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Outline

Introduction

Replacing stacking

Time symmetry

Two Statistical Problems in X-ray Astronomy
Alexander W Blocker

October 21, 2008


Outline

Introduction

Replacing stacking

Time symmetry

Outline
Introduction Replacing stacking Problem Current method Model Further development Time symmetry Problem Model Computational approach

1 2

3


Outline

Introduction

Replacing stacking

Time symmetry

Introduction

Recent projects have focused on two areas:
Analysis of faint (low-count) x-ray data with Bayesian models Analysis of events in time series

Each has presented a unique set of challenges


Outline Problem

Introduction

Replacing stacking

Time symmetry

General analysis of faint x-ray sources


Outline Problem

Introduction

Replacing stacking

Time symmetry

General analysis of faint x-ray sources

In multiwavelength x-ray studies, astronomers identify potential sources using catalogs in one waveband (typically optical or infrared) and observe the selected sources in x-rays.


Outline Problem

Introduction

Replacing stacking

Time symmetry

General analysis of faint x-ray sources

In multiwavelength x-ray studies, astronomers identify potential sources using catalogs in one waveband (typically optical or infrared) and observe the selected sources in x-rays. This frequently leads to a sample containing many faint, undetected sources.


Outline Problem

Introduction

Replacing stacking

Time symmetry

General analysis of faint x-ray sources

In multiwavelength x-ray studies, astronomers identify potential sources using catalogs in one waveband (typically optical or infrared) and observe the selected sources in x-rays. This frequently leads to a sample containing many faint, undetected sources. We want to combine information from these undetected sources to make inferences about our selected sample.


Outline Current method

Introduction

Replacing stacking

Time symmetry

Current method: stacking
Based on background subtraction For source i , observe cs ,i counts in source aperture and c counts in background aperture.
A b ,i

s Calculate net counts as cn,i = cs ,i - Ab,,ii cb,i , where As ,i and Ab,i are the effective areas for the source and background regions (taking into account exposures), respectively.


Outline Current method

Introduction

Replacing stacking

Time symmetry

Current method: stacking
Based on background subtraction For source i , observe cs ,i counts in source aperture and c counts in background aperture.
A b ,i

s Calculate net counts as cn,i = cs ,i - Ab,,ii cb,i , where As ,i and Ab,i are the effective areas for the source and background regions (taking into account exposures), respectively. ECF Calculate stacked flux as fЇ = P c , where ECF is

x

the mean energy conversion factor.

i

A

s ,i

i n ,i


Outline Current method

Introduction

Replacing stacking

Time symmetry

Current method: stacking
Based on background subtraction For source i , observe cs ,i counts in source aperture and c counts in background aperture.
A b ,i

s Calculate net counts as cn,i = cs ,i - Ab,,ii cb,i , where As ,i and Ab,i are the effective areas for the source and background regions (taking into account exposures), respectively. ECF Calculate stacked flux as fЇ = P c , where ECF is

x

the mean energy conversion factor. 1 Ї Calculate stacked luminosity as Lx = N i LCFi cn,i , where LCFi is the luminosity conversion factor for source i
LCFi =
4 d
2 ,i

i

A

s ,i

i n ,i

ECF

i

вA A s ,i

corr ,i

вK

corr ,i


Outline Current method

Introduction

Replacing stacking

Time symmetry

Problems with conventional stacking


Outline Current method

Introduction

Replacing stacking

Time symmetry

Problems with conventional stacking

Use of background subtraction Gaussian assumption; clearly inappropriate here. Above manifests as negative net counts; for sufficiently faint samples, can lead to negative stacked fluxes and luminosities.


Outline Current method

Introduction

Replacing stacking

Time symmetry

Problems with conventional stacking

Use of background subtraction Gaussian assumption; clearly inappropriate here. Above manifests as negative net counts; for sufficiently faint samples, can lead to negative stacked fluxes and luminosities. No clean measure of uncertainties on luminosities.


