Bayesian model comparison of solar flare spectraJ. Ireland1 & G. D. Holman2

1ADNET Systems, NASAʼs GSFC,Greenbelt MD, USA, 2 NASAʼs GSFC,Greenbelt MD, USA

scale = 8

Contact: Jack.Ireland@nasa.gov

Conclusions

Which model is preferred? Introduction

The Bayesian Information Criterion

The odds ratio and the BIC suggest that the NUI model is the preferred model of the two considered here.

Future work applying Bayesian data analysis techniques to RHESSI data analysis will include: other models of the X-ray emission, simultaneous fitting of the background spectrum with the model spectrum, and inclusion of data from multiple RHESSI detectors whilst taking into account systematic differences between those detectors.

We wish to determine which of two models is the preferred description for a portion of RHESSI spectral data observed during the GOES X4.3 solar eruptive event of 23 July 2002.

is derived by suitably approximating the posterior close to its maximum. The lowest value of the BIC is the preferred model. Comparing the NUI and DPL models we find

which is interpreted as positive evidence (Neath & Cavanaugh 2012) preferring NUI over DPL.

Bayesian/MCMC model fitting

Bayes’ Theorem can be used to calculate the odds ratio O12 in favor of one model 1 over model 2. If model 1 has prior probability p(M1|I) then

Bayes’ Theorem states that (b) NUI

Two models for the observed emission

Double power law (DPL) - 7 parameter fit

F0(E) = A

0 E < Ec

(E/Ep)−δ1 Ec ≤ E < Eb

(E/Ep)−δ2(Eb/Ep)δ2−δ1 Eb ≤ E < Eh

0 E ≥ Eh

Ithermal(�) ≈[EM ]

�T 1/2exp(−�/kT )

where Q(ε,E) is the bremsstrahlung cross-section differential in photon energy ε. We model the photon flux energy spectrum as the sum of emission due to a flare-injected electron spectrum interacting with a target, and emission from hot plasma with a Maxwellian distribution of speeds corresponding to some temperature T.

Solar flares accelerate electrons which can interact with the surrounding plasma to produce hard X-rays. For a general inhomogeneous optically thin source of plasma density n(r) and electron flux density energy spectrum F(E, r) (electrons cm−2s−1keV−1) in volume V for electron energy E, the bremsstrahlung photon flux energy spectrum I(ε) (photons cm−2s−1keV−1 at Earth distance R) can be written (Brown 1971) as

I(�) =nV

4πR2

� ∞

�F (E)Q(�, E)dE,

F (E) =�V n(r)F0(E, r)dV/(nV )n =

�V ndV/V

AcronymsGOES - Geostationary Operational Environmental SatelliteMAP - maximum a posterioreMCMC - Markov chain Monte CarloOSPEX - Object SPectral EXecutivePDF - probability density functionRHESSI - Reuven Ramaty High Energy Solar Spectroscopic Imager

DPL parameter

best value

(MAP)Credible intervalCredible interval PriorDPL

parameterbest value

(MAP) 68% 95%Prior

EM (1049 cm-3) 2.16 ±0.04 ±0.08 0.9→8.14

kT (keV) 3.18 ±0.01 ±0.02 0.5→8.0

† A50 0.027 -0.003, 0.002 -0.006, 0.004 0.002→0.3

δ1 3.38 -0.13, 0.09 -0.34, 0.13 1.1→50

Eb (keV) 249 -143, 556 -213, 1131 50→32000

δ2 3.91 -0.12, 0.47 -0.18, 1.48 1.1→50

Ec (keV) 32.1 -16.3, 11.7 -22.7, 19.4 0.01→50

NUI parameter

best value (MAP)

Credible intervalCredible intervalPrior

NUI parameter

best value (MAP) 68% 95%

Prior

EM (1049 cm-3) 2.18 -0.05, 0.03 -0.08, 0.06 0.9→8.14

kT (keV) 3.17 0.008, 0.014 0.019, 0.025 0.5→8.0

† A1 147552 -59348, 30367 -90876, 71678 1→10^6

δ 3.95 -0.07, 0.02 -0.13, 0.05 1.1→50

E* (keV) 83.5 -24.2, 15.1 -39.7, 32.5 1-1000

Ec (keV) 30.9 -23.6, 4.7 -29.7, 12.2 0.01→50

Funded by NASA RHESSI Data analysis WBS 667339.04.01, and via NASA NNH09ZDA001N-SHP

ReferencesAndrae, R., et al., 2010, arXiv:1012.3754 [astro-ph.IM].Brown, J. C. 1971, Sol. Phys., 18, 489.Gregory, P., 2005, Bayesian logical data analysis for the physical sciences, Cambridge University Press.Holman, G. D., et al., 2003, ApJ, 595, L97.Kass,, R. E., Raftery, A. E., 1995, Journal of the American Statistical Association, vol. 90, No. 430, 773.Kontar, E. P., et al., 2002, Sol. Phys., 2002, 210: 419.

OSPEX model fitting

The reduced-χ2 distribution has a deviation of (2/k)1/2≅0.11 (where k=N-n-1 is the number of degrees of freedom in the fit) for the model fits to the data. The minimum reduced-χ2 values found by fitt ing cannot dist inguish between these two models within one deviation (Andrae et al 2010). Figure 1: OSPEX spectral fits for (a) the double power law (DPL) and (b) the non-

uniform ionization (NUI) models.

