Stochastic modeling of AGN light curves

Credit: ESO/Kornmesser
Stochastic Modeling of Quasar Variability
Brandon C. Kelly (UCSB, CGE Fellow, bckelly@physics.ucsb.edu)
Aneta Siemiginowska (CfA), Malgosia Sobolewska (CfA), Tommaso Treu (UCSB),
Matt Malkan (UCLA), Anna Pancoast (UCSB), Jong-Hak Woo (Seoul), Jill Bechtold (Arizona)
Friday, May 24, 13
Flux
Flux
The Data Analysis Challenge: Aperiodic Lightcurves
Time
Friday, May 24, 13
Time
The Data Analysis Challenge: Aperiodic Lightcurves
Mrk 766 Vaughan & Fabian (2003)
What variability ‘features’ can we
measure for quasar lightcurves?
Friday, May 24, 13
Variance / Frequency
Quantifying Variability with the Power Spectral
Density (PSD)
Frequency [arbitrary]
Friday, May 24, 13
Time [arbitrary]
Variance / Frequency
Quantifying Variability with the Power Spectral
Density (PSD)
Frequency [arbitrary]
Friday, May 24, 13
Time [arbitrary]
Variance / Frequency
Quantifying Variability with the Power Spectral
Density (PSD)
Frequency [arbitrary]
Friday, May 24, 13
Time [arbitrary]
Variance / Frequency
Quantifying Variability with the Power Spectral
Density (PSD)
Frequency [arbitrary]
Friday, May 24, 13
Time [arbitrary]
Disadvantages of Traditional Non-parameteric
Tools for Quantifying Aperiodic Variability
Lomb-Scargle
Periodogram
Measurements
True
Mock Data
True
Empirical
Structure
Function
Friday, May 24, 13
Disadvantages of Traditional Non-parameteric
Tools for Quantifying Aperiodic Variability
Lomb-Scargle
Periodogram
Measurements
True
Mock Data
True
Empirical
Structure
Function
Friday, May 24, 13
Tools for Characterizing Aperiodic (Quasar)
Variability: What should they do?
Friday, May 24, 13
Tools for Characterizing Aperiodic (Quasar)
Variability: What should they do?
• Handle irregular/arbitrary sampling patterns and measurement errors
Friday, May 24, 13
Tools for Characterizing Aperiodic (Quasar)
Variability: What should they do?
• Handle irregular/arbitrary sampling patterns and measurement errors
• Produce interpretable results:
• Connection to physical models
• Connection to features in the power spectrum
Friday, May 24, 13
Tools for Characterizing Aperiodic (Quasar)
Variability: What should they do?
• Handle irregular/arbitrary sampling patterns and measurement errors
• Produce interpretable results:
• Connection to physical models
• Connection to features in the power spectrum
• Fast & scalable to massive time domain surveys
• By the end of LSST we will have millions to billions of multiwavelength lightcurves with ~
50-1000+ epochs
Friday, May 24, 13
Tools for Characterizing Aperiodic (Quasar)
Variability: What should they do?
• Handle irregular/arbitrary sampling patterns and measurement errors
• Produce interpretable results:
• Connection to physical models
• Connection to features in the power spectrum
• Fast & scalable to massive time domain surveys
• By the end of LSST we will have millions to billions of multiwavelength lightcurves with ~
50-1000+ epochs
• Handle multiwavelength/multivariate time series
• Account for correlations/time lags among lightcurves in different bands
Friday, May 24, 13
Two approaches to (stochastic) modeling of real
lightcurves: Frequency Domain and Time Domain
Friday, May 24, 13
Monte-Carlo Methods (Done+1992,Uttley
+2002,Emmanaloupolous 2010,2013)
• Extremely flexible, limited only by ability to do simulation
• Can be computationally expensive
• Reliance on χ2 may not provide optimal use of information in lightcurve
Friday, May 24, 13
Monte-Carlo Methods (Done+1992,Uttley
+2002,Emmanaloupolous 2010,2013)
• Extremely flexible, limited only by ability to do simulation
• Can be computationally expensive
• Reliance on χ2 may not provide optimal use of information in lightcurve
Friday, May 24, 13
Monte-Carlo Methods (Done+1992,Uttley
+2002,Emmanaloupolous 2010,2013)
• Extremely flexible, limited only by ability to do simulation
• Can be computationally expensive
• Reliance on χ2 may not provide optimal use of information in lightcurve
Friday, May 24, 13
Monte-Carlo Methods (Done+1992,Uttley
