1 A Pictorial Introduction to Bay 2 f(x) 1 0 −1 −2 0 Rasmussen + Williams 0.5 input, x 1 (a), prior Figure 1.1: Panel (a) shows four samples d (b) shows the situation after two datapoints h is shown as the solid line and four samples lines. In both plots the shaded region deno input value x.
  • 47.+ works for stochastic processes + probabilistic generative model — computationally expensive
  • 48.CARMA Continuous-time AutoRegressive Moving Average Processes
  • 49.N }n=1 are the measurement uncertainties, nm is the K The intuition behind this method is that, for this choi CARMAwith a small number o diagonal and can be computed Subsequently, Ambikasaran (2015) generalized this m exponentials k↵ (⌧nm ) = 2 n nm + J X aj exp( cj ⌧nm ) . j=1 Foreman-Mackey al, 2017; seebut also Kelly et al, 2014 , the inverse iset dense Equation (3) can still be ev the number of components in the mixture. nel function can be made even more general by in aj ! aj ± i bj and cj ! cj ± i dj . In this case, the
  • 50.CARMA The Astrophysical Journal, 788:33 (18pp), 2014 June 10 Figure 2. PSD for the light curve shown in Figure 1. The true PSD is given by the solid black line, the periodogram by the orange circles, the PSD from the maximum-likelihood estimate assuming a CARMA(5, 1) model (chosen to minimize AICc) by the blue dashed line, and the blue region contains 95% of the probability on the PSD assuming a CARMA(5, 1) model. There is a weak oscillatory feature centered at a frequency of 1/5 day−1 , which is at the measurement noise level. This feature is not obvious above the measurement error component for the periodogram, but the CARMA model is able to recover it, along with the rest of the PSD. We note that the tight errors on the PSD below the measurement noise level are due to extrapolation assuming the parametric form of the CARMA(5, 1) model, and using a higher order model would enable more flexibility and consequently broader errors below the measurement noise level. Kelly et al, 2014 Kelly et al. Figure 4. Simulated light curve from a CARMA(5, 3) process irregularly sampled over three observing seasons.The black line denotes the true values, and the blue dots denote the measured values. Also shown are interpolated and forecasted values, based on the best-fitting CARMA(5, 1) process; a CARMA(5, 1) model had the minimum AICc value. The solid blue line and cyan region denote the expected value and 1σ error bands of the interpolated and extrapolated light curve, given the measured light curve. (A color version of this figure is available in the online journal.) not reflective of the actual uncertainty on the PSD in this regime when one does not know the order of the CARMA process. Because the PSD is largely unconstrained below the
  • 51.CARMA + fast — assumes specific underlying process
  • 52.Implementations + + + Carma-pack Carma.jl Celerite
  • 53.BeyondCARMA:Gaussian Processes 6 S. Aigrain et al. Aigrain et al,Figure 20163. Same as Figure 1, but for a light curve displaying quasi-periodic variations.
