Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/ChaMPlane/quantile/paper/rev2.ps Дата изменения: Tue Jun 8 19:03:40 2004 Дата индексирования: Tue Oct 2 07:23:50 2012 Кодировка: Поисковые слова: m 5

Draft version June 7, 2004
Preprint typeset using L A T E X style emulateapj v. 16/07/00
NEW SPECTRAL CLASSIFICATION TECHNIQUE FOR X-RAY SOURCES: QUANTILE
ANALYSIS
Jaesub Hong 1 , Eric M. Schlegel 1 , and Jonathan E. Grindlay 1
Received: 2 Feb 2004, Accepted:
ABSTRACT
We present a new technique called quantile analysis to classify spectral properties of X-ray sources
with limited statistics. The quantile analysis is superior to the conventional approaches such as X-
ray hardness ratio or X-ray color analysis to study relatively faint sources or to investigate a certain
phase or state of a source in detail, where poor statistics does not allow spectral tting using a model.
Instead of working with predetermined energy bands, we determine the energy values that divide the
detected photons into predetermined fractions of the total counts such as median (50%), tercile (33%
& 67%), and quartile (25% & 75%). We use these quantiles as an indicator of the X-ray hardness or
color of the source. We show that the median is an improved substitute for the conventional X-ray
hardness ratio. The median and other quantiles form a phase space, similar to the conventional X-
ray color-color diagrams. The quantile-based phase space is more evenly sensitive over various spectral
shapes than the conventional color-color diagrams, and it is naturally arranged to properly represent
the statistical similarity of various spectral shapes. We demonstrate the new technique in the 0.3-8
keV energy range using Chandra ACIS-S detector response function and a typical aperture photometry
involving background subtraction. The technique can be applied in any energy band, provided the energy
distribution of photons can be obtained.
Subject headings:
1. introduction
A major obstacle for understanding the nature of the
faintest X-ray sources is poor statistics, which prevents
the usual practice of spectral analysis such as tting with
known models. Even for apparently bright sources, as we
investigate the sources in detail, we frequently need to
divide source photons in various phases or states of the
sources and the success of such an analysis is often limited
by statistics. The common practice to extract spectral
properties of X-ray sources with poor statistics is to cal-
culate X-ray hardness or color of the sources (e.g. Schulz
et al. 1989; Kim et al. 1992; Netzer et al. 1994; Prestwich
et al. 2003). In this conventional method, the full-energy
range is divided into two or three sub-bands and the de-
tected source photons are counted separately in each band.
The ratio of these counts is dened as X-ray hardness or
color, to serve as an indicator of the spectral properties of
the source.
In principle, one can constrain a meaningful X-ray hard-
ness or color to a source if at least one source photon is
detected in each of at least two bands. However, the equiv-
alent requirement for total counts in the full-energy band
can be rather demanding and it is strongly spectral de-
pendent. Fig. 1 shows an example of a X-ray color-color
diagram (Kim et al. 2003). In the gure, we divide the full-
energy range (0.3-8.0 keV) into three sub-energy bands;
0.3-0.9 (S), 0.9-2.5 (M), and 2.5-8.0 keV (H). The ratio of
the net source counts in S to M band is dened as the soft
X-ray color (x-axis in the gure), and the ratio of the net
counts in M to H band as the hard X-ray color (y-axis).
The energy range in this example is the region where the
Chandra ACIS-S detectors are sensitive, but the technique
described in this paper is not bounded to any particular
energy range, provided the energy distribution of photons
can be obtained.
The grid pattern in the gure represents the true loca-
tion for the sources with the power-law spectra governed
by two parameters: power-law index () and absorbing
column depths (N H ) along the line of the sight. The grid
pattern is drawn for an ideal detector response, which is
constant over the full-energy band. The grid pattern ap-
pears to be properly spaced to the changes of the spectral
parameters and such an arrangement suggests that this
color-color diagram may be an ideal way to classify sources
with poor statistics.
