Variance-Covariance data of experimentalCovariance … · Variance-Covariance data of...

1“Workshop on Nuclear Data and Uncertainty Quantification”, CCFE, Abingdon, UK, 24-25 January 2012

Variance-Covariance data of experimentalVariance Covariance data of experimental observables

in the resonance regionB. Becker, P. Schillebeeckx, S. Kopecky, J. Heyse

Joint Research Centre (JRC)IRMM - Institute for Reference Materials and MeasurementsGeel BelgiumGeel, Belgiumhttp://irmm.jrc.ec.europa.eu/http://www.jrc.ec.europa.eu/


Objective WP36

Produce accurate cross section data together withProduce accurate cross section data together with

reliable covariance information in the resonance region

Reduce bias effects

Produce reliable and realistic covariance data


Covariance on model cross sections, (, V)

ModelReaction(, V) (, V)

( , V ) determined by :

• Reaction model : = F () : model parameters• Reaction model : = Fm() , : model parameters– R-Matrix; HF+WF; Optical Model

C l l ti M th d f V• Calculation Method for V

– GLUP : V = D V DT with D = F/

MC i l ti (t f ti f i bl CEA C d h )– MC-simulations (transformation of variables, CEA Cadarache)

• + model parameters (, V): in resonance regionfully determined by experimentfully determined by experiment


Covariance matrix of model parameters : ( , V)

(, V)Model

Reaction + Experiment (Zexp, Vz, , V)

)),(GZ(V)),(GZ()( mexp1exp,Z

Tmexp

2

• Model Gm : reaction model () + experiment ()• Reaction model: R-matrix

Resonance parameters = (R’ E J )– Resonance parameters = (R , Er, J, n, , …)

• Experimental model : parameters – Normalization, Target characteristics, Resolution, …

• Determination of reliable (, V) requires full details of the experiment and data reduction procedures


Resonance shape analysis

Tm = Gm(, , )0.8

1.0

mis

sion

dE)E('T)Et()t(T ),E(totne)E('T

Gm is a model describing the observable Tm with model parameters:4

ls

0.2

0.4

0.6

Tran

msm

Mn powder n = 9.94 10-3

dE),,E(T)E,t()t(Tm e),E(T

model parameters:Resonance parameters Experimental parameters (N, TD, n, …)(t,E) response function of TOF-spectrometer 1.0

10000 20000 30000 40000 50000

-404

Res

iuda

: Least square adjustment (generalized, …)V : GLUP0.2

0.4

0.6

0.8

Tran

smis

sion

11VT

T )DVD(V

G U

MC Simulations + REFIT (CEA Cadarache)

10000 20000 30000 40000 50000

-4

4

Res

idua

ls

Sputtering target n = 1.92 10-2 Vexp,T )DVD(V

Neutron Energy / eVTime-of-flight


Model parameters from experimental dataa) reaction experiment

800

1000 Exp. Fm(x,)

s) statisticscountingonlyVZ

)),x(FZ(V)),x(FZ()( mexp1exp,Z

Tmexp

2

(1 % at peak)

)A(eA)(F 222

2)oxx(

400

600

Cou

nt ra

te /

(1/s

stat st cscou t go yexpZ ( p )

),x,A(e2

),x(F 2o

22m

010)00013.0(5x001)30.0(100A

o 4.6 4.8 5.0 5.2 5.4

0

200C

x / unit

S th ti XS d l

2)oxx(

100)0000078.0(0018.0)(

2o

22PA

Synthetic XS model: XS shape: Gaussian Peak value: 10000 counts

),x,P(eP),x(F 2o

22

)oxx(

m

58.001)52.3(437.939P

2PA correlation depends on fit region

1058.0)0000078.0(0018.0010)00013.0(5x

2o


Model parameters from experimental datareaction experiment

),x,A(e2

A),x(F 2o

22

2)oxx(

2m

),x(mFnm e),x(T

2

38001)610(100A 1.0

n = 0.01

1038.0)000024.0(0018.0010)00030.0(5x

38.001)61.0(100A

2o

0.4

0.6

0.8

ount

rate

/ (1

/s)

0 01

n = 0.001

4.6 4.8 5.0 5.2 5.40.0

0.2

Co

x / unit

n = 0.01

1082.0)000010.0(0018.0010)00011.0(5x

82.001)78.0(100A

2o

n = 0.01



)B,C(VBCZ expexpZexpexp B5B


Tmexp

2

001)33.0(100A

p

800

1000 Exp. Fm(x,)

s)

