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%%%% Statistical Challenges in Modern Astronomy IV, 2007, G. J. Babu and
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\begin{document}
\title{How To Win With Non-Gaussian Data: Poisson Goodness-of-Fit}
\author{A. Connors}
\affil{ CHASC and Eureka Scientific;
(aconnors@eurekabayes.com)}
\author{D. A. van Dyk}
\affil{ CHASC and University of California, Irvine;
(dvd@uci.edu)}
%\tableofcontents
\begin{abstract}
We propose a new method to test for goodness-of-fit of a model for
low-count Poisson data. Our approach does not resemble the usual
methods of approximations to $\chi^2$, but instead explicitly uses the
full Poisson distribution. First, we propose to use a simple
(Poisson-specific) multiscale model to characterize the ``mismatch''
between a best-fit physical model and the data. Next, we embed this
multiscale model into a probabilistic/likelihood framework (via
hierarchical Bayes), allowing us to handle statistical uncertainties.
We then use MCMC to map out the shape of the joint posterior
probability of all of the unknown parameters. Finally, we note that
this is a generalization of a problem with a known solution: whether
an additional component of known shape is justified by the data.
Hence, when the multiscale structure that we use to model the
``mismatch'' between the data and the physical model accounts for a
significant number of counts, and/or when the scale-factor for the
best-fit physics model is significantly different than one, then the
original fit was not ``good-enough''. That is, the fitted multiscale
``mismatch'' isolates the discrepency between model and data. We use
models of the Gamma-Ray sky as viewed by CGRO/EGRET as our example.
We demonstrate that our method works nicely on several examples, but
further work is needed to investigate the method's power.
The authors and CHASC acknowledge support from NSF
(DMS 04-06085) and Chandra X-Ray Center, and AISRP NAG5-12082.
It is a particular pleasure to acknowledge the SAMSI06 Special Program in Astrostatistics.
\end{abstract}
\section{Introduction: Two Tough Problems}
We begin by discussing two tough related questions that often arise in astronomical image analysis:
``Is a fitted model {\it Good Enough}?'' and ``Is there additional unknown structure in the image (or spectrum)?''
That is,
faced with a dataset, and a physical model\footnote{
Throughout this paper we use ``physical model'' to mean
a model of the photon source. It is assumed
to be a parametric model based on prior physical information, and
of (relatively) low dimension. The ``statistical model'', on the other hand,
is a model of the complete data generating mechanism. It
incorporates both the physical model and the statistical
distribution of the data including any calibration and measurement uncertainties.}
that is to be fit to the data
an astrophysicist generally confronts
four main questions:
\begin{description}
\item[\sc Question~1:] What physical model parameters best fit the data,
and what are their uncertainties;
\item[\sc Question~2:] What is the evidence for an additional
component of known shape (such as a line in an an
energy spectrum or a point-source in an image);
\item[\sc Question~3:]Is this best-fit good enough;
and
\item[\sc Question~4:] If the fit is not good enough
how does one quantify the unknown additional structure in the image
and the uncertainties in this structure?
\end{description}
Most astrophysicists know reasonably principled procedures
for {\sc Question}~1 (exploring the probability space around
the maximum likelihood estimate) and {\sc Question}~2 (such as computing limits on
the size of the possible additional component, or calibrating the
rise in log-likelihood as in Protassov et al. (2001)
and the references therein).
However outside the Gaussian realm, {\sc Questions}~3 and 4---is the fitted physical model good enough
and how does one quantify additional structure in the image---are tough.
Since we will use the structure
later, we next sketch four example problems, rather pedantically,
from simplest to more troublesome.
\vskip-0.5in
\subsection{One Bin Problems}
\paragraph{\underline{\sc Problem~1: One Bin, Gaussian Statistics.}}
In the first problem, data are a rate in a single bin, with a known physical model giving the mean (and
previously-measured instrumental characteristics giving the variance).
It is straightforward
to obtain the tail probability of the data under the Gaussian model.
There is a well-known reference distribution
(i.e., $\chi_1^2$) that does not depend on details of the parameters or the
model and that is easy to calculate once ahead of time using a standard table.
Further, any extra or missing component can be estimated
simply as the data minus the model (i.e., the residual), whether one prefers
a mean, a mode, or a median,
since they are the same. Unfortunately, the simplicity of this analysis is unique to this
simple case.
\paragraph{\underline{\sc Problem~2: One Bin, Poisson Counts.}}
In this case, the data are again in a single bin, with the known physical model
giving the mean. But now the ``rate'' is a discrete variable, i.e., the
Poisson-distributed counts in the bin (or detector) of known size (or known ``effective area'').
