bayesian methods in high energy astrophysics · •data collection in high energy astrophysics is...

Bayesian Methodsin High Energy Astrophysics

Aneta SiemiginowskaHarvard-Smithsonian Center for Astrophysics

http://hea-www.harvard.edu/AstroStat/CHASC Astro-Statistics Collaboration

and

SAMSI Astrostatistics - Sep.19, 2012 Aneta Siemiginowska 2

Outline• Data collection in High Energy Astrophysics• Statistical Model setting• CHASC Algorithms and Application Examples

• Spectral analysis with Bayes• Calibration Uncertainties• Low Counts Hardness Ratio• Significance of the diffuse structures in low counts images

NOT to talk about Solar Data, timing, feature detection, upperlimits, and more

• Stars• Supernova remnants• Hot gas galactic outflows, clusters of galaxies• Compact objects: neutron stars, accreting black holes, supermassive black holes• Relativistic jets• GRBs• etc…

Sources of High Energy Radiation

Data Collection in Space

AstronomicalObject

Telescope + Detectors Interstellar Medium

Chandra X-ray Observatory

PhysicsLoss of dataMeasurement ProcessLoss of dataInstrument characteristicsCalibration


Data Collection• Data are collected for each arriving photon:

• the (2-dimensional) location - sky coordinates• the energy• the arrival time

• All variables are discrete• High resolution -> finer discretization,• e.g., 4096 x 4096 spatial or 1024 spectral bins

• Table with photon counts for:• Spectral analysis - 1D• Spatial analysis - 2D• Timing analysis - 1D

Instrumental Effects:Detection inefficiency

• Image:• exposure map“sensitivity to photons per area”

• Spectrum:• effective area (ARF)“sensitivity to photons perenergy”

Energy

Instrumental Effects: Blurring

• Image• point source observed sizedepends on the source locationon the detector• “blurring” is described by a pointspread function (PSF)

• Spectrum• photon energy is “blurred”• a probability of detecting thephoton at given energy in adetector channel is described bya redistribution matrix (RMF)

RMF

Detector channels

Ene

rgy

PSF Simulated Images


A Model-Based Statistical Paradigm• Model Building:

• Model source spectra, image and/or time series• Model the data collection process

• Background contamination• Instrumental effects (effective area, response, pileup, psf)

• Results in a highly structured multi-level model

• Model-Based Statistical Inference• Bayesian posterior distribution• Maximum likelihood estimation

• Sophisticated Statistical Computation Methods are Required• Goals: computational stability and easy implementation• Emphasize natural link with models: The Method of Data

Augmentation


Highly Structured Models Required

van Dyk et al. 2001

Loss of data

Emitted Spectrum

Observed

Model directly the source and data collection mechanism, andinclude statistical procedure to fit the resulting highly structuredmodels and address the substantial scientific questions


flat

Bayesian Inference using Monte Carlo

• The Building Blocks of Bayesian Analysis• The sampling distribution:• The prior distribution:• The Bayes theorem and posterior distribution:

• Inference using a Monte Carlo Sample

We use MCMC(e.g. Gibbs sampler)to obtain a Monte CarloSample


Bayesian Deconvolution

• Complex data collection needs to be included in thestatistical model:

Expected photon count Instrumental

“blurring”* from calibration

Non-homogenous stochastic censoring

* from calibration

Physical SourceModel

Background Contamination

* from separate observation

Observed counts are modeled as independentPoisson variables with means of λ


Source Models

MultiScale Models forDiffuse Emission

Parametrized finite mixture models (several components)

Compound deconvolution models

simultaneous instrumental and physical deconvolution of complex sources

Smoothing prior distributions


MCMC Simulations from the Posterior

Data

Model

Compute Likelihood

prior

Draw parameters

Accept/Reject Update parameters

Calibration

Simulation from the posterior distribution requires careful andefficient algorithms:

Draw parameters from a "proposal distribution'', computelikelihood and posterior probability of the "proposed'' parametervalue given the observed data, use a Metropolis-Hastings criterionto accept or reject the "proposed" values.

pyBLoCXSPython implementation in Sherpa:Two MCMC samplers:

• Metropolis-Hastings:» centered on the best fit values

• Metropolis-Hastings mixed with Metropolis jumping rule:» centered on the current draw of parameters

Explores parameter space and summarized the full posterior or marginal posterior distributions.Computed parameter uncertainties can include calibration errors.Simulates replicate data from the posterior predictive distributions.


pyBLoCXS: Running it!


Trace of a parameter during MCMC run

3D Parameter spaceprobed with MCMC

Cummulative distribution of a parameter

median

stat

istic

s

Gamma NH


Account for Calibration UncertaintiesChandra ACIS-S Effective Area

• Non-linear errors cannot simply add tostats errors.• Include a draw from an ensemble ofeffective area curves in the simulations.