%10BBand

1005

PBwith

100)000010.0(0018.0010)00015.0(5x

)(

2o

400

600

nt ra

te /

(1/s

200

400

Cou

n

4.8 4.9 5.0 5.1 5.20

x / unit001)30.0(100A

only counting statistics

100)0000078.0(0018.0010)00013.0(5x

2o



)N,C(VCNZ expexpZexpexp )),x(FZ(V)),x(FZ()( mexp

1exp,Z

Tmexp

2

)000130(5x)98.1(9.98A

%2N800

1000 Exp.

N/N2 %s)

p

)0000080.0(00180.0)00013.0(5x

2o

%2

N

400

600

2 %

nt ra

te /

(1/s

200

400

Cou

n

4.8 4.9 5.0 5.1 5.20

x / unit001)30.0(100A


100)0000078.0(0018.0010)00013.0(5x

2o




Tmexp

2

)N,C(VCNZ expexpZexpexp

800

1000 Exp.

N/N2 %s)

p

)000130(5x)98.1(9.98A

%2N

400

600

2 % 5 %

nt ra

te /

(1/s

)0000080.0(00180.0)00013.0(5x

2o

%2

N

%5NN

)0000084.0(00179.0)00014.0(5x

)85.4(62.93A

2o

200

400

Cou

n

4.8 4.9 5.0 5.1 5.20

x / unit




Tmexp

2

)N,C(VCNZ expexpZexpexp

800

1000 Exp.

N/N2 %s)

p

)000130(5x)98.1(9.98A

%2N

400

600

2 % 5 % 10 %

nt ra

te /

(1/s

)0000080.0(00180.0)00013.0(5x

2o

%2

N

200

400

Cou

n

%5NN

)0000084.0(00179.0)00014.0(5x

)85.4(62.93A

2o

%10N

)00017.0(5x)87.8(60.78A

o

4.8 4.9 5.0 5.1 5.20

x / unit%10

N

)000010.0(00179.0)(

2o

shape is not affected


Model parameters from experimental datab) transmission experiment

)),x(FT(V)),x(FT()( mexp1exp,Z

Tmexp

2

)N,C(VCNT expexpZexpexp p

%10NN

1.0

N

0.6

0.8 Exp. Fm(x,)

nsm

issi

on

010)00030.0(5x41.001)63.0(100A

o

0.4

Tran 1041.0)000024.0(0018.0

)(2o

4.8 4.9 5.0 5.1 5.20.2

x / unit38.001)61.0(100A


1038.0)000024.0(0018.0010)00030.0(5x

2o




Tmexp

2

)B,C(VCBT expexpZexpexp p

%10BB

21

PB

1.0

B 2P

0.6

0.8 Exp. Fm(x,)

nsm

issi

on

010)00030.0(5x41.001)66.0(100A

o

0.4

Tran 1041.0)000024.0(0018.0

)(2o

4.8 4.9 5.0 5.1 5.20.2

x / unit38.001)61.0(100A


1038.0)000024.0(0018.0010)00030.0(5x

2o




Tmexp

2

)N,B,C(VCNBT expexpZexpexp p

%10BB

%10NN

21

PB

1.0

B

010)00034.0(5x61.001)2.5(0.85A

o

N 2P

0.6

0.8 Exp. Fm(x,)

nsm

issi

on

1061.0)00038.0(0018.0

)(2o

0.4

Tran

38.001)61.0(100A

only counting statistics4.8 4.9 5.0 5.1 5.20.2

x / unit

1038.0)000024.0(0018.0010)00030.0(5x

2o


Peelle’s Pertinent Puzzle Solutions

1) Normalization: use estimate E[y] for normalizationcan not always be applied in practice (chicken/egg problem)

G. D’Agostini, “On the use of covariance matrix to fit correlated data”, NIM. A346 (1994) 3062

22

1 zz

2212

2k2

2y21y

1y22y1

)yy(k

zYKZ

Yexp

2) General solution: include exp. parameters as adjustable parameters

exp

additional advantage: avoid underestimation of V when Bayes’ is applied

F. Fröhner, “Assigning uncertainties to scientific data”, NSE. 126 (1997) F Fröhner “Evaluation of data with systematic errors” NSE 145 (2003) 342F. Fröhner, Evaluation of data with systematic errors , NSE 145, (2003) 342


Reliable model parameters from experiment

F.H. Fröhner,

“Evaluation of Data with Systematic Errors”, NSE 145 (2003) 342y , ( )

« The optimal way to handle error-affected auxiliary quantities(“nuisance parameters”) in data fitting and parameter estimation is toadjust them on the same footing as the parameters of interest and tointegrate (marginalize) them out of the joint posterior distributionintegrate (marginalize) them out of the joint posterior distributionafterward. »