  • 54.Spectral Timing = consider time and energy information at the same time* *with evenly sampled data
  • 55.1 0.5 Time Lags 1 2 Energy (keV) 5 10 100 150 200 Fig. 9 The ratio spectrum of 1H0707-495 to a continuum model (Fabian et al. 2009) broad iron K and iron L band are clearly evident in the data. The origin of the soft below 1 keV in5 this source had been debatable, but in this work was found to be domi by relativistically broadened emission lines. −50 0 50 Lag (s) -ray reverberation around accreting black holes 0.5 10−5 10−4 10−3 Temporal Frequency (Hz) 0.01 Fig. 10 The frequency-dependent lags in 1H0707-495 between the continuum domi hard band at 1–4 keV and the reflection dominated soft band at 0.3–1 keV. ig. 3 Time lag (8–13 keV relative to 2–4 keV) versus frequency for a hard state obser-X-1:Nowak, 1H0707-495:Uttley et al, 2014 ation Cygnus of Cyg X-1 obtained by2000 RXTE in December 1996. The trend can be very roughly pproximated with a power-law of slope −0.7, but note the clear step-like features, which orrespond roughly to different Lorentzian features in the power spectrum (Nowak 2000). and found significant high-frequency soft lags in 15 sources. Plotting the plitude of the lags with their best-estimated black hole masses11 , revealed 11 Black hole masses used by De Marco et al. (2013); Kara et al. (2013c) and in F 997). However, given the large low-frequency lags seen in BHXRB data obwere obtained from the literature, and estimated primarily using optical broad line ained by the Rossi X-ray Timing Explorer, these mechanisms wereberation. considered In a few cases masses were estimated using the scaling relation between o o be unfeasible when taking into account the energetics of heating such a large orona (Nowak et al. 1999). To get around this difficulty Reig et al. (2003) and
  • 56.Time Lags II 6 15 a 1.5 a Lag (rad) 10 5 1.25 1 0 3 b 0.5 3 b 2 Lag (rad) count rate (s-1) 0.75 1 2 1 0 0 15 c −1 −5 10 10 5 0 0 105 2×105 Time (s) 3×105 4×105 Figure 2. et Typical light curve realizations from cases 1 (panel a) Zoghbi al, 2013 and 2 (panel b) in Fig.1. In each case, the second light curve is lagged by 1 radians with respect to first. Panel c shows typical light curves with gaps for the two cases. The y-axis is similar to panels a and b. 6 a 400 10−4 Frequency (Hz) 10−30 1 2 3 4 Normalized Histogram Figure 5. Similar to Fig. 3 but now showing lags instead of psd. High and low rms cases are shown in panels a (top) and b (bottom) respectively for light curves without gaps. The average estimated lag is shown as points. The standard deviation around the mean is shown in as error pars. The envelope dotted line shows the average estimated uncertainties. The right panels in each case show the (normalized) number of values histogram for the 1st (un-shaded) and 5th (shaded) frequency bins, marked with vertical lines in the left panels.. + coherence, lag-energy spectra, lag-frequency spectra, is similar to that in Fig. 3. The points and the errors covariance bispectra, bars are for the case spectra, with no gaps for comparison. The… red-dotted and green-dashed lines are the envelope of the standard deviation of the PSD estimates for light curves F ga cu ar lin an fo 20 iz co ca 5) th in d d
  • 57.Future Challenges - Multi-wavelength CARMA/GPs? - Generative models for time-energy data sets - GPs for Poisson-distributed data - additionaldimensions:polarization
  • 58.Help us build Stingray!* *https://github.com/StingraySoftware/stingray
  • 59.… but my data is non-stationary!
  • 60.Gaussian Processes Revisited
  • 61.Gaussian Processes Revisited Heinonen et al, 2015
  • 62.nk F IG . 3.— We test the model constructed in Section 3 on simulated data. We simulated light curves of a single spike with Fermi /GBM-like background count rates and varied the amplitude of the spike in order to test detectability. In the left panel, we show the posterior distribution of the number of spikes as a function of the signal-to-noise ratio of the spike as a box plot. The box encompasses the interquartile range (the 0.25 and 0.75 quantiles) with the median marked. The whiskers extend out to 1.5 times the interquartile range; outliers are marked as scatter points. In the other panels, we show distributions of peak position versus amplitude for the four signal-to-noise ratios of the left panel (in the same order). The position and time and amplitude of the signal injected into the light curve is marked as a dark grey cross; similarly coloured lines are added to guide the eye. If the noise in the light curve dominates the signal, the model will place a large number of low-amplitude spikes all throughout the light curve (second panel). For a signal-to-noise ratio of 3 or greater, the probability distributions over amplitude and