However, the appearance can be deceiving, which is
hinted by the uneven sizes of error bars in the gure. The
error bar near each grid node represents the central 68%
of simulation results from the spectral shape at the grid
node. Each simulation has 1000 source counts in the full-
energy range with no background counts, and thus the size
of the symbol represents the size of typical error bars for
1000 count sources in the diagram.
In the gure, there are two kinds of error bars for some
spectral shapes (e.g. = 0 and N H = 10 22 cm 2 ); the
thick error bars represent the central 68% distribution of
the simulation results that have \proper" color values (de-
ned here as at least one photon in each of all three bands),
and the thin error bars show the 68% distribution of all
the simulation trials (10,000) for each spectral model, i.e.
a distribution of the central 6827 trials. The two error
bars should be identical if all the trials produce a proper
soft and hard color.
The right panel in Fig. 1 shows the minimum counts in
the full-energy band to have at least one count in each of
1
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138
1

2
three bands. The gure indicates that the required mini-
mum counts are dramatically dierent among the models
in the grid. For the power-law spectra with = 0 and
N H  10 22 cm 2 , more than 500 counts are required in
order to have reasonable color values, while in the case
of = 2 and N H  10 20 cm 2 , a few counts are suф-
cient. Because of statistical uctuations, for some models,
even 1000 counts in the full-energy range do not guarantee
positive net counts in all three bands, which explains why
some trials fail to produce proper colors.
The spectral-model dependence of error bars in this kind
of color-color diagram is inevitable and the dependence is
determined by the choice of the sub-energy bands for a
full-energy range. In this example, the sub bands are cho-
sen so that the diagram is mostly sensitive near = 2 and
N H < 10 22 cm 2 . In fact, it is not possible to select sub
bands to have uniform sensitivity over dierent spectral
shapes.
Fig. 1 clearly indicates that the conventional color-color
diagram (CCCD) is heavily biased, which is very unfortu-
nate when trying to extract unknown spectral properties
from the sources. The heavy requirement on the total
counts for certain spectral shapes defeats the purpose of
the diagram since the large total counts may allow nom-
inal spectral analysis such as spectral tting with known
models. One might have to repeat the analysis with dif-
ferent choices of sub-bands to explore all the interesting
possibilities of spectral shapes.
2. quantiles
In order to overcome the selection eects originating
from the predetermined sub-energy bands, we propose to
use the energy value to divide photons into predetermined
fractions. We choose fractions to take full advantage of the
given statistics, such as 50% (median), 33% & 67% (ter-
cile), and 25% & 75% (quartile), although any quantile
may be used. We use the corresponding energy values -
quantiles - as an indicator of the X-ray hardness or color of
the source. In particular, we will show below that the me-
dian, Q 50 , is an improved substitute for the conventional
X-ray hardness.
Let E x% be the energy below which the net counts is
x% of the total counts and we dene quantile Q x
Q x = E x% E lo
E up E lo
;
where E lo and E up is the lower and upper boundary of
the full-energy band respectively (0.3 and 8.0 keV in this
example). The algorithm for estimation of quantiles and
their errors relevant to X-ray astronomy is given in the ap-
pendix 2 . Unlike the conventional X-ray hardness or colors,
for calculating quantiles, there is no spectral dependence
of required minimum counts. The required minimum only
depends on types of quantiles: two counts for median and
terciles, and three for quartiles.
Fig. 2 shows an example of quantile-based color-color
diagrams (QCCDs) using the median m( Q 50 ) for the
x-axis and the ratio of two quartiles Q 25 /Q 75 for the y-
axis (see x5 for the motivation of the choice of the axes).
The grid pattern in Fig. 2 is drawn for the same spectral
parameters as in Fig. 1 with ve additional cases for N H
= 5  10 22 cm 2 . Note that the approximate axes of
and N H are rotated  90deg from those in Fig. 1. The er-
ror bars are drawn for the central 68% of the same 10,000
simulation results of the spectral shape at each grid node,
and each simulation run contains 1000 source counts with
no background counts in the full-energy band. The rela-
tively similar size of the error bars for 1000 count sources
in the gure indicates that there is no spectral dependent
selection eect. Note that there is no need of distinction
for thin and thick error bars in this gure because all the
trials produce proper quantiles.