Peelle’s Pertinent Puzzle: e.g. 103Rh(n,) in URR

))T,S(FZ(V))T,S(FZ()T,S()T,S(FZ

mexp1expZ

Tmexp

2m

VZexp includes 2% normalization uncertainty

150

175 60 m 0.26 mm Full covariance

V1/2 )

125

2 ) / (

barn

eV

100

E n1/2

0 50000 10000075

Neutron energy / eV



))T,S,N(G)Y,N((V))T,S,N(G)Y,N(()T,S,N()T,S,N(G)Y,N(

mexpexp1

expY,NT

mexpexp2

m

exp,

Normalization included as model parameter

150

175 60 m 0.26 mm Fm(N,S

,T

)

V1/2 )

125

2 ) / (

barn

eV

100

E n1/2

0 25000 50000 75000 10000012500075

Neutron energy / eV



))T,S,N(G)Y,N((V))T,S,N(G)Y,N(()T,S,N()T,S,N(G)Y,N(

mexpexp1

expY,NT

mexpexp2

m

exp,

Normalization included as model parameter

150

175 30 m 0.05 mm Evaluation Full covariance

V1/2 )

150

175 60 m 0.26 mm Evaluation Full covariance

125

) / (b

arn

eV

125

100

E n1/2 )

100

0 50000 100000 15000075

Neutron energy / eV0 50000 100000 150000

75

Neutron energy / eV


Accurate cross section data (, V )

Require reliable model parameters which can only be deduced from experiment:experiment:

• Relation between experimental observables and model parameters are well defined

• All uncertainty components are identified, quantified and documented (uncorrelated & correlated)

Proposal NDS-IAEA / IRMM (Otsuka et al.)


AGS - system (Analysis of Geel Spectra)Special format for full covariance information

Analysis of Geel Spectra (AGS)

• Transforms count rate spectra into observables (transmission, yields)

• Includes all features to perform a full analysis

f• Full uncertainty propagation starting from counting statistics

• Output: complete covariance matrix

S i l f f i i• Special format for covariance matrix

– Reduce space for data storage (EXFOR)– Document the sources of uncertainties due to each step

in the data reduction process

X Z Dz Sz

SZ : correlated part

DZ : uncorrelated partn values

in the data reduction process– Verify the contribution of each quantity introducing a

correlated uncertainty component!

Z pdim. (n x k)C. Bastian, Int. Soc. Optical Eng. 2867 (1997) 611


TOF - spectrum and parameter vectorf(a,Y) only channel-channel operations

)Y,a(fZn.dim)counts(spectrumTOF )V,Y( Y

vectorparameter)a,...,a(a k1

TT GDGGVGV

)V,a( a

if

2l tdi lDV1)

YYYaaaZ GDGGVGV k

iik,aa a

fgG.g.e

2iyYY elementsdiagonalDV 1)

2) operationschannelchannelonlyf

definitepositiveandsymmetric:Va

Taaa

LLV Cholesky transformation

)

3)

py

matrixtriangularlowerLa



)Y,a(fZn.dim)counts(spectrumTOF )D,Y( Y


TT GDGGVGV

)V,a( a

YYYaaaZ GDGGVGV

Taaa

LLV diagonalDV YY aaa

Ta

Taaa

GLLGdiagonal:GY

channelchannelonlyf

TYYYZ GDGD

DZ diagonal n - values

aaZLGS

dim (n k)T

Z DSSV DZ diagonal, n - valuesdim (n,k) ZZZZ





TT GDGGVGV

)V,a( a

YYYaaaZ GDGGVGV

Taaa

LLV

TYYYZ GDGD

aaZLGS

TZ DSSV

dim (n,k)ZZZZ DSSV

AGS - format number of values to be stored dim (n k+1) number of values to be stored dim. (n, k+1)

k = total number of correlated uncertainty components





TT GDGGVGV

)V,a( a

YYYaaaZ GDGGVGV

Taaa

LLV

TYYYZ GDGD

aaZLGS

TZ DSSV componentseduncorrelatDcomponentscorrelatedS ZZZZ DSSV

AGS - format number of values to be stored dim (n k+1)

componentseduncorrelatDZcomponentscorrelatedS

Z

number of values to be stored dim. (n, k+1)k = total number of correlated uncertainty components


TOF - spectrum and parameter vector Example: YaZ


)V,a( 2aa 1.dimvectorparameter)a(a 1

TT GDGGVGV YYYaaaZ GDGGVGV

TaaaL diagonalDV YY

diagonal:GYi

ii,a y

afg

YGa

channelchannelonlyf

ayfgi

iii,Y

Y2

Z DaD

DZ diagonal

aZLYS

dim (n 1)T

Z DSSV DZ diagonaldim (n,1) ZZZZ


TOF - spectra in AGS format and parameter vector)Z,Z,b(gZ 21 b, Z1 and Z2 not correlated