Once again, the computation of a tail probability is straightforward
(if more tedious---it must be calculated rather than looked up;
see Bevington 1969, Loredo 1992). One has a choice of estimator for both the
physical model and the excess structure. The Bayesian posterior mean, mode, and
median, for example, are not the same. Nonetheless the procedures are conceptually
simple (if numerically more tedious).
\vskip-0.5in
\subsection{Multi-Bin Problems}
With more bins, we can model shape or structure in how the flux varies among the
bins as well as the total flux. At the same time, the two tough questions,
{\sc Questions}~3 and 4, require more complex treatment.
\paragraph{\underline{\sc Problem~3: Multiple Bins, Gaussian Statistics.}}
Presume one has a spectral energy distribution or image, with independent Gaussian
measurements and heteroscedastic errors (i.e., the variances are known ahead of time,
but differ among the energy channels or image pixels).
\paragraph{\sc Question~1:} {\it Parametrized physical model---best fit and uncertainties.}
As Lampton, Margon and Bowyer (1976) and Cash (1976)
suggest, one finds the mode (or modes) by maximizing the log-likelihood
function via one of a number of standard numerical optimization methods
(as in Press et al 1992). The region around the mode can then be
mapped. The correspondence between drop in log-likelihood and
interval coverage is assumed (via the Central Limit Theorem)
to be given by $\chi_\nu^2$ regardless of the details of the model.
Alternatively, a Bayesian procedure maximizes and maps out the posterior probability of
the parameters given the model and data. The probability
that a parameter is within a certain range (``credible region'') is then calculated directly
(as in Loredo 1992 and references therein).
\paragraph{\sc Question~2:}{\it Evidence for an additional component of known shape.}
Since a parametric function for the possible additional component is
known, one builds on the procedure described above for one bin in two
ways. First, one investigates the limits on the total flux (or
equivalent) of the unknown component by computing confidence or
credible intervals. Second, one investigates whether the rise in
log-likelihood upon the addition of this component signal it is
statistically significant. (We remind astrophysicists that for many
applications, often including point sources and spectral lines, the
mapping between the rise in log-likelihood and the chance significance
cannot be compared to a pre-computed reference distribution such as
$\chi^2$, but must be calibrated via appropriate Monte Carlo;
Protassov et al. 2002)
\paragraph{\sc Question~3:}{\it Is the best fit good enough?}
One can compute a $\chi^2$ on $\nu$ degrees of freedom, and compare it to
a previously tabulated ``reference distribution'' (i.e., $\chi^2_\nu$)
by invoking the Central Limit Theorem (e.g., Eadie et al. 1971 and Bevington 1969).
There is nothing technically incorrect about this simple, popular procedure.
However $\chi^2$ is almost always an approximation
to the true distribution. Hence any statistic that relies
on its tail (where the differences would be the most pronounced) should be viewed with skepticism.
Also, Loredo (1992) points out that occasionally one should {\it expect}
data that are not a ``good fit'' to one's model, by chance.
\paragraph{\sc Question~4:} {\it Quantify unknown additional structure and its uncertainties.}
If the physical model is not sufficient to describe the data, the
difference between the data and the physical model, i.e., the
residual, is a reasonable estimate of additional unknown component.
Uncertainties can be computed by appropriately scaling the pixel-wise
observational variances. If something is known about the structure of
the additional component (such as local smoothness) one could gain
additional power by using a model that formalizes this knowledge.
However there is nothing actually wrong with simply subtracting the
physical model from the data, as is usually done.
\paragraph{\underline{\sc Problem~4: Multiple Bins, Poisson Counts.}}
This is the same as {\sc Problem~3}, but with Poisson data, i.e., discrete counts.
\paragraph{\sc Question~1:} {\it Parametrized physical model---best fit and uncertainties}
and {\sc Question~2:} {\it Evidence for an additional component of known shape}.
The procedures for these questions are essentially the same as for the Gaussian case
(Cash 1979; Protassov et al. 2002).
\paragraph{\sc Question~3:} {\it Is the best fit good enough?}
Here is our challenge.
In the Poisson case, there is no pre-computed, single, reference distribution
that does not depend on the details of the model and data.
\paragraph{\sc Question~4:} {\it Quantify unknown additional structure and its uncertainties.}
If one has no good prior knowledge of a simple parametric model for
the ``mismatch'' between the data and physical model, how does one estimate
shape and significance of the mismatch?
Subtracting the model from the data is no longer a handy estimator
even of the mean. Even if the counts are relatively high, when one
is interested in low probability events---such as the
counts from the unknown additional structure---it is important to take full account
of the skew and long tail of the underlying Poisson distribution.
