Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/HEAD2004/poster_yaming_timeseries.ps Дата изменения: Mon Sep 13 05:59:45 2004 Дата индексирования: Tue Oct 2 06:08:24 2012 Кодировка: Поисковые слова: merging galaxies

Detecting
Variations
in
Source
Intensity
for
Poisson
Data
Y.
Yu,
XL.
Meng,
D.
van
Dyk,
V.
Kashyap,
and
A.
Zezas
Abstract
The
analysis
of
light
curves
is
an
especially
challenging
statistical
task
for
low-count
Poisson
data:
Variation
in
the
source
intensity
may
be
shadowed
by
the
Poisson
variation
of
the
counts.
Here
we
discuss
a
class
of
statistical
models
that
are
designed
to
capture
trends
and
autocorrelations
in
Poisson
data
that
are
collected
over
time.
We
expect
such
models
to
be
useful
not
only
in
the
analysis
of
sources
that
vary
in
their
intensities
but
also
in
joint
spectral-temporal
modeling
tasks
such
as
analyzing
the
variability
in
spectral
power
law
parameters
or
hardness
ratios
over
time.
To
tackle
these
problems,
we
propose
a
Bayesian
statistical
model
that
takes
into
account
both
the
variation
in
source
intensity
and
the
Poissonian
character
of
the
count
data.
Parameter
estimation
and
error
bars
are
computed
using
sophisticated
MCMC
(Markov
chain
Monte
Carlo)
methods.
We
illustrate
our
methods
using
several
X-ray
sources.
yu@stat.harvard.edu
1

Is
There
Variation
in
Source
Intensity?

If
not,
then
counts
in
equal
time
intervals
should
follow
independent
Poisson
distribution
with
the
same
parameter.

Use
Posterior
Predictive
P-value
(ppp),
a
Bayesian
hypothesis
test
method:
{
Simulate
replicate
datasets
using
parameters
drawn
from
the
posterior
distribution.
{
Select
a
statistic
T
.
Compute
T
obs
,
the
observed
T
for
the
original
dataset.
{
For
each
replication,
compute
T
.
Compare
T
obs
with
the
distribution
of
T
.
The
ppp
is
the
proportion
of
time
T
obs
exceeds
T
.

Example:
A
downward
trend?
.
.
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.
time
counts
0
20
40
60
80
100
360 380 400 420 440
2


Use
ppp.
Choose
T
as
the
correlation
between
counts
and
time
index.

The
trend
is
statistically
signicant:
­0.4
­0.2
0.0
0.2
0.4
0 200 400 600 800 1000
1.14%
of
time
T Correlation
of
counts
and
time
Fig.
The
posterior
predictive
distribution
of
T
,
vertical
line
being
T
obs
.

Advantage:
No
theoretical/analytical
calculations
needed;
choice
of
T
is
open.
3

Statistical
Modeling
for
the
Variation
in
Source
Intensity

We
can
design
statistical
models
that
account
for
two
levels
of
variation:
{
the
variation
in
source
intensity
,
and
{
the
Poisson
variation
of
photon
arrivals.

Model
formulation:
Let
Y
t
be
the
counts
in
bin
t,
t
=
1;






;
T
.
Y
t

Poisson(d
t
e
+t

t
);
where
d
t
is
the
width
(e.g.,
in
seconds)
of
bin
t;

t
=
e

t

2
=2
;

t
=

t
1
+
Z
t
;
where
Z
t

N(0;
(1

2
)
2
),
so
that

t

N(0;

2
)
is
a
stationary
autoregressive
process.

Note:

t
has
mean
1
and
variance

2
=
e

2
1.
For
small

2
,

2


2
.
4

Easy
Interpretation:
Trend
and
Variation
Source
intensity
has
two
(multiplicative)
components:
a
deterministic
term
e
+t
and
a
stochastic
term

t
.

When

2
=
0,
source
intensity
is
simply
e
+t
;

controls
the
trend.
(Here
we
consider
exponential
trend.
Other
choices
present
no
additional
diфculty.)

E.g.,
suppose

=
0:1,
and
counts
are
collected
by
the
day
,
then
in
a
week
(log
2=0:1
days)
we
expect
the
source
intensity
to
decrease
by
half.