The grid pattern of power-law spectra in Fig. 2 appears
to be less intuitively arranged than the one in Fig. 1. How-
ever, we believe that the proximity of any two spectra in
the quantile-based diagram accurately exhibits the simi-
larity of the two spectral shapes (as folded through the
detector response, which is constant in this example). For
example, in the case for = 0 in Fig. 2, the separation in
phase space by various N H values is much smaller than in
the case for  1. Note that the column depth (N H ) in
the considered range can change mainly soft X-rays (. 2
keV) and the spectrum for = 0 is less dominated by soft
X-rays than that for  1. In other words, the overall
eects of N H on the spectral shape would be much smaller
in the case for = 0 than in the case for  1. Therefore,
the grid spacing in Fig. 2 indeed reveals the appropriate
statistical power needed to discern these spectral shapes,
despite the degeneracy arising from utilizing only three
variables (quantiles) extracted from a spectrum.
3. color-color diagram example
Now let us consider more realistic examples. We intro-
duce the Chandra ACIS-S response function 3 ; we use the
pre- ight calibration data for energy resolution 4 and the
energy dependent eective area from PIMMS version 2.3 5 .
Fig. 3 shows the simulation results of the spectra with
100 and 50 source counts with no background counts. The
error bars again indicate the interval that encloses the cen-
tral 68% of the simulation results for each model. The re-
sults are shown for both the conventional and new color-
color diagrams in the gure. The grid patterns in the
gures are for the same power-law models and they look
dierent from the previous examples because of the Chan-
dra ACIS-S response function.
The analysis of faint sources is often limited by back-
ground uctuations. In contrast to Fig. 3, we allow for
large background counts in the source region (e.g. a point
source in a bright diuse emission region). The top two
panels in Fig. 4 shows the simulation results of the spectra
with 100 source counts and 50 background counts in the
2
The quantile analysis software (IDL and perl versions) is available at http://hea-www.harvard.edu/ChaMPlane/quantile.
3
See Chandra Proposers' Observatory Guide at http://cxc.harvard.edu/.
4
 100 { 250 eV FWHM. See http://cxc.harvard.edu/cal/Acis/.
5
Portable, Interactive Multi-Mission Simulator. See http://heasarc.gsfc.nasa.gov/docs/software/tools/pimms.html.

3
Fig. 1.| Conventional color-color diagram (CCCD): two colors are dened by the net counts (c1, c2, and c3) in three energy bands (0.3-0.9,
0.9-2.5, and 2.5-8.0 keV). The left panel shows simulation results of 20 spectral shapes for an ideal detector with a uniform response function.
The error bar represents the central 68% of simulation results for each of 20 power-law spectra governed by two parameters { power-law index
() and column depth (N H ) along the line of the sight. The thick error bars are for the 68% of the cases with proper color values (at least one
count in each of the three bands) and the thin error bars for considering 10,000 trials (see text). Each simulation run contains 1000 source
counts in the full 0.3-8.0 keV energy range. In this example, the two error bars are identical except for the case of =0 & NH =10 22 cm 2
and =4 & NH =10 20 cm 2 , where many simulation runs result in null count in at least one of sub-bands. The grid pattern represents the
true location of these spectra in the diagram. The right panel shows the required minimum counts in the full-energy band in order to have
at least one count in each of three bands. The contour lines quantize the gray (color) scale for easier interpretation. See electronic ApJ for
color version of the gure as well as other gures.
Fig. 2.| Same as Fig. 1 but quantile-based color-color diagram (QCCD) based on the median Q 50 and the ratio of two quartiles Q 25 /Q75 .
