Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/HEAD2013/nstein_13106_20130408.pdf Дата изменения: Mon Apr 22 23:12:25 2013 Дата индексирования: Fri Feb 28 10:39:01 2014 Кодировка:

.

. ..

LIRA: Low-count Image Reconstruction and Analysis
. Nathan M. Stein
Department of Statistics, Harvard University

.

April 8, 2013

Nathan M. Stein

LIRA

.


Overview

LIRA is a software package for the R statistical computing language A main focus for Alanna Connors in recent years Multiscale non-parametric image analysis for use in high energy astrophysics Based on Poisson model suitable for images with low counts Fully Bayesian analysis using Markov chain Monte Carlo Allows for quantification of uncertainty of fitted image and evaluation of goodness-of-fit of a proposed 2D model

Nathan M. Stein

LIRA


Statistical Model

Observations are modeled as independent Poisson(µi )

)6!).%)+&$-17)(& ",&."-$%'&#$&!#6)0&!&

'.*0)&,*.%"(& 1-0/!02#$3& $-00&1"+)0&

$-00&1"+)0&

)4)./5)&*()*&

µi =

"
j

PSFi, j ( !!

0, j

+!

MS, j

)

"

j

!"#$%&'!()*+& ,-$./"$&

1-0/'.*0)& ."1!"$)$%&

Nathan M. Stein

LIRA


Multiscale Component
1216 ESCH ET AL. Vol. 610



MS,i

=G

K k =1

D

klm

Dklm = split proportion at scale k Smoothing prior on Dkl

Fig. 2.--Multiscale image representation. The top layer is the total intensity, and each subsequent layer splits the blocks of its parent layer into four parts. The total intensities of the cells indicated with arrows are shown on the left.

along a grid (e.g., Willett & Nowak these strategies might lead to finer d imag e, we d o not pu rsue th em here . v elopin g me thods tha t pr ovide err or little user intervention as possible.

Figure: Esch et al. (2004)
2002). Although both of etail in the reconstructed Rather, we focus on demaps while requiring as

describing the statistical model, we describe the model fitting technique, summarize the algorithm, and describe the outputs. 3.1. The Likelihood To model the relative intensity of photon emission across the image, we begin by overlaying a grid of pixels on the source image. The photon counts originating from each of these pixels cannot actually be observed because of instrumental effects such as the PSF and other data degradation such as the effects of the exp osure m a p and of instru m ental ba ckgro und co ntamination. In the statistics literature, such unobservable

3. THE EMC2 RESTORATION TECHNIQUE The statistical model used by the EMC2 procedure is c om p r i s e d o f s e ve r a l l a y er s : a l i k e l i h o o d f u n c t i o n , a p r i o r d i st r i but ion , a nd a hy p er pr i or di st r ib ut i on, ea ch of w hi ch i s discussed in detail in the following sections. In addition to

Nathan M. Stein

LIRA


Evaluating the Goodness-of-Fit of a Proposed Model
!"#!$% !"#!$%&%'()% *+!+%
­ 45 ­

,#--%

,#--%&%'()%

­ 57 ­

Given a proposed physical null model:

Fig. 21.-- Before LIRA `Deconvolution': Comparing Data with a Null Mo del. As before, we show our Sky-Truth in the two left-most panels (1st uncolvolved, 2nd convolved with PSF); and our data(center). The Null model we have chosen to use in this sectio$%&is% !""#n ' shown in the two panels at right (1st uncolvolved, 2nd convolved with PSF).

()*#$%&'%

Simulate RunOutputs/ datasets under this model multiple Simple03_Prelim_Gauss2d1.5_PoisDatons128x128testE_Strt0.01_viaFits.out Use LIRA Simple03_Prelim_Gauss2d1.5_PoisDatons128x128testE_Strt0.01_viaFits.param to analyze simulated and observed Simple03_Prelim_Gauss2d1.5_PoisDatons128x128testE_Strt1.00_viaFits.out data Simple03_Prelim_Gauss2d1.5_PoisDatons128x128testE_Strt1.00_viaFits.param Compare structureimin multiscale "residual" · Check the age samples: from analyses of simulated vs observed data Compare posterior distributions of model parameters

Simple03_Prelim_Gauss2dSig1.5_17x17PSF_PoisDatons128x128testEvsNullModel_fits.Rout

Fig. 26.-- Averages of Images Corresp onding to Upp er (Left) and Lower (Right) 5% of chosen Summary Statistic. TOP: Sorted on Summary Statistic combining all Smoothing Hyper-parameters. BOTTOM: Sorted on Summary Statistic combining total Multi-scale counts and background pre-factor. Notice that the images are pretty well constrained.

­ 54 ­

Nathan M. Stein

LIRA


For more information. . .

Software available at: github.com/vkashyap/LIRA A. Connors and D. A. van Dyk. (2007) Statistical Challenges in Modern Astronomy IV D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk. (2004) ApJ 610:1213­1227

Nathan M. Stein

LIRA