+2002,Emmanaloupolous 2010,2013)
• Extremely flexible, limited only by ability to do simulation
• Can be computationally expensive
• Reliance on χ2 may not provide optimal use of information in lightcurve
Friday, May 24, 13
Gaussian Processes (Rybicki & Press 1992, Kelly+
2009,2011, Miller+2010)
loglik =
log |⌃|
1
(y
2
T
µ) ⌃
1
(y
• Likelihood-based approach, enables Bayesian inference
• Statistically powerful, but limited by Gaussian assumption
• In general, computationally expensive (O(n3))
Friday, May 24, 13
µ)
Gaussian Processes (Rybicki & Press 1992, Kelly+
2009,2011, Miller+2010)
loglik =
0
log |⌃|
1
⌃=@ n⇥n A
1
(y
2
T
µ) ⌃
1
(y
⌃ij = Cov(y(ti ), y(tj ))
Z 1
2⇡if |ti
=
P SD(f )e
1
• Likelihood-based approach, enables Bayesian inference
• Statistically powerful, but limited by Gaussian assumption
• In general, computationally expensive (O(n3))
Friday, May 24, 13
µ)
tj |
df
Simple and fast tool: First order continuous
autoregressive process (CAR(1), Kelly+2009)
Lightcurve
dL(t) =
dt
(L(t)
⌧
Characteristic
Time Scale
White Noise
Power Spectrum
µ) + dW (t)
1/τ
σ2
LC Mean
Frequency
• Solution provides likelihood function, enables
maximum-likelihood or Bayesian inference
• Fitting is fast! Only O(n) operations to evaluate
likelihood function (e.g., Kelly+2009, Kozlowski
+2010) or do interpolation
Friday, May 24, 13
Fitting the CAR(1) model: Illustration
τ=5 days
σ=1
τ=1 day
σ=1
Friday, May 24, 13
Fitting the CAR(1) model: Illustration
τ=5 days
σ=1
τ=1 day
σ=1
Friday, May 24, 13
CAR(1) does well on optical lightcurves with typical
sampling of current surveys
MacLeod+(2010), ~10,000 quasars from
stripe 82
Kozlowski+(2010),
Ogle-III
Kelly+(2009), AGN Watch
Friday, May 24, 13
total variability
Trends involving the CAR(1) process parameters
Kelly+(in prep), data from reverberation mapping
MacLeod+(2010), Stripe82
Friday, May 24, 13
Using the CAR(1) model to find quasars
MacLeod+(2011)
Based on Stripe82 variable
point sources
Quasars
Stars
Quasars
Butler & Bloom (2011)
Stars
Works because quasars have more
correlated variability on longer time
scales compared to stars
Friday, May 24, 13
Using the CAR(1) model as a tool for simulating/
interpolating quasar lightcurves
Reverberation Mapping
(Zu+2011, see also Brewer+,
Pancoast+2011)
Toy Models of Accretion Disks
(Dexter & Argol 2011)
See also Emmanaloupolous+(2013)
for more flexible simulation methods
Friday, May 24, 13
Is there evidence for deviations from the CAR(1)
model in the optical?
Stripe82: No
(Andrae+2013)
Friday, May 24, 13
Ogle-III: Maybe
(Zu+2013)
Kepler: Yes
(Mushotzky+2011)
Stochastic Modeling in the X-rays: Mixtures of
CAR(1) processes (Kelly+2011)
Friday, May 24, 13
Stochastic Modeling in the X-rays: Mixtures of
CAR(1) processes (Kelly+2011)
Process is solution of
stochastic diffusion equation
Diffusion Time Scale
@u(x, t)
@ 2 u(x, t) @W (x, t)
=D
+
2
@t
@x
@t
u(x, t) =
1
X
ck (x)CAR(t; ⌧k )
k=1
Potential for astrophysicallymotivated lightcurve models?
Friday, May 24, 13
Diffusion time scale
across noise field
Mixture of CAR(1) processes does well for X-rays
too
Gonzalez-Martin & Vaughan (2012),
see also McHardy+(2006)
Kelly+(2011)
Friday, May 24, 13
log Power Spectrum
Estimating Black Hole Mass from Mixed CAR(1)
process model
McHardy (2013)
log Frequency
Friday, May 24, 13
See also, e.g., Yu & Lu (2001), Nikolajuk+(2004),
Papadakis (2004), O’Neill+(2005), Miniutti+(2009),
Caballero-Garcia+(2012)
Estimating Black Hole Mass from Mixed CAR(1)
process model
log Power Spectrum
Excess Variance
Zhou+(2010)
log Frequency
Friday, May 24, 13
See also, e.g., Yu & Lu (2001), Nikolajuk+(2004),
Papadakis (2004), O’Neill+(2005), Miniutti+(2009),
Caballero-Garcia+(2012)
Estimating Black Hole Mass from Mixed CAR(1)
process model
log Power Spectrum
Excess Variance
Ponti+(2012)
log Frequency
Friday, May 24, 13
See also, e.g., Yu & Lu (2001), Nikolajuk+(2004),
Papadakis (2004), O’Neill+(2005), Miniutti+(2009),
Caballero-Garcia+(2012)
Estimating Black Hole Mass from Mixed CAR(1)
process model
log Power Spectrum
New method works with X-ray counts.