\medskip
We briefly note that although we have not discussed instrumental ``blurring" or ``smearing'',
if the blurring matrix or point spread function
is assumed to be known completely,
it is straightforward to convolve it with the sky and physical models
before comparison with the data. Thus, we mainly ignore these issues throughout this article.
(The case of uncertainty in instrumental
response is being addressed in a hierarchical way
in a separate effort; see Drake et al. 2006.)
\section{High Energy Gamma-Ray Bridge Over the Milky Way, circa 1999}
\subsection{Brief Gamma-Ray Astronomy Overview: Matter \& Energy in our Galaxy}
Gamma-ray emission (and hence detection) is an instrinsically Poisson
process (Longair 1992). At very high energies, even after almost
a decade of high-resolution satellite observations are summed, there
are many regions in the sky with very low counts per bin (see Figure
1a, Left). Gaussian approximations are not suitable.
Detailed physics theory --- such as quantum mechanics --- allows
researchers to `decode' the basic physics of distant objects via their
remotely-detected photons. Across the entire electro-magnetic
spectrum, from radio (lowest energies) through infrared, optical, UV,
X-ray, and gamma-rays (highest energies), photons tell us about
specific processes in the sky. Gamma-ray emission originates, for
example, from high-energy cosmic processes including nuclear decay
and high energy particles impinging on the magnetic fields, dense
material, or lower energy photons. In particular, gamma-rays reveal
the whereabouts of extremely energetic particles, in our Milky Way
Galaxy and beyond.
Given this, do we understand the production cycle of such highly
energetic matter and energy in our Galaxy? Or is there more ``stuff''
we are missing? From our vantage point two thirds of the way out from
the center of the Milky Way Galaxy, how well can we use the
information packed in gamma-ray photons to learn about these
processes?
\subsection{The Gamma-Ray Sky}
\paragraph{Instruments.}
These were the questions and techniques we faced in the 1990's,
when the {\it Compton
Gamma-Ray Observatory} (CGRO), the first of NASA's ``Great
Observatories", surveyed the entire sky for gamma-ray emission.
CGRO carried two survey
telescopes: COMPTEL (0.5-30 MeV, the nuclear regime); and
EGRET (30 MeV-100 GeV). It also carried two non-imaging instruments:
BATSE, which focused on gathering signals from
the whole sky as a function of time (in order to observe gamma-ray bursts); and
OSSE, which concentrated on spectra.
Of these, the instrumental response of CGRO-EGRET's Spark Chamber is
the simplest, and its particle induced background is the lowest.
Hence for this paper, where we ignore ``deconvolution'', we used
$>1$GeV EGRET Spark Chamber data (real and simulated), binned into
$1.5^{\circ}\times 2^{\circ}$ bins, a level of resolution where instrumental
blurring is minimal.
\paragraph{CGRO/EGRET data.}
In Figure~1a (on the left), we show the $>$1~GeV counts (per
$0.5^{\circ}\times 0.5^{\circ}$ bin) collected by the CGRO/EGRET telescope during its
$\sim$9+ years viewing the sky (Nolan et al. 1992, Thompson
et al. 1993, Esposito et al. 1999, Mattox et al. 1992). On the
right is the corresponding EGRET telescope effective area, in units
of $\rm{cm}^2$. Visible in the plot of gamma-ray counts are a couple
hundred ``point sources'' (Hartman et al. 1999). A number are
thought to be spatially unresolved glowing gas from supernova
remnants (SNRs) (Esposito et al. 1996). Virtually all of the others
are understood to be compact objects such as neutron stars and
accreting black holes. These include pulsars (neutron stars with
magnetic fields roughly a dozen orders of magnitude greater than that
of the Earth) and various kinds of active galaxies.
\begin{figure}
\begin{center}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 82 325 278]{cgroegretcountsg1t9g004loresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 82 325 278]{cgroegretexposrg1t9g004loresgs2.eps}
}
\caption{CGRO/EGRET data: a) Left: $>$1~GeV counts measured over the whole
sky over the lifetime of the mission (from HEASARC), plotted in Galactic coordinates in $0.5^{\circ}\times 0.5^{\circ}$ bins.
b) Right:
corresponding exposure (effective area times time).}
\label{fg:cgroegretallskydata}
\end{center}
\end{figure}
The bulk of the signal in this low-background,
low count regime, however, comes from diffuse gas
and nearby background photons along the plane
of our galaxy
(e.g., Cillis and Hartman 2005, Elwe 2003, and references therein),
glowing as they are impacted by energetic particles
called cosmic rays.
There were physics-based
models available at the time that predicted what the diffuse
glow should look like (Bertsch et al. 1993; Hunter et al. 1997,
Strong and Moskalenko 1998, Moskalenko, Strong, and Reimer 1998, Moskalenko and Strong 1998).
These models of the Galaxy included supernovae as sources of
energetic cosmic ray particles; the spatially complex gamma-ray glow
from energetic particles impinging on gas clouds (Bremmstrahlung and Pion decay); and a broader,
smoother component due to these energetic particles ``boosting''
the energy of photons from the ubiquitous low energy (microwave and infrared) background (Inverse Compton).
The components of these physical models were measured
separately. High resolution radio observations were used to map out the gas distribution,
and local measurements of cosmic ray particles were used to constrain their energy spectra
and abundances.
With the physical model and CGRO observations in place, a fundamental question was
whether the models
matched the observed data.
\subsection{Imaging Methods for the Whole Sky in Gamma-Rays}
\paragraph{CGRO era.}
In order to answer this question, Dixon et al. (1999) applied the
TIPSH (translation invariant Poisson smoothing using Haar wavelets)
method of Kolaczyk (1997) and Dixon and Kolaczyk (2000). That is, in {\sc Question}~4 under {sc Problem~4} above, they
modeled the ``mismatch'' between the physical model and the CGRO/EGRET
all-sky data with a multiscale (boxy-shaped Haar wavelet, with
cycle-spinning) model. The TIPSH ``keep or kill'' thresholds were
carefully tailored to match the Poisson statistics. At the time, they
did find a mismatch that looked rather like a bridge above the plane
of the Galaxy. They famously emphasized, however:
\begin{quote}
``The immediate question arises as to the statistical significance of
this feature. Though we are able to make rigorous statements about
the coefficient-wise and level-wise FDR [False Discovery Rate], similar quantification of
object-wise significance (e.g., `this blob is significant at the
$n$ sigma level') are difficult.''
\end{quote}
The use of multi-scale methods was a notable improvement in modeling
unknown diffuse emission, which is expected to show both the sharp and
broad features of the complicated diffuse gas, the photon field, and
sea of energetic particles (cosmic rays) from which it arises.
Previously, among astrophysicists, other prominent ``model-free'' or
non-parametric (i.e., a model with about as many parameters as data
bins) methods include Richardson-Lucy (the EM algorithm applied to an
unconstrained grid of poisson parameters; Richardson 1972, Lucy 1974,
Shepp and Vardi 1982) and Maximum-Entropy-related methods (such as MaxEnt
and successors; Skilling 1989; also Strong 2003). These methods
lacked the multiscale representation of both diffuse and sharp features.
\paragraph{Progress 1: NK.}
In the ensuing years, several potential improvements were tried. (See
also Willett 2006 and Young 2006, this volume; and the different
approaches of, say, Bourdin et al. 2001; Starck, Pantin and Murtagh
2002, Starck and Murtagh 2006 and references therein.) The TIPSH
wavelet structure was revamped into a more general multiscale
structure on certain ratios of expected counts to exploit the
statistical properties of the Poisson likelihood and to allow the
inclusion of instrument response (``smearing'' or blurring) in an
elegant (and speedy) way (Nowak and Kolaczyk 2000; Kolaczyk and Nowak
2004, 2005; henceforth this method is referred to as NK). Like
wavelets (which often use recursive dyadic partitioning), the NK
structure breaks an image into four quadrants (or two segments, in 1D)
and then successively breaks each of these quadrants into its four
quadrants until the inherent pixelization is reached. Unlike
wavelets, for which the basis function is typically some sort of
differencing function, for NK the basic shape is a constant intensity
block. Further, rather than working with the intensity itself, we
consider the ratios of the intensities of each quadrant to the whole
(see Willet 2006, this volume). These ``split probabilities''
are described by prior distributions tuned by smoothing parameters, in
analogy to the way that shrinkage (or complexity penalties) are used
to estimate the best values of coefficients in a wavelet
decomposition. Further, NK is formulated so that the problem is
convex and there is only one mode, so good convergence is assured.
Finally, as with most wavelet analyses, cycle-spinning is recommended
to remove artifacts in the fitted image (Coifman and Donoho, 1995).
Still, this used the EM algorithm to get a ``best fit'' image, and
uncertainty estimates were not available.
\paragraph{Progress 2: EMC2.}
Later, Esch et al. (2004) embedded the NK multiscale model
into a hierachical Bayesian framework.
(See Karovska et al., 2005 for applications with
beautiful graphics.) This also allows the
smoothing parameters used in NK to be fit to the data.
The range of variation of the tuning parameters
is effectively governed by a hyper-parameter
that is set ahead of time.
In order to fully map out the posterior distribution of the various
model and tuning parameters,
Esch et al. (2004)
uses a Markov Chain Monte Carlo sampler.
This procedure produces not only
a ``best fit'' image
(the mean, rather than the mode), but also a
full exploration of the probability space, via
the MCMC samples. Hence in principle one has
all the information necessary for calculating
any summary of the posterior distribution and to quantify uncertainty in the
fitted image.
The results of an EMC2 run on a Poisson image are often not as
pleasing to the eye as, say, a Gaussian-kernel-smoothed picture, or
the related popular (commercial) package {\it Pixons} (Pina and
Puetter 1993). However, since the original NK algorithm elegantly
factors the log-likelihoood, it is both complete and speedy. The
examples later in the paper each took about 40 minutes to run on a
small iBook G4. Further, in theory, one has all the information one
needs to understand the uncertainties in the fitted image.
Unfortunately, it was not immediately obvious how this information
could be synthesized to answer the questions of direct scientific
interest.
We emphasize that although we are not discussing it here,
NK and EMC2 do have the capacity to ``deconvolve'' the image,
see Esch et al. (2004) and Karovska et al. (2005).
(More precisely, EMC2 employs forward-folding of the multiscale model
convolved with the instrument response.)
\paragraph{New features.}
During the SAMSI06 Special Session on Astrostatistics,
the second author worked on modularizing,
speeding up, and adding new features to the EMC2 code. In this paper,
we focus particularly on the ability to add a shaped
baseline component and to fit this component's overall
intensity at the same time as inferring the shape
of an additional NK-type component.
The critical question was still: How can we use these methods, in
practice, to answer the question Dixon et al. (1999) posed? How can
we begin to quantify the statistical significance of features in the
fitted image and how can we assess goodness of fit of the statistical
model?
\section{How to Define a `Feature'}
\paragraph{Usual thinking.}
An astrophysicist investigating possible unknown diffuse emission (or
an unknown spectral feature) usually tries to draw a boundary around
it and then tries to estimate its significance. Whether doing simple box-car
smoothing or more robust Gaussian kernel smoothing (Ebeling, White,
and Rangarajan 2006), or even wavelet-based detection of potential
sources at different scales (Freeman et al. 2002, 2003), the usual
algorithms can be thought of as defining a region of `significant' emission by
drawing a boundary around it and calculating its significance. That
is, once the boundary is drawn, one can ask: are there a significant
number of counts in this new feature, compared to satistical
fluctuations expected from my ``baseline'', or known physical model?
(See also wavelets in an EM context in Kn\"odelseder et al. 1999).
Still, as in Dixon et al. (1999), the researcher is essentially
looking at coefficient-wise or level-wise (rather than
``object-wise'') significance. Even the first uses of EMC2 followed
(roughly) this usual thinking (Esch et al. 2004; Karovska et
al. 2005), defining the boundary at the pixel-wise $3 \sigma$
limit.
However, the larger question of how to
quantify the uncertainty in the boundary itself remained unanswered.
%%%Other useful techniques include allowing the data to determine the
%%%size of the model ``blocks'' via changepoint algorithms such
%%%as Bayesian Blocks (Scargle 2006, this volume) and the related
%%%methods based on
%%%Voronoi tesselations ({\bf voronoiRef}).
%%%In both these cases {\bf [what cases?]}, explicitly incorporating the known
%%%instrument response is theoretically possible,
%%%but inefficient in practice {\bf [I THINK..]}.
%%%\paragraph{With instrument response and background model}
\paragraph{Key breakthrough---It's in there.}
After much discussion, we realized that the multiscale model itself
had already ``chosen'' the boundaries. In real astrophysical sources,
boundaries of glowing gas clouds or photon fields are not sharply
defined, nor are they smooth or even continuous. No additional step
was needed: we could start with the best-fit physical model and add an
additional component model via the NK/EMC2 flexible multiscale
model. Since we know how to fit this model, we have, in essence,
reduced our original two tough problems ({\sc Questions}~3 and 4 under {\sc Problem~4}) to
one that we already know how to handle (i.e., {\sc Question}~2, but with a
flexible semi-parametric model as our proposed additional component,
rather than a low-dimensional physics-based model).
We can quantify the evidence for including the added multiscale
component in a simple manner by looking at the posterior distribution
of the total flux from this component using the MCMC sampler, as we
have done informally (visually) in Figures 6 and 9. A more
computationally intensive but possible approach is to run a simulation
to calibrate the distribution of the change in log-likelihood with the
addition of this component. This strategy is similar to the
calibration methods for the LRT promoted by Protossov et al. (2002).