position collapse into a sharp peak at the position and amplitude where we places the spike in the simulations (panels 3-5). tk + ⌘ An t/2 ⌧n tn (2) , where nk is the mean count rate of the nth model component in time bin k, bg is the background count rate of that bin, and the count rate nk in a bin k with width t is defined as the value of a functional form defining the shape of the model component at the mid-point of that time bin. The component is a generic shape, and can be modified by an amplitude parameter An and a parameter setting the width ⌧n , in addition to parameters such as the time offset tn and intrinsic parameters further describing the component’s shape. We will define a component model for magnetar bursts in Section 3.2 below, and restrict ourselves here to a general description of the model. The posterior probability distribution for all model parameters is givenby:Hierarchical Flare Modeling 9 25 8 20 Count rate [104 counts s 1] 7 6 N(samples) 15 5 4 },H) p(N,↵,{✓n } H) = p(y N,↵,{✓n p(y . H) 10 Here, N is the number of model components, with the corresponding set of model parameters for these components {✓n } = {✓1 , ✓2 , ..., ✓N }. Each ✓n may be a scalar, for component models with a single parameter, or a vector, for component models with multiple parameters. We build a hierarchical 3 2 5 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Time since burst start [s] 0.7 0.8 10 (3) p(N, ↵, {✓n } y, H) 12 14 16 18 Number of spikes per burst 20 11 F IG . 4.— An example burst from the magnetar SGR J1550-5418, in an observation taken on 2009 January 22 (ObsID 090122173). In the left panel, the light curve at high time resolution (black), t = 5 ⇥ 10 4 s, and model light curves for 10 random draws from the posterior distribution (in colours). On the right, the marginalised posterior distribution over the number of components in the model. The posterior for the number of components lies between 10 and 20 components. - 120 50 N(log10 (Fluence)) - magnetar bursts blazar flares solar flares 140 N(log10 (Amplitude)) - 200 features in the data; the presence of Poisson noise leads to uncertainty in the weaker features, leading to a broader posterior distribution in those 150 dimensions. There is some ambiguity for some features on whether there 150 should be a component, or whether perhaps a particular feature should be modelled as a superposition of two components, but this ambiguity is generally small. 100 in deciding whether a feature should If we were interested 100 be modelled with a spike component, we could marginalise out N(log10 (Duration)) ing (either integrating for continuous variables, or summing for discrete variables) over all nuisance parameters (e.g. the hyperparameters). In Figure 4, we show an example of a burst light curve, together with 10 random draws from the posterior distribution (left panel). The burst was chosen specifically for its multipeaked structure such that we can investigate how well the method does in inferring the properties of a single burst. Overall, the posterior distribution is narrow and peaked for bright 160 100 80 60 40 50 20 0 3 2 1 0 log10 (Duration) [s] 1 0 2 1 0 1 2 3 log10 (Amplitude) [counts/s] 0 14 12 10 8 log10 (Fluence) [erg/cm2] Huppenkothen et al, 2015 F IG . 7.— Differential distributions for the duration, count-space amplitude and fluence for all model components from 332 bursts. In each bin, we plot the mean (bars) and standard deviation (error bars) for that bin from 100 ensembles of random draws from the posterior distribution of each burst (see text for details). All
  • 63.Machine Learning?
  • 64.+ very flexible + works on large data sets + excellent at prediction problems
  • 65.+ very flexible + works on large data sets + excellent at prediction problems — deep learning often a black box — deep models hard to interpret
  • 66.
  • 67.How do we do inference with machine learning models?
  • 68.How do we do inference with machine learning models? How do we make machine learning models interpretable? seealso:Lipton, 2017
  • 69.Long-Term Evolution of Stellar-Mass Black Holes
  • 70.300 1.0 200 Long-Term Evolution of Stellar-Mass Black Holes 100 300 50200 50300 50400 50500 50600 50700 50800 50900 51000 51100 51200 51300 51400 51500 51600 51700 51800 51900 52000 52100 52200 52300 52400 52500 52600 52700 52800 52900 53000 53100 53200 53300 53400 53500 53600 53700 53800 53900 54000 54100 54200 54300 54400 54500 54600 54700 54800 54900 55000 55100 200 100 300 0.8 200 100 300 200 100 300 Count rate [counts/s] 200 0.6 100 300 200 100 300 200 0.4 100 300 200 100 300 200 100 0.2 300 200 100 300 200 100 0.0 0.0 55200 0.2 55300 0.4 55400 Time in MJD 0.6 55500 0.8 55600 1.0
  • 71.274 274 T.T.Belloni variability GRS 1915+105 T. Belloni etofofal.:'>al.: