pseudo-measurement simulations and bootstrap for the ......s. varet ( cea-dam-dif, ens cachan )...
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Pseudo-measurement simulations and bootstrapfor the experimental cross-section covariances
estimation with quality quantification
S. Varet1
P. Dossantos-Uzarralde1 N. Vayatis2 E. Bauge1
1CEA-DAM-DIF
2ENS Cachan
WONDER-2012 September 25-28
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 1 / 29
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Motivations: experimental cross-sections
12 13 14 15 16 17 18 19 200.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Energy (MeV)
n2n
cros
s se
ctio
n
2555Mn n2n cross section
EXFOR data
Available informations:the measurementstheir uncertainty
→ covariance matrix estimation and its inverse
Example
Evaluated cross sections uncertainty: generalized χ2 ([VDUVB11])
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 2 / 29
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Motivations: experimental cross-sections covariances
HOW?
Empirical estimator: only one measurement per energy
cov(Fi, Fj) ≈1n
n∑k=1
(F(k)i − F̄i)(F(k)j − F̄j)
Conventionnal approach: propagation error formula [IBC04],[Kes08]
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 3 / 29
-
Motivations: experimental cross-sections covariances
HOW?
Empirical estimator: only one measurement per energyConventionnal approach: propagation error formula [IBC04],[Kes08]
cov(Fi, Fj) ≈∑
param. exp. (qk, ql)
∂Fi∂qk
.∂Fj∂ql
.cov(qk, ql)
ex. : fission fragments yield, recoil protons yield,
→ the needed informations are rarely available→ linearity assumption
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 3 / 29
-
Motivations: experimental cross-sections covariances
HOW?
Empirical estimator: only one measurement per energyConventionnal approach: propagation error formula [IBC04],[Kes08]
QUALITY OF THE COVARIANCES ESTIMATION?
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 3 / 29
-
Outline
1 Notations
2 Experimental covariances estimation: new method
3 Validation
4 Quality measure of the obtained estimation
5 Conclusion
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 4 / 29
-
Notations
Outline
1 Notations
2 Experimental covariances estimation: new method
3 Validation
4 Quality measure of the obtained estimation
5 Conclusion
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 5 / 29
-
Notations
Notations
Schematic representation of the experimental cross sections
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 6 / 29
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Experimental covariances estimation: new method
Outline
1 Notations
2 Experimental covariances estimation: new method
3 Validation
4 Quality measure of the obtained estimation
5 Conclusion
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 7 / 29
-
Experimental covariances estimation: new method
Pseudo-measurements method
Given the available information (the measurements and theiruncertainty)
The idea:
Repeat the available information in order to artificially increase thenumber of measurements
The method:
1 Construction of a regression model h (SVM, polynomial,...)2 Generation of r pseudo-measurements (S(1), ..., S(r)): gaussian
noise centered on h: N ((h(E1), ..., h(EN))t, diag(σ1, ..., σN))3 Σ̂F: empirical estimator of ΣF4 Diagonal terms are imposed
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 8 / 29
-
Experimental covariances estimation: new method
Pseudo-measurements method
Given the available information (the measurements and theiruncertainty)
The idea:
Repeat the available information in order to artificially increase thenumber of measurements
The method:
1 Construction of a regression model h (SVM, polynomial,...)2 Generation of r pseudo-measurements (S(1), ..., S(r)): gaussian
noise centered on h: N ((h(E1), ..., h(EN))t, diag(σ1, ..., σN))3 Σ̂F: empirical estimator of ΣF4 Diagonal terms are imposed
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 8 / 29
-
Experimental covariances estimation: new method
The SVM regression principle [SS04], [VGS97]
Linear case:Assume Fi = h(Ei) = w.Ei + b with Ei ∈ Rs and w ∈ Rs (here s = 1).The svm regression model is a solution of the constraint optimisationproblem:
flat function: minimize12‖w‖2
errors lower than ε: subject to{
Fi − (w.Ei + b) ≤ ε(w.Ei + b)− Fi ≤ ε
Non linear case:Find a map of the initial space of energy, (into a higher dimensionnalspace) such as the problem becomes linear in the new space(mapping via Kernel).