In summary, our method starts with the physical model and adds a
multiscale model. We aim to investigate whether any statistically
significant structure above the supposed physical model is captured by
the multiscale component (detection). This strategy simultaneously gives us a
model for the unknown feature's shape (characterization). Finally, we
need a strategy for determining if the original physical model is
``good enough". That is, we need a procedure that gives a ``goodness
of fit" measure for our best-fit physical model and describes the
shape and significance of the ``residuals".
\begin{figure}
\begin{center}
%\vspace{1.80in}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{modeltruediffonlyloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons000loresgs2.eps}
}
\caption{Quantifying Nothing---Model+Data:
a) Left: Physical model of all-sky diffuse emission $>$1~GeV;
b) Right: Poisson data simulated from the physical model with no extra component.
Top and bottom are padded with zeros to make an even 128x128 bins.
}
\label{fg:datons000msk100model}
\end{center}
\end{figure}
For this final objective we suggest calibrating the null distribution
(for what no added feature looks like) by simulating several datasets
from the physical model (our ``baseline'', or null hypothesis). On
each of these simulated data sets, we run the EMC2+baseline fitting
procedure. The baseline corresponds to the physical model and the
multiscale model is used for the added model component or feature. We
then compare the joint distribution of the physical model scale factor
and the total expected counts form the multiscale component for each
simulation with that of the real data. If the multiscale component is
more important when fit to the real data than to the simulations, we
conclude the new extra component may be real.
\section{Worked Example: Simulations of All-Sky CGRO/EGRET Data}
\paragraph{Simulation summary.}
We began with simulated data
based on the CGO EGRET summed phase 1-9 exposure and all-sky count-rates,
courtesy of HEASARC.\footnote{
{\bf cossc.gsfc.nasa.gov/docs/cgro/egret/egret\_doc.html}. }
\begin{figure}
\begin{center}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons000msk100meanloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons000msk100sigmloresgs2.eps}
}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons000msk100skewloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons000msk100kurtloresgs2.eps}
}
\caption{Quantifying Nothing---Results:
a) Top Left: mean value in each pixel from 900 draws of multiscale model (i.e., EMC2 run);
b) Top Right: sigma of same;
c) Bottom Left: skew;
d) Bottom Right: kurtosis.
}
\label{fg:datons000msk100mean}
\end{center}
\end{figure}
We use the
output of the physics-based model GALPROP
(version {\bf fits\_49\_700405})
provided via A. Strong \footnote{{\bf www.gamma.mpe.mpg.de/\~~aws/aws.html}}
(Strong, Moskalenko, and Reimer 2004; Strong, et al. 2004; 2005);
together with the CGRO/EGRET exposure map of Figure 1b,
to simulate several physically realistic maps of the gamma-ray sky $>$1~GeV.
We simulated Poisson datasets that matched the physical model
assuming an integration time of about two weeks,
as well as datasets with extra components and missing pieces.
Our procedure is:
\begin{enumerate}
\item The physical model is used as a baseline in EMC2.
\item An additional component of unknown shape is modeled using the NK/EMC2 multiscale model.
\item The normalization of the baseline physical model is fit.
\item If the multiscale model component contains a significant number of counts, and/or
if the normalization of the baseline is significantly different than one,
then we conclude the physical model is not a good enough fit.
\item Significance is assessed by comparison to the posterior distributions
produced by running EMC2 on simulations from the uncontaminated physical model
(i.e, the baseline or null model).
\end{enumerate}
\begin{figure}
\begin{center}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{modeltruediffonlyloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons539loresgs2.eps}
}
\caption{Bright Unknown (Discontinuous) Source---Model+Data:
a) Left: Again, the baseline is the physical model of all-sky diffuse emission $>$1~GeV;
b) Right: Poisson data simulated from the physical model with a bright added component.
Top and bottom are padded with zeros to make an even 128x128 bins.
}
\label{fg:datons539msk100model}
\end{center}
\end{figure}
\paragraph{Measuring nothing (no added feature).}
In the first simulation study, we demonstrate how this procedure works
when our data has no extra component above the physical model
(i.e., baseline or null model; Figure~2). Even when started far from
the true value, the MCMC sampler converges correctly, showing no
significant structure or counts in our multiscale component, see
Figure~3. There is no evidence for multiple modes; the inferred mean
(and mean/sigma) are very small compared to the counts (and baseline
rate); and the skew (and kurtosis) are large, indicating the posterior
distribution of the multiscale count is highly skewed towards zero, as
it should be. In order to spread out the sharply peaked posterior
probability of our two parameters of interest (the baseline scale factor and
total expected count from the multiscale component) we have plotted them
on a $\log_{10}$ scale.
\begin{figure}
\begin{center}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons539msk100meanloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons539msk100sigmloresgs2.eps}
}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons539msk100skewloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons539msk100kurtloresgs2.eps}
}
\caption{Bright Unknown (Discontinuous) Source---Results:
a) Top Left: mean value in each pixel from 900 draws of multiscale model (i.e., EMC2 run);
b) Top Right: sigma of same;
c) Bottom Left: skew;
d) Bottom Right: kurtosis.
