type ia sn light curve inference: hierarchical …hea-2011/01/25 · type ia sn light curve...

Kaisey S. MandelCfA Supernova GroupAstrostatistics Seminar

25 January 2011

Type Ia SN Light Curve Inference:Hierarchical Models for Nearby SN

in the Rest-Frame Near Infrared and Optical

1Tuesday, January 25, 2011

Related Papers

Mandel, K., G. Narayan, R.P. Kirshner. Type Ia Supernova Light Curve Inference: Hierarchical Models in the Optical and Near Infrared. 2011 submitted to ApJ, arXiv:1011.5910

Mandel, K., W.M. Wood-Vasey, A.S. Friedman, R.P. Kirshner. Type Ia Supernova Light Curve Inference: Hierarchical Bayesian Analysis in the Near Infrared. 2009, ApJ, 704, 629-651

Blondin, S., K. Mandel, R.P. Kirshner.Do Spectra improve distance measurements of SN Ia?2011, A&A 526, A81

Wood-Vasey, et al. Type Ia Supernovae are Good Standard Candles in the Near Infrared: Evidence from PAIRITEL.2008, ApJ, 689, 377-390


Outline• Type Ia SN and Cosmology

• Statistical Inference with SN Ia Light curves

• Hierarchical Framework for Structured Probability Models for Observed Data

• Describing Populations & Individuals, Multiple Random Effects, Covariance Structure

• Statistical Computation with Hierarchical Models

• BayeSN (MCMC)

• Application & Results: Hierarchical Model for Nearby CfA NIR (PAIRITEL) and Optical (CfA3) SN Ia light curves

3


Standard Candle Principle

1. Know or Estimate Luminosity L of a Class of Astronomical Objects

2. Measure the apparent brightness or flux F

3. Derive the distance D to Object using Inverse Square Law: F = L / (4π D2)

4. Optical Astronomer’s units: m = M +μ


Type Ia Supernovae areAlmost Standard Candles

• Progenitor: C/O White Dwarf Star accreting mass leads to instability

• Thermonuclear Explosion: Deflagration/Detonation

• Nickel to Cobalt to Iron Decay + radiative transfer powers the light curve

• SNe Ia progenitors have nearly same mass, therefore energy

Credit: FLASH Center

5


Type Ia Supernova ApparentLight Curve

!10 0 10 20 30 40 50 60

6

8

10

12

14

16

18

20

22

B + 2SN2005el (CfA3+PTEL)

V

R ! 2

I ! 4

J ! 7

H ! 9

Obs. Days Since Bmax

Ob

s.

Ma

g. !

kc !

mw

x


Supernova Cosmology:Constraining Cosmological Parameters

using Luminosity Distance

vs. Redshift

AAS 215

!"##$%&! !""#$%&#'()(#&*'+,*'#!""#$%&#'()(#&*'+,*'#! -*)&(././0#1234#5.)6#-*)&(././0#1234#5.)6#

"(&0*#"7589#:*)#)7#;*))*&#"(&0*#"7589#:*)#)7#;*))*&#:*<(&()*#,7"7&#(/'#'+:)#:*<(&()*#,7"7&#(/'#'+:)#(/'#&*'+,*#:%:)*=().,:(/'#&*'+,*#:%:)*=().,:

! >?(=././0#@#;(/'#>?(=././0#@#;(/'#(/7=("%(/7=("%

! A&7<(0()*#<&7;(;.".)%#A&7<(0()*#<&7;(;.".)%#:+&B(,*#B7&#*(,6#4C#)7#:+&B(,*#B7&#*(,6#4C#)7#,7:=7"70%#D#;*))*&#,7:=7"70%#D#;*))*&#*:).=()*:#7B#:%:)*=().,#*:).=()*:#7B#:%:)*=().,#

! !"#$%&%'(")##,7/:)&(./):#,7/:)&(./):#7/#,7/:)(/)#57/#,7/:)(/)#5

*+,+-./01+*+,+-./01+EE+./.2+345657++./.2+345657+EE+./89+34:47+./89+34:47

• Nearby Hubble law is linear• High-z depends on cosmology• Host Galaxy Dust is a Major Confounding Factor

Credit: Gautham Narayan


Cosmological Energy Content

Dark Energy Equation of state P = wρIs w + 1 = 0? Cosmological Constant


Reading the Wattage of a SN Ia:Empirical Correlations

• Width-Luminosity Relation: an observed correlation (Phillips)

• Observe optical SN Ia Light Curve Shape to estimate the peak luminosity of SN Ia: ~0.2 mag