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 9 / 29
-
Experimental covariances estimation: new method
Pseudo-measurements method
Given the available information (the measurements and theiruncertainty)
The idea:
Repeat the available information in order to artificially increase thenumber of measurements
The method:
1 Construction of a regression model h (SVM, polynomial,...)2 Generation of r pseudo-measurements (S(1), ..., S(r)): gaussian
noise centered on h: N ((h(E1), ..., h(EN))t, diag(σ1, ..., σN))3 Σ̂F: empirical estimator of ΣF4 Diagonal terms are imposed
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 10 / 29
-
Experimental covariances estimation: new method
Pseudo-measurements method
Σ̂F term at the ith line and jth column, for i and j ∈ {1, ..., N}:
Cij =σiσjn + r
n∑k=1
(F(k)i − h(Ei))(F(k)j − h(Ej))√
V̂i√
V̂j
+σiσjn + r
r∑k=1
(S(k)i − h(Ei))(S(k)j − h(Ej))√
V̂i√
V̂j
and
Cii = σ2i
where n = 1.
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 11 / 29
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Experimental covariances estimation: new method
Number r of pseudo-measurements?
Constraint on r
0
r too small:Σ̂F non invertible
r too large:Σ̂F diagonal matrix
r = 0
Σ̂F invertible Σ̂F non invertible
no pseudo-measurements r such as Σ̂F invertible
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 12 / 29
-
Validation
Outline
1 Notations
2 Experimental covariances estimation: new method
3 Validation
4 Quality measure of the obtained estimation
5 Conclusion
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 13 / 29
-
Validation
Toy model
Goal:Compare Σ̂F with ΣF: real case: ΣF is unknown
⇒ we build a toy model with ΣF fixed ⇒ numerical validation
Toy Model:
M :=
M1
...MN
(≡ F) ↪→ NN(mM,ΣM)number of M realisations n = 1, N = 5
θ =
0BBBB@1.25−3.49−0.67−7.41−2.29
1CCCCA ΣF =0BBBB@
9.49 −1.20 2.39 −0.03 0.07−1.20 3.64 −3.21 0.03 −0.092.39 −3.21 5.27 0.48 0.10−0.03 0.03 0.48 6.79 0.010.07 −0.09 0.10 0.01 5.72
1CCCCA
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 14 / 29
-
Validation
Pseudo-measurements with the ’Toy Model’
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−20
−15
−10
−5
0
5
10
Variable (ex : energy)
Mea
sure
men
ts (e
x : X
s)
SVM regression on the toy model
mM
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29
-
Validation
Pseudo-measurements with the ’Toy Model’
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−20
−15
−10
−5
0
5
10
Variable (ex : energy)
Mea
sure
men
t (ex
: X
s)
SVM regression on the toy model
mM
Toy measurement
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29
-
Validation
Pseudo-measurements with the ’Toy Model’
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−20
−15
−10
−5
0
5
10
Variable (ex : energy)
Mea
sure
men
t (ex
: X
s)
SVM regression on the toy model
SVM regressionmM
Toy measurement
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29
-
Validation
Pseudo-measurements with the ’Toy Model’
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−20
−15
−10
−5
0
5
10
Variable (ex : energy)
Mea
sure
men
ts (e
x : X
s)
SVM regression on the toy model
SVM regressionmMToy measurementr=1 pseudo−measurement
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29
-
Validation
Pseudo-measurements with the ’Toy Model’
N=5, r = 1 sample of pseudo-measurements
1 2 3 4 5
1
2
3
4
5
1 2 3 4 5
1
2
3
4
5−2
0
2
4
6
8
Real matrix Estimation
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29
-
Validation
5525Mn: (n,2n) cross section
Values extracted from [MTN11]N=11, r=1, Σ̂F is a 11x11 matrix
12 13 14 15 16 17 18 19 200.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Energy (eV)
Cro
ss s
ectio
n (m
b)
SVM modelExperimental cross sectionsPseudo−measurements
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
11 −0.4
−0.2
0
0.2
0.4
0.6
0.8
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 16 / 29
-
Validation
5525Mn: total cross section
Values extracted from EXFOR [EXF]N=399, r=1, Σ̂F is a 399x399 matrix
0 5 10 15x 106
2.5
3
3.5
4
Energy (eV)
Cro
ss s
ectio
n (m
b)
SVM modelExperimental dataPseudo−measurements
50 100 150 200 250 300 350
50
100
150
200
250
300
350
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
How far is the estimation from the real matrix?
