New Spectral Classification Technique for Faint X-ray Sources: Quantile Analysis JaeSub Hong Spring,...
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New Spectral Classification Techniquefor Faint X-ray Sources:
Quantile Analysis
JaeSub Hong Spring, 2006
J. Hong, E. Schlegel & J.E. Grindlay, ApJ 614, 508, 2004
The quantile software (perl and IDL) is available at http://hea-www.harvard.edu/ChaMPlane/quantile.
Extracting Spectral Properties or Variationsfrom Faint X-ray sources
• Hardness RatioHR1 =(H-S)/(H+S) or HR2 =
log10(H/S)
e.g. S: 0.3-2.0 keV, H: 2.0-8.0 keV
• X-ray colors C21 = log10(C2/C1) : soft colorC32 = log10(C3/C2) : hard color
e.g. C1: 0.3-0.9 keV, C2: 0.9-2.5 keV,
C3: 2.5-8.0 keV
Hardness Ratio
Pros• Easy to calculate • Require relatively low statistics (> 2 counts)• Direct relation to Physics (count flux)
Cons• Different sub-binning among different analysis• Many cases result in upper or lower limits• Spectral bias built in sub-band selection
Pros• Easy to calculate • Require relatively low statistics (> 2 counts)• Direct relation to Physics (count flux)
Cons• Different sub-binning among different analysis• Many cases result in upper or lower limits• Spectral bias built in sub-band selection
Hardness Ratio
Pros• Easy to calculate • Require relatively low statistics (> 2 counts)• Direct relation to Physics (count flux)
Cons• Different sub-binning among different analysis• Many cases result in upper or lower limits • Spectral bias built in sub-band selection
Pros• Easy to calculate • Require relatively low statistics (> 2 counts)• Direct relation to Physics (count flux)
Cons• Different sub-binning among different analysis• Many cases result in upper or lower limits • Spectral bias built in sub-band selection
e.g. simple power law spectra (PLI = )on an ideal (flat) response
S band : H band ~ 0 ~ 1 ~ 2
0.3 – 4.2 : 4.2 – 8.0 keV = 1:1 4:1 27:1
0.3 – 1.5 : 1.5 – 8.0 keV = 1:5 1:15:1
0.3 – 0.6 : 0.6 – 8.0 keV = 1:24 1:41:1
Hardness Ratio
Pros• Easy to calculate • Require relatively low statistics (> 2 counts)• Direct relation to Physics (count flux)
Cons• Many cases result in upper or lower limits • Spectral bias built in sub-band selection
Pros• Easy to calculate • Require relatively low statistics (> 2 counts)• Direct relation to Physics (count flux)
Cons• Many cases result in upper or lower limits • Spectral bias built in sub-band selection
e.g. simple power law spectra (PLI = )on an ideal (flat) response
S band : H band Sensitive to (HR~0)0.3 – 4.2 : 4.2 – 8.0 keV ~ 00.3 – 1.5 : 1.5 – 8.0 keV ~ 10.3 – 0.6 : 0.6 – 8.0 keV ~ 2
X-ray Color-Color Diagram
C21 = log10(C2/C1) C32 = log10(C3/C2)
C1 : 0.3-0.9 keVC2 : 0.9-2.5 keVC3 : 2.5-8.0 keV
Power-Law : & NH
Intr
insi
cally
Hard
More
Absorption
X-ray Color-Color Diagram
• Simulate 1000 count sources with spectrum at the grid nods.
• Show the distribution (68%) of color estimates for each simulation set.
• Very hard and very soft spectra result in wide distributions of estimates at wrong places.
X-ray Color-Color Diagram
• Total counts required in the broad band (0.3-8.0 keV) to have at least one count in each of three sub-energy bands
• Sensitive to C21~0 and C32~0
Use counts in predefined sub-energy bins.
• Count dependent selection effect• Misleading spacing in the diagram
Use counts in predefined sub-energy bins.
• Count dependent selection effect• Misleading spacing in the diagram
Hardness ratio & X-ray colors
Use counts in predefined sub-energy bins.
• Count dependent selection effect• Misleading spacing in the diagram
Use counts in predefined sub-energy bins.
• Count dependent selection effect• Misleading spacing in the diagram
Hardness ratio & X-ray colors
e.g. simple power law spectra (PLI = )on an ideal (flat) response
S band, H band Sensitive to Median0.3 – 4.2, 4.2 – 8.0 keV ~ 0 4.2 keV0.3 – 1.5, 1.5 – 8.0 keV ~ 1 1.5 keV0.3 – 0.6, 0.6 – 8.0 keV ~ 2 0.6 keV
Search energies that divide photons
into predefined fractions.
: median, terciles, quartiles, etc
Search energies that divide photons
into predefined fractions.
: median, terciles, quartiles, etc
How about Quantiles?
e.g. simple power law spectra (PLI = )on an ideal (flat) response
S band, H band Sensitive to Median0.3 – 4.2, 4.2 – 8.0 keV ~ 0 4.2 keV0.3 – 1.5, 1.5 – 8.0 keV ~ 1 1.5 keV0.3 – 0.6, 0.6 – 8.0 keV ~ 2 0.6 keV
Quantiles
• Quantile Energy (Ex%) and Normalized Quantile (Qx)
x% of total counts at E < Ex%
Qx = (Ex%-Elo) / (Elo-Eup), 0<Qx<1
e.g. Elo = 0.3 keV, Eup=8.0 keV in 0.3 – 8.0 keV
• Median (m=Q50) Terciles (Q33, Q67) Quartiles (Q25, Q75)
Quantiles
• Low count requirements for quantiles: spectral-independent
2 counts for median3 counts for terciles and
quartiles
• No energy binning required• Take advantage of energy resolution• Optimal use of information
Hardness Ratio
HR1 = (H-S)/(H+S)
-1 < HR1 < 1
HR1 = (H-S)/(H+S)
-1 < HR1 < 1 HR2 = log10[ (1+HR1)/(1-
HR1) ]
m=Q50= (E50%-Elo)/(Eup-Elo)
0 < m < 1
m=Q50= (E50%-Elo)/(Eup-Elo)
0 < m < 1
Median
HR2 = log10(H/S)
- < HR2 <
HR2 = log10(H/S)
- < HR2 <
qDx= log10[ m/(1-m) ]
- < qDx <
qDx= log10[ m/(1-m) ]
- < qDx <
Hardness ratio simulations (no background)
S:0.3-2.0 keV H:2.0-8.0 keV
Fractional cases withupper or lower limits
Hardness Ratio vs Median(no background)
Hardness Ratio0.3-2.0-8.0 keV
Median0.3-8.0 keV
Hardness Ratio vs Median(source:background = 1:1)
Hardness Ratio0.3-2.0-8.0 keV
Median0.3-8.0 keV
Quantile-based Color-Color Diagram (QCCD)
• Quantiles are not independent
• m=Q50 vs Q25/Q75
• Power-Law : & NH
• Proper spacing in the diagram
• Poor man’s Kolmogorov -Smirnov (KS) testAn ideal detector
03-8.0 keV
IntrinsicallyHard
More
Absorp
tion
E50%=
Overview of the QCCD phase space
Color estimate distributions (68%) by simulationsfor 1000 count sources
Quantile Diagram0.3-8.0 keV
Conventional Diagram0.3-0.9-2.5-8.0 keV
E50%=
Realistic simulations
ACIS-S effective area & energy resolutionAn ideal detector
E50%=
100 count source with no background
Quantile Diagram0.3-8.0 keV
Conventional Diagram0.3-0.9-2.5-8.0 keV
100 source count/ 50 background count
Quantile Diagram0.3-8.0 keV