Outline Current method

Introduction

Replacing stacking

Time symmetry

Problems with conventional stacking

Use of background subtraction Gaussian assumption; clearly inappropriate here. Above manifests as negative net counts; for sufficiently faint samples, can lead to negative stacked fluxes and luminosities. No clean measure of uncertainties on luminosities. Solution: model data as Poisson


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking"
Observation Model For source i , we assume that cn,i Pois(n,i ) Also assume c
b ,i n ,i

Pois(

Ab,i b ,i As ,i

)

Finally, cs ,i - c

Pois(b,i )


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking"
Observation Model For source i , we assume that cn,i Pois(n,i ) Also assume c Intensity Model If redshifts are known, can model luminosities directly & assume Li Lognormal(µL , L ) (or Li (L , L ))
b ,i n ,i

Pois(

Ab,i b ,i As ,i

)

Finally, cs ,i - c

Pois(b,i )


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking"
Observation Model For source i , we assume that cn,i Pois(n,i ) Also assume c Intensity Model If redshifts are known, can model luminosities directly & assume Li Lognormal(µL , L ) (or Li (L , L )) Otherwise, can apply analogous framework to flux fi .
b ,i n ,i

Pois(

Ab,i b ,i As ,i

)

Finally, cs ,i - c

Pois(b,i )


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking"
Observation Model For source i , we assume that cn,i Pois(n,i ) Also assume c Intensity Model If redshifts are known, can model luminosities directly & assume Li Lognormal(µL , L ) (or Li (L , L )) Otherwise, can apply analogous framework to flux fi . Generally assume
b ,i b ,i n ,i

Pois(

Ab,i b ,i As ,i

)

Finally, cs ,i - c

Pois(b,i )

(b , b )


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking"
Observation Model For source i , we assume that cn,i Pois(n,i ) Also assume c Intensity Model If redshifts are known, can model luminosities directly & assume Li Lognormal(µL , L ) (or Li (L , L )) Otherwise, can apply analogous framework to flux fi . Generally assume
b ,i b ,i n ,i

Pois(

Ab,i b ,i As ,i

)

Finally, cs ,i - c

Pois(b,i )

(b , b )

Using noninformative priors on hyperparameters (Jefferys)


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking", continued

Key assumptions


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking", continued

Key assumptions For luminosity-based inference, assuming that redshifts are known
Relatively plausible for spectroscopic; not as much for photometric


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking", continued

Key assumptions For luminosity-based inference, assuming that redshifts are known
Relatively plausible for spectroscopic; not as much for photometric

Assuming the spectra of sources are know & identical
Typically assume power law with photon index 1.7


Outline Model

Introduction

Replacing stacking

Time symmetry

A hierarchical Bayesian model for "stacking", continued

Key assumptions For luminosity-based inference, assuming that redshifts are known
Relatively plausible for spectroscopic; not as much for photometric

Assuming the spectra of sources are know & identical
Typically assume power law with photon index 1.7

Attempting to make inferences only on selected sample, for now; not dealing with selection effects, etc.


Outline Model

Introduction

Replacing stacking

Time symmetry

Computation


Outline Model

Introduction

Replacing stacking

Time symmetry

Computation
- Using data augmentation algorithm with as missing data cn


Outline Model

Introduction

Replacing stacking

Time symmetry

Computation
- Using data augmentation algorithm with as missing data cn For hyperdistributions, using Metropolis-Hastings step within Gibbs sampler to draw &


Outline Model

Introduction

Replacing stacking

Time symmetry

Computation
- Using data augmentation algorithm with as missing data cn For hyperdistributions, using Metropolis-Hastings step within Gibbs sampler to draw & For Lognormal hyperdistribution, using Gibbs step to draw µ - & L ; Metropolis-Hastings step used to draw n

L


Outline Model

Introduction

Replacing stacking

Time symmetry

Computation
- Using data augmentation algorithm with as missing data cn For hyperdistributions, using Metropolis-Hastings step within Gibbs sampler to draw & For Lognormal hyperdistribution, using Gibbs step to draw µ - & L ; Metropolis-Hastings step used to draw n

L

MH step here is very efficient; using Haley's method to identify posterior modes in parallel and tune proposal distribution.