)V,b( b bkb.dim

Input Output

)V,b( b bkb.dim

)Z,Z,b(fZ 21

T

Tbbb

LLV

)V,Z( 1Z1 )Y,a(fZ 1111 11 ka.dim Z

TZZZ DSSV

TcomponentseduncorrelatD

Z

componentscorrelatedSZ

1ZT1Z1Z1Z DSSV

)V,Z( 2Z2 )Y,a(fZ 2222 22 ka.dim

pZ

2ZT2Z2Z2Z DSSV

2Z2Z2Z


AGS : )Z,Z,b(gZ 21 b, Z1 and Z2 not correlatedZ1 and Z2 in AGS-format

1ZT1Z1Z1Z DSSV

2ZT2Z2Z2Z DSSV

)k(Sdi )k(Sdi

Tbbb LLV

TTTZ GVGGVGGVGV

)k,n(Sdim 11Z )k,n(Sdim 2

2Zbkb.dim

2Z2Z2Z1Z1Z1ZbbbZ GVGGVGGVGV

T2Z2Z2Z

T1Z1Z1Z

T2Z

T2Z2Z2Z

T1Z

T1Z1Z1Z

Tb

TbbbZ

GDGGDGGSSGGSSGGLLGV

T2Z2Z2Z

T1Z1Z1ZZ

GDGGDGD

2Z2Z2Z1Z1Z1Z2Z2Z2Z2Z1Z1Z1Z1ZbbbbZ

2Z2Z1Z1ZbbZ SGSGLGS

componentseduncorrelatDZ

dim. (n x k) k = kb + k1 + k2

componentscorrelatedSZ

ZTZZZ

DSSV


AGS, File format

ZTZZZ

DSSV

n : number of data points (TOF)

k b f titi i t d i l t dk : number of quantities introducing correlateduncertainty components

DZ : uncorrelated part (n values)Z p ( )

SZ : matrix dimension (n x k)contains the contribution of each quantitycreating a correlated uncertainty component

X Z DZ SZ

creating a correlated uncertainty component

C. Bastian, Int. Soc. Optical Eng. 2867 (1997) 611

Observable Z (dimension n) with k sources of correlated uncertainties


AGS, File format

ZTZZZ

DSSV

Storage spaceCovariance matrix

n2 elements (e.g. 32k x 8 bytes 8 Gb) ( g y )AGS representation

n (k+1) elements (32k, 20 corr. 5 Mb)

InformationCovariance matrix

no separation between components

AGS representationseparates uncorrelated and correlated components(avoid PPP)

X Z DZ SZ

C. Bastian, Int. Soc. Optical Eng. 2867 (1997) 611

Observable Z (dimension n) with k sources of correlated uncertainties

(avoid PPP)verify impact of each individual component


AGS commands

Write only commands ags_mpty Create an empty AGS file

tA I t t f th AGS filags_getA Import spectra from another AGS fileags_getE Import/interpolate evaluated data from an ENDF file ags_getXY Import histogram data from an ASCII file Read/Write commands : Operations on spectra ags_addval Add a constant value to all Y-values of a spectrum ags_avgr Average Y values per channel ags_func Calculates the Y values for a special function ags_idtc Determine the dead time correction of a TOF-spectrum ags_divi Divide a spectrum by another ags mult Multiply a spectrum with anotherg _ p y pags_lico Linear combination of n spectra with n constants ags_ener Build energy from TOF X-vector ags_fit Non-linear fit of spectra ags_fxyp User-programmed function

Read Only commands Read Only commandsags_edit Edit constants and scalers attached to a spectrum ags_list List Y values of spectra with common X values ags_putX Export final result to an ASCII file ags_scan Scan the contents of an AGS file


AGS, Script for transmission data

# create ags-file ags_mpty TRFAK # read sample out scaler=TOout,CMoutscaler TOout,CMout ags_getXY TRFAK /SCALER=$scaler /FROM=spout.his /ALIAS=SOUT # read sample in scaler=TOin,CMin ags_getXY TRFAK /SCALER=$scaler /FROM=spin.his /ALIAS=SIN /LIKE=A01SOUT