Mass estimates with ~0.3 dex precision.
Kelly+(in prep)
log Frequency
Friday, May 24, 13
See also, e.g., Yu & Lu (2001), Nikolajuk+(2004),
Papadakis (2004), O’Neill+(2005), Miniutti+(2009),
Caballero-Garcia+(2012)
Current Work: More Flexible Stochastic Models
dLp (t)
dLp 1 (t)
+ ↵p
+ . . . + ↵1 L(t) =
p
1
dt
dt
dq ✏(t)
+
q
q
dt
q
dq 1 ✏(t)
+ . . . + ✏(t)
1
q
1
dt
• Continuous-time autoregressive moving average models (CARMA(p,q)) provide
flexible modeling of variability
• Power spectrum is a sum of Lorentzian functions (cf. X-ray binaries)
• Can still derive likelihood function of the data, enables maximum-likelihood or
Bayesian inference
• Still very fast, computational complexity scales as O(n), but more complex
algorithms needed to perform do the ‘fit’
Friday, May 24, 13
Example: Quasar vs Variable Stars
Friday, May 24, 13
Example: Quasar vs Variable Stars
Quasar
Kelly+(in prep)
LPV, AGB
LPV, RGB
68% Probability
Intervals
Friday, May 24, 13
Credit: ESO/Kornmesser
Main Takeaway Point:
Stochastic modeling provides a useful and powerful framework to quantify
quasar variability that can be applied to lightcurves of arbitrary sampling and
with measurement error.
Friday, May 24, 13
Time Domain Stochastic Modeling: Outstanding
issues and directions for future work
Friday, May 24, 13
Time Domain Stochastic Modeling: Outstanding
issues and directions for future work
• Multivariate lightcurves:
• Explicit modeling of time lags and correlation structure between different wavelengths
• Vector-valued CARMA(p,q) processes may provide general framework
Friday, May 24, 13
Time Domain Stochastic Modeling: Outstanding
issues and directions for future work
• Multivariate lightcurves:
• Explicit modeling of time lags and correlation structure between different wavelengths
• Vector-valued CARMA(p,q) processes may provide general framework
• Moving beyond a single stationary Gaussian process:
• Direct modeling of stochastic process + flares, other ‘state’ changes (Sobolewska+, in prep)
• Using alternatives to Gaussian noise (Emmanaloupolous+2013)
• Non-stationary and non-linear models
Friday, May 24, 13
Time Domain Stochastic Modeling: Outstanding
issues and directions for future work
• Multivariate lightcurves:
• Explicit modeling of time lags and correlation structure between different wavelengths
• Vector-valued CARMA(p,q) processes may provide general framework
• Moving beyond a single stationary Gaussian process:
• Direct modeling of stochastic process + flares, other ‘state’ changes (Sobolewska+, in prep)
• Using alternatives to Gaussian noise (Emmanaloupolous+2013)
• Non-stationary and non-linear models
Uttley+(2005)
Friday, May 24, 13
Time Domain Stochastic Modeling: Outstanding
issues and directions for future work
• Multivariate lightcurves:
• Explicit modeling of time lags and correlation structure between different wavelengths
• Vector-valued CARMA(p,q) processes may provide general framework
• Moving beyond a single stationary Gaussian process:
• Direct modeling of stochastic process + flares, other ‘state’ changes (Sobolewska+, in prep)
• Using alternatives to Gaussian noise (Emmanaloupolous+2013)
• Non-stationary and non-linear models
Friday, May 24, 13
Time Domain Stochastic Modeling: Outstanding
issues and directions for future work
• Multivariate lightcurves:
• Explicit modeling of time lags and correlation structure between different wavelengths
• Vector-valued CARMA(p,q) processes may provide general framework
• Moving beyond a single stationary Gaussian process:
• Direct modeling of stochastic process + flares, other ‘state’ changes (Sobolewska+, in prep)
• Using alternatives to Gaussian noise (Emmanaloupolous+2013)
• Non-stationary and non-linear models
• Building astrophysically-motivated stochastic models
• Stochastic partial differential calculations + accretion flow models?
Friday, May 24, 13