Outline Model

Introduction

Replacing stacking

Time symmetry

Computation
- Using data augmentation algorithm with as missing data cn For hyperdistributions, using Metropolis-Hastings step within Gibbs sampler to draw & For Lognormal hyperdistribution, using Gibbs step to draw µ - & L ; Metropolis-Hastings step used to draw n

L

MH step here is very efficient; using Haley's method to identify posterior modes in parallel and tune proposal distribution.

From posterior simulations, can retain posterior mean & standard deviation of each source flux (and luminosity, if available) in addition to hyperparameter samples. This provides a great deal of information that is not available with conventional stacking in addition to estimates of sample properties with uncertainties.


Outline Further development

Introduction

Replacing stacking

Time symmetry

Potential directions for further work


Outline Further development

Introduction

Replacing stacking

Time symmetry

Potential directions for further work

Currently have a very fast method that requires no more data than conventional stacking (and makes few additional assumptions).


Outline Further development

Introduction

Replacing stacking

Time symmetry

Potential directions for further work

Currently have a very fast method that requires no more data than conventional stacking (and makes few additional assumptions). Room for improvement in some areas:
Explicit handling of the PSF


Outline Further development

Introduction

Replacing stacking

Time symmetry

Potential directions for further work

Currently have a very fast method that requires no more data than conventional stacking (and makes few additional assumptions). Room for improvement in some areas:
Explicit handling of the PSF Incorporation of spectral uncertainties


Outline Further development

Introduction

Replacing stacking

Time symmetry

Potential directions for further work

Currently have a very fast method that requires no more data than conventional stacking (and makes few additional assumptions). Room for improvement in some areas:
Explicit handling of the PSF Incorporation of spectral uncertainties Incorporation of photometric redshift uncertainties


Outline Problem

Introduction

Replacing stacking

Time symmetry

Testing time symmetry for astronomical events

We have a set of x-ray light curves like the above, each of which is believed to contain an event (in this case, an occultation).


Outline Problem

Introduction

Replacing stacking

Time symmetry

Testing time symmetry for astronomical events

We have a set of x-ray light curves like the above, each of which is believed to contain an event (in this case, an occultation). Interested in testing if the event (a dimming, in this case) is time-symmetric.


Outline Problem

Introduction

Replacing stacking

Time symmetry

Testing time symmetry for astronomical events

Even for the Gaussian case, this is not entirely straightforward.


Outline Problem

Introduction

Replacing stacking

Time symmetry

Testing time symmetry for astronomical events

Even for the Gaussian case, this is not entirely straightforward.
Question of how much structure to place on shape of event. Taking maximum over possible centers of event for less structured approach complex distribution of test statistic.


Outline Problem

Introduction

Replacing stacking

Time symmetry

Testing time symmetry for astronomical events

Even for the Gaussian case, this is not entirely straightforward.
Question of how much structure to place on shape of event. Taking maximum over possible centers of event for less structured approach complex distribution of test statistic.

With Poisson data, we really need a structured model.


Outline Model

Introduction

Replacing stacking

Time symmetry

Intensity model

Define t to be the intensity (count-rate) of our source at time t


Outline Model

Introduction

Replacing stacking

Time symmetry

Intensity model

Define t to be the intensity (count-rate) of our source at time t We model t as: t = c - g (t ; , ) where limt g (t ; , ) = limt - g (t ; , ) = 0 and supR g (t ; , ) = g ( ; , ) = 1


Outline Model

Introduction

Replacing stacking

Time symmetry

Intensity model

Define t to be the intensity (count-rate) of our source at time t We model t as: t = c - g (t ; , ) where limt g (t ; , ) = limt - g (t ; , ) = 0 and supR g (t ; , ) = g ( ; , ) = 1 Thus, c characterizes our baseline source intensity, characterizes the extent of the deviation from this baseline during the event, and g (t ; , ) characterizes the shape of the event itself.