Conventional Diagram0.3-0.9-2.5-8.0 keV
50 count source without background
Quantile Diagram0.3-8.0 keV
Conventional Diagram0.3-0.9-2.5-8.0 keV
50 source count/ 25 background count
Quantile Diagram0.3-8.0 keV
Conventional Diagram0.3-0.9-2.5-8.0 keV
Energy resolution and Quantile Diagram
• Elo = 0.3 keV Ehi = 8.0 keV
• E/E = 10% at 1.5 keV
• E50%: from Elo+ f Elo to Ehi– f Ehi
from ~ 0.4 keVto ~ 7.8 keV
Energy resolution and Quantile Diagram
• Elo = 0.3 keV Ehi = 8.0 keV
• E/E = 20% at 1.5 keV
• E50%: from Elo+ f Elo to Ehi– f Ehi
from ~ 0.4 keVto ~ 7.6 keV
Energy resolution and Quantile Diagram
• Elo = 0.3 keV Ehi = 8.0 keV
• E/E = 50% at 1.5 keV
• E50%: from Elo+ f Elo to Ehi– f Ehi
from ~ 0.5 keVto ~ 7.0 keV
Energy resolution and Quantile Diagram
• Elo = 0.3 keV Ehi = 8.0 keV
• E/E = 100% at 1.5 keV
• E50%: from Elo+ f Elo to Ehi– f Ehi
from ~ 0.7 keVto ~ 6.5 keV
Energy resolution and Quantile Diagram
• Elo = 0.3 keV Ehi = 8.0 keV
• E/E = 200% at 1.5 keV
• E50%: from Elo+ f Elo to Ehi– f Ehi
from ~ 1.0 keVto ~ 6.0 keV
Energy resolution and Quantile Diagram
• Elo = 0.3 keV Ehi = 8.0 keV
• E/E = 500% at 1.5 keV
• E50%: from Elo+ f Elo to Ehi– f Ehi
from ~ 1.2 keVto ~ 5.0 keV
E/E = 10% at 1.5 keV E/E = 100% at 1.5 keV
Energy resolution and Quantile Diagram
Sgr A* (750 ks Chandra)
Sgr A* (750 ks Chandra)
Sgr A* (750 ks Chandra)
Sgr A* (750 ks Chandra)
Sgr A* (750 ks Chandra)
Swift XRT Observation of GRB Afterglow
• GRB050421 : Spectral softening with ~ constant NH
• GRB050509b : Short burst afterglow, softer than the host Quasar
Spectral Bias
Stability
Sub-binning
Phase Space
Sensitivity
Energy Resolution
Physics
Quantile Analysis
None
Good
No Need
Meaningful
Evenly Good
Sensitive
Indirect
X-ray Hardness Ratio or Colors
Yes
Upper/Lower Limits
Required
Misleading?
Selectively Good
Insensitive
Direct
Score Board
Future Work
• Find better phase spaces.
• Handle background subtraction better.
• Find better error estimates: half sampling, etc.
• Implement Bayesian statistics?
Conclusion: Quantile Analysis
• Stable spectral classification with limited statistics
• No energy binning required
• Take advantage of energy resolution
• Quantile-based phase space is a good indicator of spectral sensitivity of the detector.
• The basic software (perl and IDL) is available at http://hea-www.harvard.edu/ChaMPlane/quantile.
• In principle, by simulations: slow and redundant
• Maritz-Jarrett Method : bootstrapping
• Q25 & Q75: not independentMJ overestimates by ~10%
• 100 count source:consistent within ~5%
Quantile Error Estimates
Quantile Error Estimatesby Maritz-Jarrett Method
• PL: =2, NH=5x1021cm-2
• >~30 count : within ~ 10%
• <~30 count : overestimate up to ~50%
• MJ requires 3 counts for Q50
5 counts for Q33, Q67
6 counts for Q25, Q75
mj/sim