Recall these moments display only the shape
of the unknown component. Significance
is assessed as in Figure 6.
}
\label{fg:datons539msk100mean}
\end{center}
\end{figure}
~
\begin{figure}
\begin{center}
\mbox{
% \plotfiddle{PSFILE}{VSIZE}{ROTANG}{HSCALE}{VSCALE}{HTRANS}{VTRANS}
% \plotfiddle{sample.eps}{2.6in}{-90.}{32.}{32.}{-250}{225}
% where HSCALE and VSCALE are percentages and HTRANS and VTRANS are
% in PostScript units, 72 PS units = 1 inch.
\includegraphics[height=2.4in]{dat539FitScatter.eps}
\includegraphics[height=2.4in]{dat000539FitScatter.eps}
}
%\vskip-0.21in
\caption{Calibrating Results---Null vs Bright Unknown Component:
Scatter plot of $\log_{10}$ of total number of counts inferred
to be in the multscale model for each of the draws, versus
$\log_{10}$ of the baseline scale factor.
a) Left: Bright unknown component;
b) Right: Same, compared to results on the baseline (null).
Bullets indicate data with no signal (null hypothesis), while `+' indicates
results from data with a bright additional component.
}
\label{fg:datons000539msk100scatter}
\end{center}
\end{figure}
\paragraph{Bright discontinuous unknown.}
The results appear very different when we add a
bright but discontinuous, irregular, and low surface brightness
second component to the physical model, see Figure~4.
The difference between the baseline physical
model (Figure~4a) and simulated data (Figure~4b) is visible to the unaided eye.
Yet from plots of the mean (and sigma) counts per pixel (Figure~5a and 5b),
one sees the rate per pixel is not significant.
But the overall multiscale component is hugely significant (Figure~ 6).
Again, even when starting far from
the true value, the MCMC sampler converges correctly and there is no evidence
for multiple modes. The unknown component's shape---i.e., the difference between
areas where the the two irregular diffuse components
are statistically significant and where they are not---is easily seen in the plots of the
mean, sigma, skew, and kurtosis.
Again, in order to spread out the sharply peaked posterior probability of
our two parameters of interest, the baseline scale factor and the total expected counts
due to the multiscale component,
we use the $\log_{10}$ scale.
\begin{figure}
\begin{center}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{modelHlfSquishICloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datons000loresgs2.eps}
}
\caption{Faint Unknown Source, Baseline Shape Mis-Specicified --- Model+Data:
a) Left: Baseline is physical model of all-sky diffuse emission $>$1~GeV, but with
intensity distribution of IC component gradually suppressed above the Galactic plane;
b) Right: Poisson data simulated from same spatial shape as the physical model
in Figure 2.
}
\label{fg:datonsHSqmsk100model}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
%\vskip0.5in
\mbox{
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datonsHSqFixmsk100meanloresgs2.eps}
\includegraphics[height=1.5in,width=2.4in,bb= 36 80 325 281]{datonsHSqFixmsk100sigmloresgs2.eps}
}
\caption{Faint Unknown Source, Baseline Shape Mis-Specified---Results, Scale Fixed:
a) Left: mean value in each pixel from 900 draws of multiscale model (i.e., EMC2 run);
b) Right: sigma of same.
(The skew and kurtosis follow the same pattern as in the previous example.)
Recall these moments display only the shape
of the unknown component. Significance
is assessed as in Figure 9.
}
\label{fg:datonsHSqFixmsk100mean}
\end{center}
\end{figure}
\paragraph{Faint unknown source, baseline shape mis-specified.}
In our final simulation, the total difference
between the model and simulated data is only about $\sim10^2$ counts
(rather than $\sim10^3$ counts, as above).
Further,
as careful visual comparison of the baseline (Figure~7a)
and the simulated data (Figure~7b) suggests,
the Poisson counts were simulated from a distribution
with the same spatial shape as the baseline physical model,
but with overall intensity of the Inverse Compton (IC) component suppressed
via a power-law ($\sim b^{-2}$) as one moves above the Galactic Center.
This roughly represents a realistic mistake in the physics model.
We have processed these data two ways: with the baseline scale
component fixed; and with it allowed to float.
The multiscale component
picks up the missing broad glow quite easily
when the baseline scale factor is fixed; with the
brightest parts being where the exposure is greatest (Figure 8).