• Color-Luminosity Relation

• Methods:

•

• MLCS, Abs LC vs Dust

• SALT, App. Color single factor

Intrinsically Brighter SN Ia have broader light curves

and are slow decliners

9

!m15(B)


Statistical inference with SN Ia

• SN Ia cosmology inference based on empirical relations

• Statistical models for SN Ia are learned from the data

• Several Sources of Randomness & Uncertainty

1. Photometric errors

2. “Intrinsic Variation” = Population Distribution of SN Ia

3. Random Peculiar Velocities in Hubble Flow

4. Host Galaxy Dust: extinction and reddening.

• Apparent Distributions are convolutions of these effects

• How to incorporate this all into a coherent statistical model? (How to de-convolve?)

10


Advantages of Hierarchical Models• Incorporate multiple sources of randomness & uncertainty

• Express structured probability models adapted to data

• Hierarchically Model (Physical) Populations and Individuals simultaneously: e.g. SN Ia and Dust

• Intrinsic Covariance: Color/Luminosity/Light Curve Shape

• Dust Reddening/Extinction

• Full (non-gaussian) probability distribution = Global, coherent quantification of uncertainties

• Completely Explore & Marginalize Posterior trade-offs/degeneracies/joint distributions

• Deals with incomplete/missing data problems

• Make best inference/estimate for the observed data

• Modularity


Directed Acyclic Graph for SN Ia Inferencewith Hierarchical Modeling

AppLC

#NAbsLC

#N

Av, Rv

#N

Dust

Pop

AbsLC

Pop

µN

D1

z1

zN

DN

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

• Intrinsic Randomness• Dust Extinction & Reddening• Peculiar Velocities • Measurement Error

“Training” - Learn about Populations

12

Generative Model

Global Joint Posterior Probability

Density Conditional on all

SN Data


Representing SN Ia Light curves:Differential Decline rates

• Gaussian Process over Decline Rates at different Wavelengths / Phases and Peak Luminosities

• Goal: Infer the Intrinsic Covariance Structure of SN Ia light curves over multiple wavelengths and phases

• Use to make “best” distance predictions

!10 0 10 20 30 40 50 60

!28

!26

!24

!22

!20

!18

!16

!14 B + 2

V

R ! 2

I ! 4

J ! 7

H ! 9


Absolu

te M

agnitude


Positive Dust only Dims and Reddens

0

50

100

SN

#

0 0.5 1 1.5 20

1

2

3

AV ! Expon(" = 0.37 ± 0.04)

Dust Extinction AV

Rela

tive F

requency

!20

!19

!18

!17

!16

MB vs B!H

MV vs V!H

!2 !1 0 1 2

!20

!19

!18

!17

!16

MR

vs R!H

!2 !1 0 1 2

MH

vs J!H

Intrinsic

Apparent


Directed Acyclic Graph for SN Ia Inference:Distance Prediction

15

zs

Ds

µs

AppLCs

s = 1, . . . , NSN

AsV , Rs

V

AbsLCs

Training

PredictionApV , Rp

Vµp

DpAppLCpAbsLCp

DustPop

SN IaAbsLC

Pop


Statistical Computation with Hierarchical SN Ia Models: The BayeSN Algorithm

• Strategy: Generate a Markov Chain to sample global parameter space (populations & all individuals) => seek a global solution

• Chain explores/samples trade-offs/degeneracies in global parameter space

Multiple chains globally converge from random

initial values

0

0.5

1

1.5

2

2.5

3

AV

SN2001az

SN2001ba

100

101

102

103

0

0.5

1

1.5

2

2.5

3

MCMC Sample

AV

SN2006cp

BayeSN MCMC Convergence

100

101

102

103

MCMC Sample

SN2007bz


BayeSN

• Metropolis-Hastings within Gibbs Sampling Structure to exploit conditional structure

• Requires (almost) no tuning of jump sizes

• Generalized Conditional Sampling to speed up exploring trade-off between dust and distance: (Av, μ) → (Av, μ) + γ(1, -x)

• Run several (4-8) parallel chains and compute Gelman-Rubin ratio to diagnose convergence