2
,
the
variance
of

t
,
controls
the
variation
of
source
intensity
.
Note

2
is
a
monotone
increasing
function
of

2
.

E.g.,
suppose

2
=
0:01,
then
we
expect
a
uctuation
of
about
10%
(
p
0:01)
in
source
intensity
around
the
general
trend.
5

Easy
Interpretation:
Correlation


measures
the
association
between

t
and

t
1
;

controls
the
correlation
between
source
intensities.

E.g.,
the
correlation
r
between
source
intensities
at
t
and
t
1
(also
the
correlation
between

t
and

t
1
):
r
=
(e

2
1)=
2
{
r
=
1
if

=
1;
r
=
0
if

=
0;
and
r
=
e

2
if

=
1.
{
r
is
a
monotone
increasing
function
of
;
for
small

2
,
r

.
6

Possible
Source
Intensity
Patterns
Produced
by
this
Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. .
.
mu=0.1,
rho=0.3,
tau2=1.0
time
source intensity
0
20
40
60
80
100
0 2 4 6 8
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mu=­0.1,
rho=0,
tau2=0.001
time
source intensity
0
20
40
60
80
100
0.0 0.1 0.2 0.3
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mu=0,
rho=0.9,
tau2=0.001
time
source intensity
0
20
40
60
80
100
0.35 0.37
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mu=0,
rho=­0.9,
tau2=1.0
time
source intensity
0
20
40
60
80
100
0 1 2 3
7

Computation
and
Application

To
learn
about
the
parameters
of
interest
;
;

2
,
use
posterior
simulation
by
Markov
chain
Monte
Carlo
(MCMC).
Here
we
use
a
Gibbs
sampler
with
Metropolis
steps.

A
real
example:
The
isolated
neutron
star/quark
star
candidate
RX
J1856.5-3754
observed
by
Chandra
HRC-S.
It
doesn't
seem
to
vary
,
or
does
it?
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..
.
.
index
counts
0
20
40
60
80
100
130 140 150 160
Fig.
Counts
Y
t
(exposure
time
55476
seconds,
divided
into
100
equal
bins).
8

­0.0010
0.0
0.0010
0 500 1000 1500
posterior
of
mu
0.0
0.0010
0.0020
0.0030
0 1000 2000 3000 4000
posterior
of
sig2
Fig.
Posterior
distributions
of
;

2
.
­0.05
0.0
0.05
0.10
0.15
0.20
0.25
0 500 1000 1500
posterior
of
rho
Fig.
Posterior
distribution
of
.
9

The
Binning
Issue
Model
interpretation
depends
on
the
binning:
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.
coarse
counts
0
20
40
60
80
100
350 400 450
.
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medium
counts
0
50
100
150
200
160 180 200 220 240 260
.
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.
fine
counts
0
100
200
300
400
80 100 120 140
Fig.
Counts
data
for
another
observation
of
RX
J1856.5-3754,
with
various
binning.
10

Parameter
estimates
for
dierent
binning
schemes:



2
coarse
0.000360
(0.000233)
0.0674
(0.0448)
0.00151
(0.000585)
medium
0.000181
(0.000103)
0.113
(0.0531)
0.00178
(0.000641)
ne
9.55e-05
(4.87e-05)
0.193
(0.0583)
0.00219
(0.000752)
Note:
posterior
standard
deviations
in
parentheses.

Estimates
of
,
the
trend
parameter,
agree,
after
taking
into
account
the
scale
dierence.

Estimates
of

are
dierent.
For
the
ne
binning,
the
estimate
shows
a
noticeable
albeit
small
autocorrelation
between
source
intensities
at
consecutive
time
points.
11

Work
in
Progress

More
eфcient
Markov
chain
algorithm.
Faster
convergence.

Joint
spectral-temporal
modeling,
e.g.,
the
variability
of
power
law
parameters
or
hardness
ratios
over
time.

Rigorous
procedures
to
check
goodness-of-t
of
the
statistical
model.
Conventional
tests
such
as
the

2
test
does
not
apply
because
of
the
stochastic
component
included
in
the
source
intensity
formula.
12