However, interestingly, when the scale factor is allowed to float,
the procedure prefers to increase the scale factor
rather than put counts into a new component
(Figure 9).
\begin{figure}
\begin{center}
%\vskip0.25in
\mbox{
\includegraphics[height=2.4in]{dat000HSqFitScatter.eps}
\includegraphics[height=2.4in]{dat000HSqFixHistogr.eps}
}
\vskip-0.21in
\caption{Calibrating results---Null vs Faint Unknown:
a) Left --- Baseline scale floats free: Scatter plot of $\log_{10}$ of total number of counts inferred
to be in the multscale model for each of the draws, versus
$\log_{10}$ of the baseline scale factor. Bullets indicate
no signal (null hypothesis). A `+' indicates
results from data witha faint additional component.
b) Right --- Baseline scale fixed (at unity): Histogram of $\log_{10}$ of total number of counts inferred
to be in the multiscale model for the null (white) and data with
faint additional componenent (gray).
}
\label{fg:datonsHSqmsk100scatter}
\end{center}
\end{figure}
\section{Conclusions: General Principals, and Further Work}
\subsection{Brief Mention of Other Methods}
There are many methods not mentioned here (e.g., Rice, 2006, these
proceedings). In a general sense, they share with our proposed
technique an overarching strategy of non- or semi- parametrically
looking for regions of fluctuations in the data beyond those expected
by chance from the underlying distribution of the data. Our method,
however, is specifically tailored to Poisson images and energy
spectra. We note that other Poisson multiscale (or very flexible)
models could also be used in place of EMC2 (e.g., Bourdin et al. 2001,
Starck and Murtagh 2006, and references therein).
\subsection{Further Work}
The simulations described here, though promising are just the
begining. We hope to develop methods to quickly calibrate and
precisely quantify the improvement offered by the addition multiscale
component. Rather than a simple visual inspection of the posterior
distribution, such methods would offer a precise numerical summary.
The power of the method to detect weak added source is another area
that we hope to explore. (Preliminary tests, however, show that where
the total difference between our usual baseline physical model---which
contains several thousand counts over the whole sky---and the
simulated data is a factor of 10 lower than the faintest example shown
here, we see no visible detection.) Finally, although the capability
for incorporating an instrument response has been built into the code,
it has not been extensively tested on these data.
There are many practical directions in which we would like to
improve the functionality of out method.
For example,
using all-sky spherical data rather than nested square pixels;
allowing for a PSF or instrument response that varies over the field of view;
incorporating simple parametric fitting in the baseline physical model;
using the full energy information together with the imaging information;
incorporating uncertainties in the physical model and/or ``instrument" model;
and including more complex prior physical information
(say, high resolution radio data).
We expect to work on some of these in the near future.
\subsection{Soapbox}
Traditionally, physicists receive very little formal training
in probability and statistics.
Our cultural habit is to think in terms of
Gauss-Normal statistics and $\chi^2$.
(``How many sigma is that result?", we ask of a detection
where, say, the expected background counts are 0.2 and measured source plus
background counts are 2.)
At the same time, physicists tend to be comfortable with
likelihood methods, and well-versed in Monte Carlo ---
and so have understanding of the basic building blocks
for tackling non-Gauss-Normal problems directly,
rather than by approximations to older Gauss-Normal
statistics, such as $\chi^2$.
Here, we have tackled Poisson data. It is good as one specific example
of the Non-Gaussian regime.
This forces us (astronomers and physicists) to {\it think}.
We cannot rely on the symmetry and limiting properties
that make, say, ``filtering" or ``subtracting" or ``correcting" {\it data}
mathematically/probabilistically/statistically equivalent
to a thorough likelihood analysis.
Instead we must pay careful attention to the
unfolding of the {\it model}:
astrophysical + instrumental + statistical.
We must leave the {\it data} alone; our tampering will not improve them.
In this paper, we have used hierarchical methods of taking apart
and putting together models and data. This enabled us to
propose a practical solution
to two tough long-standing problems for Poisson data:
``good-enough'' fit and querying
the data for evidence of an unpredicted component
of unknown shape.
We have at least qualified success in testing a solution.
But we argue the larger lesson is:
many supposedly intractable problems
can be analyzed using this same framework
of carefully breaking the difficult problem apart
into smaller, more manageable pieces.
Likelihood-based methods such as Hierarchical Bayes offer ideal tools to this end.
\acknowledgements
{The authors and CHASC acknowledge support from NSF
(DMS 04-06085) and Chandra X-Ray Center.
AC acknowledges AISRP NAG5-12082 ``Astrostatistical Tools in Python'', and the work
of her co-investigators (T. Loredo and T. Oliphant).
It is a particular pleasure to acknowledge the SAMSI06 Special Program in Astrostatistics.
This research has made use of data obtained from the High Energy
Astrophysics Science Archive Research Center (HEASARC), provided by
NASA's Goddard Space Flight Center.}
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