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 17 / 29
-
Quality measure of the obtained estimation
Outline
1 Notations
2 Experimental covariances estimation: new method
3 Validation
4 Quality measure of the obtained estimation
5 Conclusion
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 18 / 29
-
Quality measure of the obtained estimation
Estimator quality
Criterion choice: quality of Σ̂F and/or quality of Σ̂F−1
?⇒ distance criterion
1 Distance between Σ̂F and ΣF: Frobenius norm
IE(‖Σ̂F − ΣF‖fro) = IE
N∑
i=1
N∑j=1
|Ĉij − Cij|2 12
2 Distance between Σ̂F
−1and Σ−1F : Kullback-Leibler distance [LRZ08]
KL(Σ̂F,ΣF) = trace((Σ̂F)−1.ΣF)− log(det(Σ̂F−1
.ΣF))− N
Criterion estimation → parametric bootstrap
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 19 / 29
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Quality measure of the obtained estimation
Bootstrap principle
F : θF, ΣF: unknown
(F(1), ..., F(n)) : h, Σ̂F: sample measurements
(F′(1), ..., F′(n))1, ...,(F′(1), ..., F′(n))B resample measurements
̂̂ΣF1 ̂̂ΣFBResampling allows to make statistics on Σ̂F:
ÎE(Σ̂F), V̂ar(Σ̂F), B̂ias(Σ̂F),...
Resample strategy:classical: draw with replacement in the initial sample (F(1), ..., F(n))parametric: simulations with a fixed law (here N (h, Σ̂F))
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 20 / 29
-
Quality measure of the obtained estimation
Estimation of IE(‖Σ̂F − ΣF‖fro) and KL(Σ̂F, ΣF)
1st case: with r = 0, Σ̂F invertibleUsual parametric bootstrap:(F′(1), ..., F′(n))1, ..., (F′(1), ..., F′(n))B where F′(i) ↪→ N (H, Σ̂F) withH = (h(E1), ..., h(EN))t.ccΣFb term at the ith line and jth column, for i and j ∈ {1, ..., N} and for b = 1, ..., B:
Cbij =σiσj
n
nXk=1
(F′(k)bi − h(Ei))(F′(k)bj − h(Ej))q bV ′i q bV ′j and C
bii = σ
2i
IE(‖Σ̂F − ΣF‖) ≈1B
B∑b=1
‖̂̂ΣFb − Σ̂F‖KL(Σ̂F,ΣF) ≈
1B
B∑b=1
KL(̂̂ΣFb, Σ̂F)S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 21 / 29
-
Quality measure of the obtained estimation
Estimation of IE(‖Σ̂F − ΣF‖fro) and KL(Σ̂F, ΣF)
2nd case: with r = 0, Σ̂F non invertible2 simultaneous parametric bootstraps:(F′(1), ..., F′(n))1, ..., (F′(1), ..., F′(n))B where F′(i) ↪→ N (H, cΣF) et(S′(1), ..., S′(r))1, ..., (S′(1), ..., S′(r))B where S′(i) ↪→ N (H, diag(cΣF))ccΣFb term at the ith line and jth column, for i and j ∈ {1, ..., N} and for b = 1, ..., B:Cbij =
σiσjn + r
nXk=1
(F′(k)bi − h(Ei))(F′(k)bj − h(Ej))q bV ′i q bV ′j +
σiσjn + r
rXk=1
(S′(k)bi − h(Ei))(S′(k)bj − h(Ej))q bV ′i q bV ′j
and Cbii = σ2i
IE(‖Σ̂F − ΣF‖) ≈1B
B∑b=1
‖̂̂ΣFb − Σ̂F‖KL(Σ̂F,ΣF) ≈
1B
B∑b=1
KL(̂̂ΣFb, Σ̂F)S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 22 / 29
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Quality measure of the obtained estimation
Bootstrap on the ’Toy Model’
N = 5, n = 1, B = 30, ‖ΣF‖ = 15.6680, r = 1,
0 2 4 6 8 10 1210
12
14
16
18
20
Index of the 1−sample of measures
Em
piric
al m
ean
and
varia
nce
of
||C2 F
− Σ F
|| its
boo
tstra
p es
timat
ion
Empirical mean and variance of||C2F − ΣF|| its bootstrap estimation (B=30).
||C2F − ΣF||
Σb=1B ||C2F
b − C2F||/B
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 23 / 29
-
Quality measure of the obtained estimation
Bootstrap on the ’Toy Model’
N = 5, n = 1, B = 30, ‖ΣF‖ = 15.6680, r = 1,
0 2 4 6 8 10 1210
12
14
16
18
20
Index of the 1−sample of measures
Em
piric
al m
ean
and
varia
nce
of
||C2 F
− Σ F
|| its
boo
tstra
p es
timat
ion
Empirical mean and variance of||C2F − ΣF|| its bootstrap estimation (B=30).
||C2F − ΣF||
Σb=1B ||C2F
b − C2F||/B
0 2 4 6 8 10 122
3
4
5
6
7
Index of the 1−sample of measures
Em
piric
al m
ean
and
varia
nce
of K
L an
d its
boo
tstra
p es
timat
ion
Empirical mean and variance of KL and its bootstrap estimation (B=30).