See x5 for the origin of the logarithmic form for the x-axis. The energy scale across the top shows the median energy values (E 50% ). The
simulation results are given from the same simulation sets in Fig. 1 (1000 source counts). Note that there are ve more spectral shapes with
NH = 5  10 22 cm 2 than in Fig. 1. The grid pattern represents the true location of these spectra in the diagram for an ideal detector. Note
the relative uniformity of error bars, compared to Fig. 1.

4
source region, and the bottom two panel shows the case
of 50 source counts and 25 background counts. For back-
ground subtraction, we set the background region to be
ve times larger than the source region. The background
region contains 250 (top panels) and 125 background (bot-
tom panels) counts respectively. Note these are relatively
high backgrounds for a typical Chandra source and thus
illustrate the worst case of a point source superimposed in
background diuse emission. The background photons are
sampled from a power-law spectrum with = 0 and N H =
0, which is folded through the Chandra ACIS-S response
function.
In the case of the CCCD in Figs. 3 and 4, one can no-
tice that some of the error bars lie away from their true
location (grid node). The severe requirement for the total
counts (> 100) for some of the spectral models results in
only a lower or upper limit for the color in many simula-
tion runs. Cases with proper colors for these models are
greatly in uenced by statistical uctuations because of low
source counts in one or two sub-bands, and thus the esti-
mated colors fail to produce the true value. It is evident
that this conventional diagram is more sensitive towards
 2 and N H . 10 22 cm 2 .
In the case of the new quantile-based diagram, the error
bars stay at the correct location and the size of the error
bars are relatively uniform across the model phase space,
indicating that the phase space is properly arranged. The
quantile-based diagram shows more consistent results re-
gardless of background.
4. hardness ratio example
When we explore the spectral properties of sources with
an even smaller number of detected photons, the only
available tool is the use of a single hardness ratio, which
requires only two sub-energy bands. Here we use net
counts S in a 0.3-2.0 keV soft band and net counts H
in a 2.0-8.0 keV hard band. Two popular denitions for
hardness ratio exist in the literature: HR = H=S and
HR=(H S)=(H + S). We compare the median with the
conventional hardness ratio using the rst denition.
The left panel in Fig. 5 shows the performance of the
conventional hardness ratio as a function of the total
counts folded through the ACIS-S response. The plot con-
tains three spectral shapes with = 2. The shaded regions
represent the central 68% of the simulation results for a
given total net counts (except for the case of H or S  0).
The simulation is done for the case without background
(top panels) and for the case that the source region con-
tains an equal number of source and background counts
(bottom panels), where we perform a background subtrac-
tion identical to the one described in the previous section.
The background follows the same spectrum as in the pre-
vious examples.
Similar to the CCCD, the required minimum of the to-
tal counts for the conventional hardness ratio depends on
spectral shape. The right panels in Fig. 5 show the frac-
tion of the simulations that have at least one count in both
the soft and hard bands (otherwise HR = 0 or 1). The
plot indicates that one cannot assign a proper hardness
ratio value in a substantial fraction of cases when the to-
tal net counts are less than 100. In the case of N H = 10 20
cm 2 , many simulation runs result in H = 0 (too soft),
and for N H = 5  10 22 cm 2 , S = 0 (too hard). A set of
predetermined bands keeps the hardness ratio meaningful
only within a certain range of spectral shapes. In order to
compensate for such a limitation, one needs to repeat the
analysis with dierent choices of bands, which in turn will
have a dierent limitation.
Even for the cases with a proper hardness ratio value
(the left panel in Fig. 5), the hardness ratio distribution
of two spectral types (N H = 10 22 and 510 21 cm 2 ) over-
laps substantially when the total net counts are less than
40 in the case of high background. Below 10 counts, it is
diфcult to relate the distributions to their true value.
Fig. 6 shows the performance of the new hardness de-
ned by the median. First, there is no loss of events. Sec-
ond, three spectra are well separated down to less than 10
counts regardless of background. Even at two or three net
counts, each distribution is distinct and well related to its
true value.
5. discussion
The newly dened hardness and color-color diagrams
using the median and other quantiles are far superior to
the conventional ones. One can choose specic quantiles
and their combinations relevant to one's needs. We believe
the new phase space by the median and quartiles is very
useful in general, as summarized in Fig. 7.
For a given spectrum, various quantiles are not indepen-
dent variables, unlike the counts in dierent energy bands.
However, the ratio of two quartiles is mostly independent
from the median, which makes them good candidate pa-
rameters for color-color diagrams. In essence, the median
Q 50 shows how dense the rst (or second) half of the spec-
trum is and the quartile ratio Q 25 /Q 75 shows a similar
measure of the middle half of the spectrum.
For the x-axis in Fig. 7, one can simply use m ( Q 50 )
on a log scale to explore the wide range of the hardness.
However, a simple log scale compresses the phase space for
relatively hard spectra (m > 0:5; note 0  m  1). Our
expression 6 , log(m=(1 m)), shows both the soft and hard
phase space equally well.
In Fig. 7, a at spectrum lies at x=0 and y=1 in the
diagram (0 < y < 3). The spectrum changes from soft to
hard as one goes from left to right in the diagram, and it
changes from concave-upwards to concave-downwards, as
one goes from bottom to top. The examples in the pre-
vious sections explore a soft part of this phase diagram,
which is modeled by a power law.
In the case of a narrow energy range, where an emission
or absorption line is expected on a relatively at contin-
uum, one can use the QCCD to explore a spectral line
feature even with limited statistics that normally forbids
normal spectral tting. In this case, the median (x-axis)
in the QCCD can be a good measure of line shift, and
the quartile ratio (y-axis) can be a good measure of line
broadening.
The right panel in Fig. 7 shows the overlay of the grid
patterns for typical spectral models - power law, thermal
Bremsstrahlung, and black body emission. For high count
6
The inverse of hyperbolic tangent;  log(m) when m ! 0 , and  log(1 m) when m ! 1.

5
Fig. 3.| Realistic example of the color-color diagram with no background: 100 (top panels) and 50 (bottom panels) source counts for each
spectral model folded through the Chandra ACIS-S response function. The left two panels show the CCCD and the right two panels show the
QCCD. The grid pattern of both cases are dierent from the previous example because of the ACIS-S response function. In the case of the
CCCD, the average of the simulation results are often severely dierent from the true value (grid node) due to the selection eect inherent to
the band choice. The selection eect also causes the large dierence between thick and thin error bars, which indicates that only a fraction
of simulations produce proper colors. The fraction can be even smaller than 68% of the total 10,000 runs, in which case the thin error bar
represents the full range of the simulation outcomes with proper colors. The quantile-based diagram provides the correct results regardless of
spectral shapes.

6
Fig. 4.| Realistic example of the color-color diagram with relatively high background: 100 count source with 50 background counts (top
panels) and 50 count source with 25 background counts (bottom panels). The left two panels show the CCCD, and the right two panels
show the QCCD. The background follows a at spectrum, and both source and background photons are folded through the Chandra ACIS-S
response function. The diagrams are generated after the background subtraction (see the appendix). Similar to the previous results, the new
technique performs well throughout the phase space. In particular, note the relatively uniform error bars across the entire grid as well as the
relative lack of points outside the grid.

7
Fig. 5.| Conventional hardness ratio calculation. The soft band (S) is 0.3-2.0 keV and hard band (H) 2.0-8.0 keV, and the Chandra
ACIS-S response function is used. The left two panels show a conventional hardness distribution as a function of total net counts and the
right two panels show the fraction of the simulations that have H > 0 and S > 0 for a given total net count. The top two panels are for
the case of no background, and in the bottom two panels, for a given net count, the source region contains an equal number of source and
background counts, and we perform a background subtraction identical to the one described in the previous examples. Three spectral shapes
( = 2) are simulated, and the solid horizontal lines in the left panels represent the true hardness ratio of each spectrum. The shaded region
represents the central 68% of simulated results (only for H > 0 and S > 0) at a given total net count. A large fraction of trials do not produce
a proper hardness ratio when the total counts are less than 100. Below 10 counts, the hardness distribution hardly bears any relation to its
true value. Blending of the central 68% distributions starts at  40 counts for NH = 10 20 and 5  10 21 cm 2 for the case with background.