Draw effective area

Data Model

Compute Likelihood

prior

Draw parameters

Accept/Reject Update parameters

Calibration

Drake et al. 2006 Proc. SPIE, 6270,49


Effects of Calibration Uncertainties

Lee et al. 2011

Effect of the ARF uncertainty on fittedparameters and error bars:

Calibration uncertainty exceeds thestatistical errors in high S/N case

Simulations of 105 counts using A0

fit using each of colored ARFs Sim1: Γ = 2 NH=1e23Sim2: Γ = 1 NH=1e21

Sim1 Sim2

Deviations from the default effective area ARF (A0)

30 ARFs with thelargest deviations


Extreme Low Counts: Hardness Ratios

• Comparison of counts in two bands.• The lowest resolution spectral analysis (often used e.g., surveys large,

classifying faint sources).• Ignores the instrumental effects - only valid for the same instrument

• Standard methods based on Gaussian assumptions fail for lowcounts.

• Replace Gaussianlikelihood

with thePoisson likelihood

• Use Bayesian Framework => combine the likelihood with a priordistribution to compute the posterior distributions of hardness ratios


true

modeclassic

Bayes

mean

30 3

Hardness Ratio BEHR Simulations Study

• BEHR - uses Poisson models, background contaminated soft and hard counts• Classic Gaussian vs. Bayes Poisson approach

Park et al. 2006

• With large counts, theBayesian method providesmore precise error bars forhardness ratio than theclassical method - both yieldsimilar results

• The classical method fails inlow counts, error bars areunreliable

Simple ratio fails => Color or HR better


BEHR Application

Inte

nsity

Col

or

Time Color

Inte

nsity

Evolution of spectral hardness

Park et al 2006

samples


Low Counts Images

Original binned X-ray image Smoothed image

How significant is the diffuse emission?


Image Analysis:Complex Statistical Data

Smooth Extended Source• Flexible MultiScale Model

“Point” Sources• Model the location, intensity andperhaps extent and shape

Connors and van Dyk 2007


LIRA: Simulation Study

Known Physical Model Simulated Image

• Simulate the data under the assumed physical modelPhysical Model

• Fit two type of models: 1/ Physical Model

2/ Physical Model + MultiScale Residual (Extra Emission)



LIRA: Simulation Study

Known Physical Model

Simulated Image Posterior Mean of Residuals


null

interesting


Evidence for a StructureExamine joint posterior distribution of • Baseline scale factor: α• Expected total multi-scale counts: β


nullextra


Summary

• Data collection in high energy astrophysics is complex.

• Bayesian framework provides a natural way to includecomplex instrumental characteristics and initialknowledge about the astronomical sources:

• Spectral analysis• Calibration uncertainties• Hardness ratios• Image analysis

• MCMC to explore the posterior distributions.


Conclusions

• The search for highly irregular and unexpected structurein astronomical data possess many statistical challenges.

• Model-based methods allow us to make progress onformalizing and answering scientific questions.

• More sophisticated computational methods and methodsfor summarizing high dimensional posterior distributionsare yet to be explored.


Further Reading1. van Dyk, D. A., Connors, A., Kashyap, V. L., Siemiginowska, A. (2001) Analysis of Energy

Spectra with Low Photon Counts via Bayesian Posterior Simulation. The Astrophysical Journal ,548, 224-243.

2. van Dyk, D. A., Connors, A., Esch, D. N., Freeman, P., Kang, H., Karovska, M., Kashyap, V.,Siemiginowska, A., and Zezas, A. (2006). Deconvolution in High Energy Astrophysics: Science,Instrumentation, and Methods (with discussion). Bayesian Analysis, 1, 189-236.

3. Park, T., Kashyap, V. L., Siemiginowska, A., van Dyk, D. A., Zezas, A. Heinke, C. andWargelin, B. J. (2006). Hardness Ratios with Poisson Errors: Modeling and Computations. TheAstrophysical Journal, 652, 610-628.

4. Connors, A. & van Dyk, D.A., (2007), How To Win With Non-Gaussian Data: PoissonGoodness-of-Fit, SCMA IV, SCMA IV, (Eds. G.J.Babu and E.D.Feigelson), ASPC, 371,101

5. Lee, H., Kashyap, V. L., van Dyk, D. A., Connors, A., Drake, J. J., Izem, R., Meng, X. L., Min,S., Park, T., Ratzlaff, P., Siemiginowska, A., and Zezas, A. (2011). Accounting for CalibrationUncertainties in X-ray Analysis: Effective Areas in Spectral Fitting. The Astrophysical Journal,731,126-145.

6. Kashyap, V. L., van Dyk, D. A., Connors, A., Freeman, P. E., Siemiginowska, A., Xu, J., andZezas, A. (2010). On Computing Upper Limits to Source Intensities. The Astrophysical Journal,719, 900-914.

7. Protassov, R., van Dyk, D. A., Connors, A., Kashyap, V. L. and Siemiginowska, A. (2002).Statistics: Handle with Care, Detecting Multiple Model Components with the Likelihood RatioTest. The Astrophysical Journal, 571, 545-559

bayesian methods in high energy astrophysics · •data collection in high energy astrophysics is...

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