(a) DPLSPEX HESSI Count Flux vs Energy with Fit Function

10 100Energy (keV)

0.0001

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

coun

ts s

-1 c

m-2 k

eV-1

Detectors: 4F23-Jul-2002 00:30:00.000 to 00:30:20.250 (Data-Bk)

23-Jul-2002 00:30:00.000 to 00:30:20.250 (Bk)

21-Nov-2012 09:38

template 2.11 nuc1pileup_mod 1.00,0.900,1.00,10.0,3.00,0.500 thick2_vnorm 0.0277,3.39,256.,3.92,32.0,3.20e+04,50.0 vth 2.16,3.18,1.00 full chian 1.26e-04vth+thick2_vnorm+pileup_mod+template

Fit Interval 0 Chi-square = 0.95

χ2 =0.95

(a) DPLdarker = higher probability

Spearman rank

correlation coefficient

† AƐ is the electron flux at ‘Ɛ’ keV, in units of 1035 electrons (sec keV)-1

BIC = −2 ln [max(L)] + k lnN

BICNUI −BICDPL = −4.4,

RHESSI observes incident hard X-ray flux I by measuring the energy lost by the photon in the detector. This observed number of detector counts Di in energy bin 1≤i≤m are assumed to be drawn from a Poisson probability density function (PDF) p(Di) with mean Ci in energy bin i via a detector response matrix Rij and incident photon flux Ij. By assuming a model spectrum for the accelerated electrons and a model for the interaction between it and the target plasma, the incident photon flux at RHESSI can be described by a parameter set θ (Eq. 3). Hence for data D = (D1,...,Dm),

where B12 is the Bayes’ factor,

and

ONUI/DPL ≈ 13Using the Kass & Raftery (1995) interpretation of odds ratio values, the NUI model is substantially to strongly preferred.

B12 = L(M1)/L(M2).

where L is the likelihood of the data D given a model M, parameterized by θ, and information I.

O12 = p(M1|I)1−p(M1|I)B12

Two models are considered and fit to the data using OSPEX and Bayesian/MCMC model fitting.

Non-uniform target ionization (NUI) - 6 parameter fit

Thermal X-ray emission (Eq. 4) - EM, kT.Flare accelerated electrons: double power law (Eq. 5) - A, δ1, δ2, Eb, Ec.Thick target interaction (Brown 1971). See Holman et al., 2003 for a similar model.

Thermal X-ray emission (Eq. 4) - EM, kTFlare accelerated electrons: single power law (Eq. 5 with no break Eb) - A, δ, Ec

NUI model: step function (Kontar et al. 2002) - E*. In this model the target plasma transitions from non- to fully- ionized at some atmospheric depth, which can be rewritten as an energy E*. See Kontar et al., 2003 for a similar model.

We assume that the hard X-ray emission is well described by a thick target model, an isothermal background parameterized by emission measure [EM] and scaled temperature [kT] and a power law flare-injected electron spectrum that can include a single break, with low energy cutoff Ec, break energy Eb, and power-law indices δ1 and δ2 below and above Eb respectively.

OSPEX approximates Eq. (2) by replacing each Poisson distribution in the product with a Gaussian distribution with mean and variance equal to the best fit from the previous iteration (starting with a ‘best guess’). The fits obtained are shown in Figure 1.

(1)

(2) (3)

(4)

(5)

(b) NUISPEX HESSI Count Flux vs Energy with Fit Function

10 100Energy (keV)

0.0001

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

coun

ts s

-1 c

m-2 k

eV-1

Detectors: 4F23-Jul-2002 00:30:00.000 to 00:30:20.250 (Data-Bk)

23-Jul-2002 00:30:00.000 to 00:30:20.250 (Bk)

21-Nov-2012 10:16

thick_nui 9.30e+04,3.89,65.8,0.00,31.6,1.00e+04 template 1.89 brd_nucpileup_mod 0.987,0.900,1.00,12.7,4.03,0.500 vth 2.24,3.16,1.00 full chian 1.26e-04vth+pileup_mod+template+thick_nui

Fit Interval 0 Chi-square = 0.92

χ2 =0.92 Number of data points (energy bins): N = 150

Assuming that we have no prior information to prefer either the NUI or DPL models we obtain

is the global likelihood of the model M.

L(M) =

�dθp(θ|M)L (D|M(θ), I)

L(D|M(θ), I) =m�

i

CDii

Di!e−Ci Ci = RijIj(θ)

p(M,D|I) = p(θ|M, I)p(D|I) L(D|M(θ), I)

where p(M|D, I) is the posterior PDF for the model M given the data D. The quantity p(θ|M, I) is the prior probability that the model

parameters have a given value, and p(D| I) is a constant. We assume that each variable has a constant probability within a reasonable range (Tables 1, 2). A MCMC algorithm is used to sample the posterior PDF. Figure 2 shows that for both models all parameters are either moderately or strongly correlated / anti-correlated. Tables 1 & 2 show that some parameters have asymmetric uncertainty estimates.

Figure 2: all two-dimensional probability

density distributions for the (a) DPL and

(b) NUI models.

Tables 1 & 2: Model parameters: values and uncertainty estimates

(6)

(7)

(8)

(9)

(10)number of parameters n=7

number of parameters n=6

Kontar, E. P., et al., 2003, Ap. J., 595, L123.Neath, A. A., Cavanaugh, J. E., 2012, WIREs Comput. Stat., 4:199-203. doi: 10.1002/wics.199.

(b)