8
Fig. 6.| The median distribution for three power-law spectra with = 2. Compare this with Fig. 5. Regardless of background, the three
distributions are well separated down to less than 10 counts, bearing a tight relation with their true value. In addition, there is no loss of
events in this diagram.
Fig. 7.| Overview of various spectral shapes (approximated as concave or straight lines) in the QCCD using Q 50 and Q 25 /Q 75 (left),
and the grid patterns for power law (PL), thermal Bremsstrahlung (TB), black body (BB) models in a 0.3-8.0 keV range ideal detector or
incoming ux space (right). For PL, the grid patterns are for = 4, 3, 2, 1 & 0 along the positive x-axis direction, for TB, kT = 0.2, 0.5, 1,
2, 4, & 10.0 keV, and for BB, kT = 0.1, 0.2, 0.5, 1, 2, & 4 keV. For all the models, NH = 0.1, 1, 5, & 10 10 21 cm 2 along the positive y
direction.

9
sources, one can use such a diagram to nd relevant models
for the spectra before detailed analysis.
Finally, we investigate how the detector energy resolu-
tion aects the performance of the quantile-based diagram.
The left panel in Fig. 8 shows the grid patterns of the
power-law emission models for detectors with various en-
ergy resolutions but with the same detection eфciency of
the Chandra ACIS-S detector. Starting with the energy
resolution of the Chandra ACIS-S detector in the previ-
ous examples (Figs. 3, 4, 5 and 6; FWHM
E  150
eV at 1.5 keV and  200 eV at 4.5 keV), we have suc-
cessively decreased the energy resolution by multiplying
the energy resolution at each energy by a constant factor.
Each pattern is labeled by the energy resolution (
E=E)
at 1.5 keV. As the energy resolution decreases, the grid
pattern shrinks. The pattern will shrink down to a point
(x; y)=(0,1) in the diagram for a detector with no energy
resolution.
The right panel in Fig. 8 shows the 68% of the simula-
tion results for a detector with
E=E = 100% at 1.5 keV.
The size of the 68% distribution in this example is more
or less similar to that in the case of the regular Chan-
dra ACIS-S detector with
E=E = 10% (the top-right
panel in the Fig. 3). The similarity is due to the fact
that, unlike the grid pattern, the dispersion of the sim-
ulation results (or the error size of a data model point)
for each model is mainly due to statistical uctuations for
low count sources. In summary, as the energy resolution
decreases, the relative distance between various models in
the diagram decreases, but the error size of the data re-
mains roughly the same, and thus the spectral sensitivity
of the diagram decreases.
Note that the overall size of the grid patterns in the left
panel of Fig. 8 is more or less similar when
E=E . 20%
(at 1.5 keV). We expect that for a detector with energy
resolution
E, the overall size is dependent on a quantity

E=(E up E lo ) since quantiles are determined over the
full energy range. Therefore, the quantile-based diagram
is quite robust against nite energy resolutions of typical
X-ray detectors.
6. acknowledgement
This work is supported by NASA grant AR4-5003A
for our on-going Chandra Multiwavelength plane (ChaM-
Plane) survey of the galactic plane.
REFERENCES
Babu, G. J. 1999 "Breakdown theory for estimators based
on bootstrap and other resampling schemes" in Asymptotics,
Nonparameterics, and Time Series (eg. S. Ghosh), Stat. Text.
Monogr. 158, Dekker:NY, pp 669-681.
