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New Spectral Classification Technique for 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.

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slide1

New Spectral Classification Technique

for 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.

slide2

Extracting Spectral Properties or Variations

from Faint X-ray sources

  • Hardness Ratio
  • HR1 =(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 color
  • C32 = 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
slide3

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
slide4

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

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 1:1 5:1

0.3 – 0.6 : 0.6 – 8.0 keV = 1:24 1:4 1:1

slide5

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

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 ~ 0

0.3 – 1.5 : 1.5 – 8.0 keV ~ 1

0.3 – 0.6 : 0.6 – 8.0 keV ~ 2

slide6

X-ray Color-Color Diagram

C21 = log10(C2/C1)

C32 = log10(C3/C2)

C1 : 0.3-0.9 keV

C2 : 0.9-2.5 keV

C3 : 2.5-8.0 keV

Power-Law :  & NH

Intrinsically

Hard

More

Absorption

slide7

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.
slide8

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
slide9

Hardness ratio & X-ray colors

  • Use counts in predefined sub-energy bins.
    • Count dependent selection effect
    • Misleading spacing in the diagram
slide10

Hardness ratio & X-ray colors

  • Use counts in predefined sub-energy bins.
    • Count dependent selection effect
    • Misleading spacing in the diagram

e.g. simple power law spectra (PLI = )

on an ideal (flat) response

S band,H band Sensitive to Median

0.3 – 4.2,4.2 – 8.0 keV ~ 0 4.2 keV

0.3 – 1.5,1.5 – 8.0 keV ~ 1 1.5 keV

0.3 – 0.6,0.6 – 8.0 keV ~ 2 0.6 keV

slide11

How about Quantiles?

Search energies that divide photons

into predefined fractions.

: median, terciles, quartiles, etc

e.g. simple power law spectra (PLI = )

on an ideal (flat) response

S band,H band Sensitive to Median

0.3 – 4.2,4.2 – 8.0 keV ~ 0 4.2 keV

0.3 – 1.5,1.5 – 8.0 keV ~ 1 1.5 keV

0.3 – 0.6,0.6 – 8.0 keV ~ 2 0.6 keV

slide12

Quantiles

  • Quantile Energy (Ex%) andNormalized 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)
slide13

Quantiles

  • Low count requirements for quantiles:
  • spectral-independent
    • 2 counts for median
    • 3 counts for terciles and quartiles
  • No energy binning required
  • Take advantage of energy resolution
  • Optimal use of information
slide14

Hardness Ratio

HR1 = (H-S)/(H+S)

-1 < HR1 < 1

HR2 = log10(H/S)

- < HR2 < 

HR2 = log10[ (1+HR1)/(1-HR1) ]

Median

m=Q50= (E50%-Elo)/(Eup-Elo)

0 < m < 1

qDx= log10[ m/(1-m) ]

- < qDx <

slide15

Hardness ratio simulations (no background)

S:0.3-2.0 keV

H:2.0-8.0 keV

Fractional cases with

upper or lower limits

slide16

Hardness Ratio vs Median

(no background)

Hardness Ratio

0.3-2.0-8.0 keV

Median

0.3-8.0 keV

slide17

Hardness Ratio vs Median

(source:background = 1:1)

Hardness Ratio

0.3-2.0-8.0 keV

Median

0.3-8.0 keV

slide18

Quantile-based Color-Color Diagram (QCCD)

E50%=

  • Quantiles are not independent
  • m=Q50 vs Q25/Q75
  • Power-Law :  & NH
  • Proper spacing in the diagram
  • Poor man’s Kolmogorov -Smirnov (KS) test

More

Absorption

Intrinsically

Hard

An ideal detector

03-8.0 keV

slide20

Color estimate distributions(68%) by simulations

for1000 count sources

E50%=

Quantile Diagram

0.3-8.0 keV

Conventional Diagram

0.3-0.9-2.5-8.0 keV

slide21

Realistic simulations

E50%=

ACIS-S effective area

& energy resolution

An ideal detector

slide22

100 count source with no background

Quantile Diagram

0.3-8.0 keV

Conventional Diagram

0.3-0.9-2.5-8.0 keV

slide23

100 source count/ 50 background count

Quantile Diagram

0.3-8.0 keV

Conventional Diagram

0.3-0.9-2.5-8.0 keV

slide24

50 count source without background

Quantile Diagram

0.3-8.0 keV

Conventional Diagram

0.3-0.9-2.5-8.0 keV

slide25

50 source count/ 25 background count

Quantile Diagram

0.3-8.0 keV

Conventional Diagram

0.3-0.9-2.5-8.0 keV

slide26

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 keV
  • to ~ 7.8 keV
slide27

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 keV
  • to ~ 7.6 keV
slide28

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 keV
  • to ~ 7.0 keV
slide29

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 keV
  • to ~ 6.5 keV
slide30

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 keV
  • to ~ 6.0 keV
slide31

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 keV
  • to ~ 5.0 keV
slide32

Energy resolution and Quantile Diagram

E/E = 10% at 1.5 keV

E/E = 100% at 1.5 keV

slide33

Sgr A*

(750 ks Chandra)

slide34

Sgr A*

(750 ks Chandra)

slide35

Sgr A*

(750 ks Chandra)

slide36

Sgr A*

(750 ks Chandra)

slide37

Sgr A*

(750 ks Chandra)

slide39

Swift XRT Observation of GRB Afterglow

  • GRB050421 : Spectral softening with ~ constant NH
  • GRB050509b : Short burst afterglow, softer than the host Quasar
slide40

Score Board

  • 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
slide41

Future Work

  • Find better phase spaces.
  • Handle background subtraction better.
  • Find better error estimates: half sampling, etc.
  • Implement Bayesian statistics?
slide43

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.
slide44

Quantile Error Estimates

  • In principle, by simulations:
  • slow and redundant
  • Maritz-Jarrett Method : bootstrapping
  • Q25 & Q75: not independent
    • MJ overestimates by ~10%
  • 100 count source:
  • consistent within ~5%
slide45

Quantile Error Estimates

by 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

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