Outline Model

Introduction

Replacing stacking

Time symmetry

Observation model

Given our series of intensities t , we then model the observed counts at time t as: nt Pois (t )


Outline Model

Introduction

Replacing stacking

Time symmetry

Observation model

Given our series of intensities t , we then model the observed counts at time t as: nt Pois (t )

This approach generalizes easily to the high count regime with only minor modifications.


Outline Model

Introduction

Replacing stacking

Time symmetry

Testing
We can then test the hypothesis of time symmetry by placing the appropriate restrictions on and calculating a likelihood-ratio test statistic. The challenge is then to find a parsimonious yet flexible form for the "event profile" g (t ; , ).


Outline Model

Introduction

Replacing stacking

Time symmetry

Testing
We can then test the hypothesis of time symmetry by placing the appropriate restrictions on and calculating a likelihood-ratio test statistic. The challenge is then to find a parsimonious yet flexible form for the "event profile" g (t ; , ). One possibility: a "bilogistic" event profile
-ht kt

g (t ; , h1 , h2 , k1 , k2 ) =

1+e

1 + e kt h1 t < ht = h2 t kt = k1 t < k2 t

|t - |-ht


Outline Model

Introduction

Replacing stacking

Time symmetry

Testing
We can then test the hypothesis of time symmetry by placing the appropriate restrictions on and calculating a likelihood-ratio test statistic. The challenge is then to find a parsimonious yet flexible form for the "event profile" g (t ; , ). One possibility: a "bilogistic" event profile
-ht kt

g (t ; , h1 , h2 , k1 , k2 ) =

1+e

1 + e kt h1 t < ht = h2 t kt = k1 t < k2 t

|t - |-ht


Outline Model

Introduction

Replacing stacking

Time symmetry

Testing, continued

Can also use Gaussian profile for event; tradeoff between degrees of freedom to characterize event and computational requirements.


Outline Model

Introduction

Replacing stacking

Time symmetry

Testing, continued

Can also use Gaussian profile for event; tradeoff between degrees of freedom to characterize event and computational requirements. Because data is non-Gaussian, still need to simulate under null hypothesis to obtain actual distribution of test statistic (cannot necessarily rely on 2 approximation).


Outline Computational approach

Introduction

Replacing stacking

Time symmetry

Maximizing the likelihood
Another challenge: maximizing the likelihood for this model


Outline Computational approach

Introduction

Replacing stacking

Time symmetry

Maximizing the likelihood
Another challenge: maximizing the likelihood for this model It is very multimodal (lots of small, annoying, local maxima)


Outline Computational approach

Introduction

Replacing stacking

Time symmetry

Maximizing the likelihood
Another challenge: maximizing the likelihood for this model It is very multimodal (lots of small, annoying, local maxima) The good news: only the location parameter is truly troublesome


Outline Computational approach

Introduction

Replacing stacking

Time symmetry

Maximizing the likelihood
Another challenge: maximizing the likelihood for this model It is very multimodal (lots of small, annoying, local maxima) The good news: only the location parameter is truly troublesome A solution:
1

2

3

Randomly draw a set of starting values for (possibly based on scan statistics or another simple method). For each starting value, run a fast, local optimization algorithm (such as Gauss-Newton) until convergence. Take the maximum of the values given by the local algorithms.


Outline Computational approach

Introduction

Replacing stacking

Time symmetry

Maximizing the likelihood
Another challenge: maximizing the likelihood for this model It is very multimodal (lots of small, annoying, local maxima) The good news: only the location parameter is truly troublesome A solution:
1

2

3

Randomly draw a set of starting values for (possibly based on scan statistics or another simple method). For each starting value, run a fast, local optimization algorithm (such as Gauss-Newton) until convergence. Take the maximum of the values given by the local algorithms.

This approach parallelizes extremely well, making it ideal for use in a cluster environment (such as Odyssey).