Harrell, F. E. & Davis, C. E., 1982, Biometrika, 69, 635-640
Kim, D. W., Fabbiano, G., & Trinchieri, G. 1992, ApJ, 393, 134
Kim, D. W., et al., 2003 ApJ, in press (astro-ph/0308493)
Maritz, J. S. & Jarrett, R. G., 1978, Journal of the American
Statistical Association, 73, 194-196
Netzer, H., Turner, T. J., George, I. M. 1994, ApJ, 435, 106
Prestwich, A. H., Irwin, J. A., Kilgard, R. E., Krauss, M. I.; Zezas,
A., Primini, F., Kaaret, P., Boroson, B. 2003, ApJ, 595, 719
Schulz, N. S., Hasinger, G., & Trumper, J. 1989, A&A, 225, 48
Wilcox, R. R., Introduction to Robust Estimation and Hypothesis
Testing, Academic Press, 1997
APPENDIX
Many routines have been developed to estimate quantiles and the related statistics (Wilcox 1997). We use a simple
interpolation technique based on order statistics to estimate quantiles. For real data, we also need to have a reliable
estimate for the errors of quantile values, for which we employ the technique by Maritz & Jarrett (1978). For simplicity,
we assume that we measure the energy value of each photon.
quantile estimation using order statistics
In the literature, many quantile estimation algorithms often assume that the lowest value (energy) of the given distri-
bution is the (equivalent) lower bound of the distribution and the highest value the upper bound. In X-ray astronomy,
the lower (E lo ) and upper bound (E up ) of the energy range is usually set by the instrument or user selection, where
these bounds may or may not be the lowest and highest energies of the detected photons. We can explicitly impose this
boundary condition by assigning 0% and 100% quantiles to E lo and E up respectively.
We sort the detected photons in ascending order of energy, and we assign E x% to the energy E i of the i-th photon as
E i = E x% ; where x
100 = 2i 1
2N ;
and N is the total number of net counts. Using E lo =E 0% and E up =E 100% , one can interpolate any quantiles from the
above relation of x and E i . With the denitions of E lo =E 0% and E up =E 100% , the above interpolation is very robust.
One can even calculate quantiles without any detected photons (although not meaningful), the result of which is identical
to the case of a at spectrum. In the case of only one detected photon at E, the above relation reduces to E=E 50% .
Therefore, the distribution of the median from one count sources with the same spectra is the source spectrum itself.
The above interpolation for quantile E x% essentially uses only two energy values, E i and E i+1 , where (2i 1)=2N <
x=100 < (2i + 1)=2N . In order to take advantage of other energy values, one can use more sophisticated techniques like
that of Harrell & Davis (1982). In many cases, the Harrell-Davis method estimates quantiles with smaller uncertainties
than the simple order statistics technique, but because of smoothing eects in the Harrell-Davis method, there are cases
that the simple order statistics performs better, such as distributions containing discontinuous breaks (Wilcox 1997). In
real data, the nite detector resolution tends to smooth out any discontinuity in the spectra. Our simulation shows that
about 10-15% better performance is achieved by the Harrell-Davis method compared to the simple order statistics for the
cases of 50 source photons and 25 background photons. The results in the paper are generated by the order statistics
technique.

10
Fig. 8.| The grid patterns for various energy resolutions with the same Chandra ACIS-S eфciency (left) and the 68% distribution of the
simulation results for 100 count sources for a detector with
E=E = 100% (
E =FWHM) at 1.5 keV (right). Each grid pattern in the left
panel is generated for a dierent energy resolution, labeled by the energy resolution (
E=E) at 1.5 keV. As the resolution decreases, the
pattern shrinks, and if there is no energy resolution in the detector, the pattern shrinks to a point at (x; y) = (0,1). The right gure shows the
eects of limited photon statistics dominate those of energy resolution (compare with Fig. 3, with
E=E =10% at 1.5 keV) in determining
the 68% error bar sizes. See the electronic ApJ for color version of the gure which distiguished the model grids more easily.
Fig. 9.| Accuracy of the error estimates by the Maritz-Jarrett method: the ratio of the error estimate by the Maritz-Jarrett method ( mj )
to that from simulations ( sim , the 68% distribution) for the cases with no background (left) and for the cases with high background (right).
See the \background subtraction" section in the appendix for the background subtraction routine.