BayeSN in Graphs

AppLC

#NAbsLC

#N

Av, Rv

#N

Dust

Pop

AbsLC

Pop

µN

D1

z1

zN

DN

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

AppLC

#NAbsLC

#N

Av, Rv

#N

Dust

Pop

AbsLC

Pop

µN

D1

z1

zN

DN

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

18


BayeSN in Graphs

AbsLC

#N

AbsLC

Pop

AbsLC

#1

AbsLC

#N

AbsLC

Pop

AbsLC

#1

18


BayeSN in Graphs

Av, Rv

#N

Dust

Pop

Av, Rv

#1

Av, Rv

#N

Dust

Pop

Av, Rv

#1

18


BayeSN in Graphs

AbsLC

Pop

D1

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

AbsLC

Pop

D1

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

18


BayeSN in Graphs

AbsLC

Pop

z1

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

AbsLC

Pop

z1

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

18


BayeSN in Graphs

Dust

Pop

AbsLC

Pop

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1Dust

Pop

AbsLC

Pop

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

18


BayeSN in Graphs

AppLC

#NAbsLC

#N

Av, Rv

#NAbsLC

Pop

µN

DNAppLC

#NAbsLC

#N

Av, Rv

#NAbsLC

Pop

µN

DN

18


BayeSN in Graphs

AppLC

#NAbsLC

#N

Av, Rv

#NAbsLC

Pop

µN zN

AppLC

#NAbsLC

#N

Av, Rv

#NAbsLC

Pop

µN zN

18


BayeSN in Graphs

AppLC

#NAbsLC

#N

Av, Rv

#N

Dust

Pop

AbsLC

Pop

µN

AppLC

#NAbsLC

#N

Av, Rv

#N

Dust

Pop

AbsLC

Pop

µN

18


BayeSN in Graphs

AppLC

#NAbsLC

#N

Av, Rv

#N

Dust

Pop

AbsLC

Pop

µN

D1

z1

zN

DN

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

AppLC

#NAbsLC

#N

Av, Rv

#N

Dust

Pop

AbsLC

Pop

µN

D1

z1

zN

DN

AbsLC

#1

AppLC

#1

Av, Rv

#1

µ1

18


Practical Application of Hierarchical Model: NIR SN IaWhy are SN Ia in NIR interesting?

• Host Galaxy Dust presents a major systematic uncertainty in supernova cosmology inference

• Dust extinction has significantly reduced effect in NIR bands

• NIR SN Ia are good standard candles (Elias et al. 1985, Meikle 2000, Krisciunas et al. 2004+, Wood-Vasey et al. 2008, Mandel et al. 2009).

• Observe in NIR!: PAIRITEL /CfA

19

1989ApJ...345..245C


Nearby SN Ia in the NIR: PAIRITEL

Credit: Michael Wood-Vasey, Andrew Friedman

20

Observed in NIRJ (λ=1.2 μm)H (λ=1.6 μm)Ks (λ=2.2 μm)


Figure 1: 142 CfA Light curves from 2000-2004 (UBV RI) and 2004-2007 (UBV ri)

CfA3:183 Optical SN Ia

Light Curves(Hicken et al. 2009)


Optical+NIR Hierarchical Model InferencePTEL+CfA3 Light-curves Marginal Posterior of Dust

!10 0 10 20 30 40 50 60

6

8

10

12

14

16

18

20

22B + 2SN2005eq (CfA3+PTEL)

V

R ! 2

I ! 4

J ! 7

H ! 9


Ob

s. M

ag

. !

kc !

mw

x

0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

Dust Law Slope RV

!1

Ext

inct

ion (

mag)

AV

0.3 0.4 0.5 0.6R

V

!1

SN2005eq

AH


Optical+NIR Hierarchical Model InferencePTEL+CfA3 Light-curves Marginal Posterior of Dust

!10 0 10 20 30 40 50 60

6

8

10

12

14

16

18

20

22

B + 2SN2006ax (CfA3+PTEL)

V

R ! 2

I ! 4

J ! 7

H ! 9


Obs.

Mag. !

kc !

mw

x

0.2 0.3 0.4 0.5 0.60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Dust Law Slope RV

!1

Ext

inct

ion (

mag)

AV

0.2 0.3 0.4 0.5 0.6R

V

!1

SN2006ax

AH


SN NIR Population Inference: Peak Absolute Magnitudes

−18.35 −18.15

0.05

0.1

0.15

0.2

0.25

0.3

σ (

MX)

µ(MJ)

J

µ(MJ) = −18.26σ (MJ) = 0.17

−18.1 −17.9µ(MH)

H

µ(MH) = −18.00σ (MH) = 0.11

−18.35 −18.15µ(MKs)

Ks

µ(MKs) = −18.24σ (MKs) = 0.18

Marginal Distributions of SN Ia NIRAbsolute Magnitude Variances

Mandel et al. 2009 Kasen 200624Tuesday, January 25, 2011

Optical and Near InfraredLuminosity vs. Decline Rate

0.8 1 1.2 1.4 1.6

!20

!19.5

!19

!18.5

!18

!17.5

!17

!16.5

!16

! m15

(B)