KLKL boot
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 23 / 29
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Quality measure of the obtained estimation
5525Mn: (n,2n) cross section [MTN11]
12 13 14 15 16 17 18 19 200.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Energy (eV)
Cro
ss s
ectio
n (m
b)
SVM modelExperimental cross sectionsPseudo−measurements
1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
8
9
10
11 −0.4
−0.2
0
0.2
0.4
0.6
0.8
IE(‖Σ̂F − ΣF‖) ≈ 0.019 IE(KL(Σ̂F)) ≈ 6.834Var(‖Σ̂F − ΣF‖) ≈ 1.2 10−5 Var(KL(Σ̂F)) ≈ 2.351
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 24 / 29
-
Quality measure of the obtained estimation
5525Mn: total cross section
0 5 10 15x 106
2.5
3
3.5
4
Energy (eV)
Cro
ss s
ectio
n (m
b)
SVM modelExperimental dataPseudo−measurements
50 100 150 200 250 300 350
50
100
150
200
250
300
350
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
IE(‖Σ̂F − ΣF‖) ≈ 2.385 IE(KL(Σ̂F)) ≈ 386.36Var(‖Σ̂F − ΣF‖) ≈ 4.3 10−5 Var(KL(Σ̂F)) ≈ 6.19
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 25 / 29
-
Conclusion
Outline
1 Notations
2 Experimental covariances estimation: new method
3 Validation
4 Quality measure of the obtained estimation
5 Conclusion
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 26 / 29
-
Conclusion
Conclusion
ΣF estimation
1 Classical approaches
X one measurement per energyX experimental informations rarely availableX quite unrealistic assumptions
2 Our approach:
X only measurements are neededX quality measure of the estimation
Estimator quality
Numerical validation
Application
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 27 / 29
-
Conclusion
Conclusion
ΣF estimation
Estimator quality
Estimation of ‖bΣF − ΣF‖ with BootstrapEstimation of KL(bΣF) with Bootstrap
Numerical validation
Application
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 27 / 29
-
Conclusion
Conclusion
ΣF estimation
Estimator quality
Numerical validation
Toy model
Application
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 27 / 29
-
Conclusion
Conclusion
ΣF estimation
Estimator quality
Numerical validation
Application
Application to 5525Mn
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 27 / 29
-
Conclusion
Future work
New approach
Optimisation: shrinkage approach (in progress)
Other approaches
Kriging: cf poster session
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 28 / 29
-
Conclusion
References
Experimental Nuclear Reaction Data EXFOR, http://www.oecd-nea.org/dbdata/x4/.
M. Ionescu-Bujor and D. G. Cacuci, A comparative review of sensitivity and uncertaintyanalysis of large-scale systems −i: Deterministic methods, Nuclear Science andEngineering 147 (2004), 189–203.
Grégoire Kessedjian, Mesures de sections efficaces d’actinides mineurs d’intérêt pour latransmutation, Ph.D. thesis, Bordeaux 1, 2008.
E. Levina, A. Rothman, and J. Zhu, Sparse estimation of large covariance matrices via anested lasso penalty’, The Annals of Applied Statistics 2 (2008), no. 1, 245–263.
A. Milocco, A. Trkov, and R. Capote Noy, Nuclear data evaluation of 55 mn by the empirecode with emphasis on the capture cross-section, Nuclear Engineering and Design 241(2011), no. 4, 1071–1077.
A.J. Smola and B. Schölkopf, A tutorial on support vector regression, Statistics andComputing 14 (2004), 199–222.
S. Varet, P. Dossantos-Uzarralde, N. Vayatis, and E. Bauge, Experimental covariancescontribution to cross-section uncertainty determination, NCSC2 Vienna, 2011.
V. Vapnik, S. Golowich, and A. Smola, Support vector method for function approximation,regression estimation, and signal processing, Advances in Neural Information ProcessingSystems 9 (1997), 281–287.
S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 29 / 29
NotationsExperimental covariances estimation: new methodValidationQuality measure of the obtained estimationConclusion
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