11
quantile error estimation using the maritz-jarrett method
Once quantile values are acquired, one can always rely on simulations to estimate their uncertainty (or error) using a
suspected spectral model with parameters derived from the QCCD. If no single model stands out as a primary candidate,
one can derive a nal uncertainty of the quantiles by combining the results of multiple simulations from a number of
models. However, error estimation by simulations can be slow, cumbersome, and even redundant, considering that the
quantile errors are more sensitive to the number of counts than the choice of spectral shapes (cf. Figs. 3 and 4). A
quick and rough error estimate is often suфcient, and so we introduce a simple quantile error estimate technique { the
Maritz-Jarrett method. The results (error bars and shades in Fig. 1 to 6) in the main text indicate the interval that
encloses the central 68% of the simulation results, and were not driven by the Maritz-Jarrett method.
The Maritz-Jarrett method uses a technique similar to the Harrell-Davis method of quantile estimation. Both methods
rely on L-estimators (linear sums of order statistics) using Beta functions. We sort the photons in the ascending order of
energy from i = 1 to N . Then, we apply the Maritz-Jarrett method with small modications 7 as follows. For E x% , we
set
M = Nx+ 0:5
a = M 1
b = N M:
We then dene W i using the incomplete beta function B,
W i = B(a; b; y i+1 ) B(a; b; y i )
B(a; b; y) = (a+ b)
(a)(b)
Z y
0
t a 1 (1 t) b 1 dt
where y i = i=N and (n+ 1) = n! (the gamma function). Using W i and E i , we calculate the L-estimators C k ,
C k =
N
X
i=1
W i E k
i ; and then
(E x% ) =
q
C 2 C 2
1 :
The error estimate by the Maritz-Jarrett method requires at least 3 counts for medians, 5 counts for terciles, and 6 counts
for quartiles (a; b  1) .
Fig. 9 shows the accuracy of the error estimates by the Maritz-Jarrett method for = 2 and N H = 510 21 cm 2 . In the
case of no background, the Maritz-Jarrett method is accurate when the total net counts are greater than  30. It tends to
overestimate the errors when the total net counts are below  30. In the case of high background, it tends to underestimate
the errors overall since the adopted background subtraction procedure (see below) does not inherit background statistics
for the error estimates. We nd that multiplying by an empirical factor
p
1 +N bkg =N src can compensate, approximately,
for the underestimation (N src : total source counts, N bkg : total background counts).
For the error of the ratio Q 25 /Q 75 , since these two quartiles are not independent variables, the simple quadratic
combination from two quartile errors overestimates the error of the ratio ( 20 { 30% for the examples in Figs. 3 and 4).
One can nd more sophisticated techniques such as Bayesian statistics to estimate quantile errors in the literature. For
example, Babu (1999) and the reference therein discuss the limitation of bootstrap estimation and show other techniques
such as the half-sample method.
background subtraction
For background subtraction, we calculate quantiles at a set of nely stepped fractions separately for photons in the
source region and the background region. Then, by a simple linear interpolation, we establish the integrated counts
IC(>E) (number of photons with energy greater than E) as a function of energy for both regions. Now at each energy,
one can subtract the integrated counts of the background region from that of the source region with a proper ratio
factor of the area of the two regions. Because of statistical uctuations, the subtracted integrated net counts may not be
monotonically increasing from E lo to E up . Therefore we force a monotonic behavior by setting IC(>E)  IC(>E 0 ) if
IC(>E) < IC(>E 0 ) and E > E 0 . Such a requirement can underestimate the quantiles overall, so we repeat the above
using IC( x = N + IC(>E) IC( 2N
E x% = E
where N is the total net counts. If there is no statistical uctuation, IC(>E) = N IC( we need to know the energy of each source photon, which will be lost from background subtraction. So we generate a set
of N energy values for source photons matching the above quantile relation and then apply the above error estimation
technique using the Maritz-Jarrett method.
7
Originally M = [Nx + 0:5], where [z] is the integer part of z.