Peak

Magnitu

de

0.8 1 1.2 1.4 1.6! m

15(B)

MB

B0!µ

MH

H0!µ


Population Analysis

Intrinsic Correlation

Map for Abs Magnitudes

and Decline Rates

H-band provides nearly uncorrelated information on

luminosity distance

B V R I J H

B

V

R

I

J

H

! m

15(F

)

B V R I J H

B

V

R

I

J

H

! m15

(F)

B V R I J H

B

V

R

I

J

H

Peak MF

Peak

MF

!1

!0.5

0

0.5

1

!1

!0.5

0

0.5

1

!1

!0.5

0

0.5

1


Host Galaxy Dust• Previous Analyses assumed all SN host

galaxy dust has same Rv

• Estimated Rv = 1-1.7 (Astier06, Conley07) if attribute all color variation to dust

• But Rayleigh Scattering: Rv = 1.2

• But Hicken09 found Rv = 1.7 with MLCS

• For individual high Av SN, Rv < 2

• But Rv may have a distribution, or depend on Av (e.g. grain growth)


(Av, Rv) for Host Galaxy DustAssuming Linear Correlation

• Apparent Correlation of High Av / Low Rv• Low Av Rv ≈ 2.5 : High Av has Rv ≈ 1.7• Circumstellar dust at High Av ? • Multiple Scattering (Goobar 2008)

!0 (intercept R

V

!1 as AV " 0)

!1 (

slo

pe

)

P(RV

!1|AV)= N( R

V

!1 | !0+ !

1A

V, #

r

2)

P( !1 < 0) < 6e!04

0.25 0.3 0.35 0.4 0.45 0.5 0.55!0.05

0

0.05

0.1

0.15

0.2

2

3

4

RV

0 0.5 1 1.5 2

2

3

4

Dust Extinction AV

RV

P(RV

!1|AV)= N( R

V

!1 | 0.36+0.14AV, !

r

2=0.042)

Marginal Estimate


Bootstrap Cross-validationLow-z SN

data

Training Set{Ds, zs}

Bootstrap

Test Set{Dt}

Test Set{zt}

SN Ia

Model Predict{µt}

Hubble{µ(zt)}

Compare

• Test Sensitivity of Statistical Model to Finite Sample

• Avoid using data twice for training and distance prediction

• Prediction/Generalization Error


Nearby Optical+NIR Hubble Diagram

(Opt Only) rms Distance Prediction Error = 0.15 mag(Opt+NIR) rms Distance Prediction Error = 0.11 mag

Aggregate Precision ~ (0.15/0.11)2 ≈ 2

104

31

32

33

34

35

36

37

38

µ(p

red

)

h = 0.72

127 BVRI(JH) SN Ia (CfA3+PTEL+CSP+lit)

3000 5000 7000 10000 15000!1

!0.5

0

0.5

1

Velocity [CMB+Virgo] (km/s)

Diff

ere

nce

CV Pred Err (All, cz > 3000 km/s) = 0.14 mag (0.134 ± 0.010 intr.)

CV Pred Err (Opt+NIR & cz > 3000 km/s) = 0.11 mag (0.113 ± 0.016 intr.)CV Pred Err (Opt only & cz > 3000 km/s) = 0.15 mag (0.142 ± 0.013 intr.)

OpticalOptical+NIR


33 33.2 33.4 33.6 33.8 34 34.2 34.40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

SN2005el

Distance Modulus µ

PD

F

P(µ | BV)

P(µ | BVRI)

P(µ | BVRIJH)

E(µ |z) ± (300 km/s)

0.2

0.4

0.6

BV

SN2005el

0.2

0.4

0.6E

xtin

ctio

n A

V

BVRI

33.3 33.4 33.5 33.6 33.7 33.8 33.9 34 34.1 34.2

0.2

0.4

0.6

Distance Modulus µ

BVRIJH

30.5 31 31.5 32 32.50

0.5

1

1.5

2

2.5

3

3.5

4

SN2002bo

Distance Modulus µ

PD

F

P(µ | BV)

P(µ | BVRI)

P(µ | BVRIJH)

E(µ |z) ± (300 km/s)