'out

'out

'in

'in

BCBCT

# dead time correction dtcoef=DTCOEF ags_idtc TRFAK,A01SOUT /DTIME=$dtcoef /LPSC=1 ags_idtc TRFAK,B01SIN /DTIME=$dtcoef /LPSC=1 # normalize to central monitor and divide by bin width ags_avgr TRFAK,C01SOUT /CMSC=2 ags_avgr TRFAK,D01SIN /CMSC=2 #calculate background contribution ags func TRFAK /FUN=f01 /PARFILE=PAROUT /ALIAS=SBOUT /LIKE=A01SOUTags_func TRFAK /FUN f01 /PARFILE PAROUT /ALIAS SBOUT /LIKE A01SOUTags_func TRFAK /FUN=f01 /PARFILE=PARIN /ALIAS=SBIN /LIKE=A01SOUT #subtract background ags_lico TRFAK,E01SOUT,G01SBOUT /ALIAS=SOUTNET /PAR=1.0,-1.0 ags_lico TRFAK,F01SIN,H01SBIN /ALIAS=SINNET /PAR=1.0,-1.0 #create transmission factor ags_divi TRFAK,I01SOUTNET,J01SINNET /ALIAS=TRFAK


Calculation of transmission

105

Cout

Bout

Sample out105

Cin

Bin

Sample in

103

104

Cou

nts

103

104

Cou

nts

'out

'out

'in

'in

expBC

BCNT

1 10 100 1000102

Time / s1 10 100 1000

102

Time / s 0 8

1.0

to

r

Time / sTime / s

0 4

0.6

0.8

mis

sion

fact

0.2

0.4

Tran

s

1 10 100 10000.0

Time / s


Output AGS_PUTX

Bin / Bin : 10.0 %Bout / Bout : 5.0 %N / N : 0 5 %

CZ = DZ + S ST XL XH Z Z Zu DZ S

VZ = DZ + S ST

N / N : 0.5 % XL XH Z Z Zu DZ S

Zu2 Bin Bout N

800 1600 0.999 0.79E-2 0.59E-2 0.35E-4 0.14E-2 -0.08E-2 0.50E-2 1600 2400 0.999 0.86E-2 0.67E-2 0.45E-4 0.18E-2 -0.10E-2 0.50E-2 2400 3200 0.999 0.92E-2 0.73E-2 0.54E-4 0.21E-2 -0.12E-2 0.50E-2 3200 4000 0.999 0.97E-2 0.78E-2 0.61E-4 0.24E-2 -0.13E-2 0.50E-2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

16000 16800 0 899 1 30E 2 1 07E 2 1 15E 4 0 51E 2 0 25E 2 0 45E 2 16000 16800 0.899 1.30E-2 1.07E-2 1.15E-4 0.51E-2 -0.25E-2 0.45E-2 16800 17600 0.818 1.24E-2 1.02E-2 1.04E-4 0.53E-2 -0.24E-2 0.41E-2 17600 18400 0.701 1.15E-2 0.93E-2 0.86E-4 0.54E-2 -0.21E-2 0.35E-2 18400 19200 0.594 1.06E-2 0.84E-2 0.71E-4 0.55E-2 -0.18E-2 0.30E-2 19200 20000 0.501 0.98E-2 0.76E-2 0.57E-4 0.56E-2 -0.15E-2 0.25E-2 20000 20800 0.504 1.00E-2 0.77E-2 0.59E-4 0.57E-2 -0.16E-2 0.25E-2 20800 21600 0.581 1.09E-2 0.85E-2 0.73E-4 0.58E-2 -0.19E-2 0.29E-2 21600 22400 0.707 1.22E-2 0.98E-2 0.97E-4 0.60E-2 -0.23E-2 0.35E-2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

964000 972000 0 999 5 91E 2 3 75E 2 14 06E 4 3 98E 2 2 18E 2 0 50E 2 964000 972000 0.999 5.91E-2 3.75E-2 14.06E-4 3.98E-2 -2.18E-2 0.50E-2 972000 980000 1.037 6.09E-2 3.89E-2 15.13E-4 4.04E-2 -2.31E-2 0.52E-2 980000 988000 1.001 6.01E-2 3.80E-2 14.46E-4 4.05E-2 -2.23E-2 0.50E-2 988000 996000 1.010 5.92E-2 3.77E-2 14.23E-4 3.96E-2 -2.20E-2 0.50E-2


AGS Output -> Covariance matrix

ght /

ns

e-of

-flig

Tim

e

Time-of-flight / ns


Reliable model parameters from experiment

Requirements:

Well documented experimental observables in EXFOR

Including:

- experimental details (TD, n, N, …)

- all uncertainty components (correlated and uncorrelated, AGS-

format)

- + Response function of TOF-spectrometers

Recommendation IAEA / IRMM based on the AGS - system

Otsuka et al., ND2010, JKPS 59 (2011)

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