Time Series in High Energy Astrophysics Brandon C Kelly HarvardSmithsonian Center for Astrophysics Lightcurve shape determined by time and parameters Examples SNe γ ray bursts Can use ID: 272434
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Slide1
Commentary on Time Series in High Energy Astrophysics
Brandon C. Kelly
Harvard-Smithsonian Center for AstrophysicsSlide2
Lightcurve
shape determined by time and parameters
Examples: SNe, γ-ray burstsCan use parameteric methods to characterize shapeE.g., Rise and decay time scale of a pulseCan use non-parameteric methods to characterize shapeJoey: Projected spline fits of LC onto diffusion map spaceAshish: Used dm/dt to obtain simple parameterization of trends for sparse LC
Trends
Spline
fits to
SNe
LCs
in multiple bands
(Richards+, 2011)Slide3
LC shape determined by time and parameters, repeats regularly
Examples: Eclipsing binaries, Pulsars
Characterized by period, shape of pulsePavlos: Described methods for estimating periods in noisy and irregularly sampled time seriesEric: Use H-test on photon arrival times to find periods, regularly sampled time seriesPeriodic Behavior
Pulsar
Eclipsing Binary
Credit: ESASlide4
Stochastic Process: LC depends stochastically on previous values of LC and time between observations
Examples: Random-walk behavior, accretion flows, jets
Joey: Used parameters that summarize stochastic fluctuations for classificationStochastic VariationsWilms+ (2007)
Cygnus X-1
X-ray
RadioSlide5
Characterizing Stochastic Processes
Mrk
766, Vaughan & Fabian (2003)
What similarities exist
between these two
segments?Slide6
How to quantify features of stochastic processes?
More variable on
long timescales
More variable on
short timescales
Higher Variability
Lower Variability
C
ompare to
C
ompare toSlide7
Use features in the autocorrelation function (ACF) and power spectral density (PSD)
ACF: Correlation of variations in time series as a function of
ΔtPSD: Variability power per frequency intervalACF and PSD are Fourier transforms of each other (modulo a normalization factor)Weak Stationarity: Mean and ACF of do not change with timeCharacterizing Variability of Stochastic Processes
V
ariability power concentratedon
longer
time scales
V
ariability power concentrated
on
shorter
time scalesSlide8
Non-
parameteric
methods (e.g., periodogram) require high-quality dataNo Assumptions about functional form of PSD or ACFNoisySuffer biases from sampling patternParameteric modeling of PSD possible through monte-carlo methods (e.g., Uttley+ 2002)Can fit arbitrary PSDs for any sampling patternComputationally expensiveNot most efficient use of information in dataParameteric modeling in time domain
Use a parameteric model for time-evolution of stochastic processFitting is done via maximum-likelihood or Bayesian methods
Computationally cheap for many models
Parameteric
Modeling of Stochastic ProcessesSlide9
AGN optical and X-ray
LCs
well described by one or more OU processes (Kelly+2009,2011, MacLeod+2010):Likelihood function derived from solutionExample: Ornstein-Uhlenbeck Process (i.e., Damped Random Walk)
Frequency
PSD
Time, MacLeod+ 2010
Time, Kelly+ 2011Slide10
Use the various variability metrics (parameters) for classification
Classification of variables (
Joey):Improve source selection/classification based on color and other quantitiesFind new classes of objectsIdentification of short-lived (transient) phenomena (Ashish)Time-sensitive, so classification must be done in real time with sparse time seriesCan use variability + other info to identify best strategy for follow-up observations (Joey, Ashish)Classifying Variable SourcesSlide11
Improve modeling of stochastic processes in time-domain (e.g., through Stoch
. Diff. Eq.)
Incorporate multi-wavelength correlations, lagsInclude stochastic flaring activityDevelop time-domain models for arbitrary PSDsImproved computational methods/strategies (Eric)Future massive data sets will either require efficient programs and efficient use of memory or analyzing a subset of the dataImprovement in handling of differences between training and test sets for classification (Joey)Directions for Future Work