0.4

0.6

0.8

1

1.2

1.4

BV

SN2002bo

0.4

0.6

0.8

1

1.2

1.4

Extinction A

V

BVRI

31.2 31.4 31.6 31.8 32 32.2 32.4

0.4

0.6

0.8

1

1.2

1.4

Distance Modulus µ

BVRIJH


Improved Distance Precision

for Individual Opt+NIR LCs

• Precision = 1/Variance

• On avg, 2.2x better BVRI vs BV

• 3.6x better BVRIJH vs BV

• 60% better BVRIJH vs BVRI

34 34.5 35 35.50

0.5

1

1.5

2

2.5

3

3.5

4

SN2006cp

Distance Modulus µ

PDF

P(µ | BV)P(µ | BVRI)P(µ | BVRIJH)E(µ |z) ± (300 km/s)

0.20.40.60.8

BV SN2006cp

0.20.40.60.8

Extin

ctio

n A V

BVRI

34.4 34.6 34.8 35 35.2 35.4

0.20.40.60.8

Distance Modulus µ

BVRIJH


Summary• Hierarchical models are useful statistical methods for

discerning multiple random effects

• BayeSN: an efficient MCMC Sampler for computing inferences with SN hierarchical models

• Apparent differential trend of Rv vs Av (local dust at high Av?)

• NIR Light Curves have low correlation with optical, provide independent information on distance

• SN Ia Optical with NIR: Better dust and distance estimates than with Optical alone

33


Future Work & Problems

• Application to Larger Sample of Opt+NIR SN Ia

• Application to high-z SN Ia & Cosmological Inference

• Accounting for Selection Effects

• Using Auxiliary Information

• Host Galaxy Information (e.g. P. Kelly, et al. 2010)

• Spectral? Blondin, Mandel, & Kirshner 2011

• Foley & Kasen 2011 (Color / Ejecta Velocity)


Spectral Info correlate with SN Ia luminosity and light curves?

Blondin, Mandel, Kirshner 2011

Multiple Comparisons Problem


Correlating Spectral Ratios with Luminosity

A&A 526, A81 (2011)

WRMS residuals [mag] Absolute Pearson correlation

R

(x1,R)

(c,Rc)

(x1, c,Rc)

Fig. 7. Results from 10-fold cross-validation on maximum-light spectra. From top to bottom: R only; (x1,R); (c,Rc); (x1, c,Rc). The left columnis color-coded according to the weighted rms of prediction Hubble residuals, while the right column corresponds to the absolute Pearson cross-correlation coe!cient of the correction terms with uncorrected Hubble residuals.

results in a Hubble diagram with lower scatter when comparedto the standard (x1, c) model. Using a flux ratio alone, Baileyet al. (2009) find R(6420/4430) as their most highly-rankedratio, while we find R(6630/4400) (see Table 2). The wave-length bins are almost identical, and in any caseR(6420/4430) is

amongst our top 5 ratios. For this ratio we find ! = !3.40±0.10,in agreement with ! = !3.5± 0.2 found by Bailey et al. (2009)3.3 In fact Bailey et al. (2009) find ! = +3.5 ± 0.2, but this is due to atypo in their equation for the distance modulus: !R really appears as anegative term in their paper (S. Bailey 2010, priv. comm.).

A81, page 10 of 24

A&A 526, A81 (2011)

WRMS residuals [mag] Absolute Pearson correlation

R

(x1,R)

(c,Rc)

(x1, c,Rc)

Fig. 7. Results from 10-fold cross-validation on maximum-light spectra. From top to bottom: R only; (x1,R); (c,Rc); (x1, c,Rc). The left columnis color-coded according to the weighted rms of prediction Hubble residuals, while the right column corresponds to the absolute Pearson cross-correlation coe!cient of the correction terms with uncorrected Hubble residuals.

results in a Hubble diagram with lower scatter when comparedto the standard (x1, c) model. Using a flux ratio alone, Baileyet al. (2009) find R(6420/4430) as their most highly-rankedratio, while we find R(6630/4400) (see Table 2). The wave-length bins are almost identical, and in any caseR(6420/4430) is

amongst our top 5 ratios. For this ratio we find ! = !3.40±0.10,in agreement with ! = !3.5± 0.2 found by Bailey et al. (2009)3.3 In fact Bailey et al. (2009) find ! = +3.5 ± 0.2, but this is due to atypo in their equation for the distance modulus: !R really appears as anegative term in their paper (S. Bailey 2010, priv. comm.).

A81, page 10 of 24

K-fold Cross-ValidationMultiple Comparisons

Blondin, Mandel, Kirshner 2011


Open Problems• Photometric Classification of SN Light Curves

• Classification of SN by Spectra


type ia sn light curve inference: hierarchical …hea-2011/01/25 · type ia sn light curve...

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