Pseudo-measurement simulations and bootstrap for the experimental - - PowerPoint PPT Presentation

▶
pseudo measurement simulations and bootstrap for the
SMART_READER_LITE
LIVE PREVIEW

Pseudo-measurement simulations and bootstrap for the experimental - - PowerPoint PPT Presentation

Mar 25, 2024 32 likes •432 views

Pseudo-measurement simulations and bootstrap for the experimental cross-section covariances estimation with quality quantification S. Varet 1 . Dossantos-Uzarralde 1 N. Vayatis 2 E. Bauge 1 P 1 CEA-DAM-DIF 2 ENS Cachan WONDER-2012 September

slide-1
SLIDE 1

Pseudo-measurement simulations and bootstrap for 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

slide-2
SLIDE 2

Motivations: experimental cross-sections

12 13 14 15 16 17 18 19 20 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Energy (MeV) n2n cross section

25 55Mn n2n cross section

EXFOR data

Available informations: the measurements their 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

slide-3
SLIDE 3

Motivations: experimental cross-sections covariances

HOW? Empirical estimator: only one measurement per energy

cov(Fi, Fj) ≈ 1 n

n

  • k=1

(F(k)

i

− ¯ Fi)(F(k)

j

− ¯ Fj)

Conventionnal approach: propagation error formula [IBC04],[Kes08]

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 3 / 29

slide-4
SLIDE 4

Motivations: experimental cross-sections covariances

HOW? Empirical estimator: only one measurement per energy Conventionnal 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

slide-5
SLIDE 5

Motivations: experimental cross-sections covariances

HOW? Empirical estimator: only one measurement per energy Conventionnal approach: propagation error formula [IBC04],[Kes08] QUALITY OF THE COVARIANCES ESTIMATION?

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 3 / 29

slide-6
SLIDE 6

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

slide-7
SLIDE 7

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

slide-8
SLIDE 8

Notations

Notations

Schematic representation of the experimental cross sections

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 6 / 29

slide-9
SLIDE 9

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

slide-10
SLIDE 10

Experimental covariances estimation: new method

Pseudo-measurements method

Given the available information (the measurements and their uncertainty) The idea: Repeat the available information in order to artificially increase the number 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

  • ΣF: empirical estimator of ΣF

4

Diagonal terms are imposed

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 8 / 29

slide-11
SLIDE 11

Experimental covariances estimation: new method

Pseudo-measurements method

Given the available information (the measurements and their uncertainty) The idea: Repeat the available information in order to artificially increase the number 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

  • ΣF: empirical estimator of ΣF

4

Diagonal terms are imposed

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 8 / 29

slide-12
SLIDE 12

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 optimisation problem: flat function: minimize 1 2w2 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 dimensionnal space) 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

slide-13
SLIDE 13

Experimental covariances estimation: new method

Pseudo-measurements method

Given the available information (the measurements and their uncertainty) The idea: Repeat the available information in order to artificially increase the number 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

  • ΣF: empirical estimator of ΣF

4

Diagonal terms are imposed

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 10 / 29

slide-14
SLIDE 14

Experimental covariances estimation: new method

Pseudo-measurements method

  • ΣF term at the ith line and jth column, for i and j ∈ {1, ..., N}:

Cij = σiσj n + r

n

  • k=1

(F(k)

i

− h(Ei))(F(k)

j

− h(Ej)) Vi

  • Vj

+ σiσj n + r

r

  • k=1

(S(k)

i

− h(Ei))(S(k)

j

− h(Ej)) Vi

  • Vj

and Cii = σ2

i

where n = 1.

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 11 / 29

slide-15
SLIDE 15

Experimental covariances estimation: new method

Number r of pseudo-measurements?

Constraint on r r too small:

  • ΣF non invertible

r too large:

  • ΣF diagonal matrix

r = 0

  • ΣF invertible
  • ΣF non invertible

no pseudo-measurements r such as ΣF invertible

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 12 / 29

slide-16
SLIDE 16

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

slide-17
SLIDE 17

Validation

Toy model

Goal: Compare ΣF 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 = B B B B @ 1.25 −3.49 −0.67 −7.41 −2.29 1 C C C C A ΣF = B B B B @ 9.49 −1.20 2.39 −0.03 0.07 −1.20 3.64 −3.21 0.03 −0.09 2.39 −3.21 5.27 0.48 0.10 −0.03 0.03 0.48 6.79 0.01 0.07 −0.09 0.10 0.01 5.72 1 C C C C A

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 14 / 29

slide-18
SLIDE 18

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 5 10 Variable (ex : energy) Measurements (ex : Xs) SVM regression on the toy model mM

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 15 / 29

slide-19
SLIDE 19

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 5 10 Variable (ex : energy) Measurement (ex : Xs) SVM regression on the toy model mM Toy measurement

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 15 / 29

slide-20
SLIDE 20

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 5 10 Variable (ex : energy) Measurement (ex : Xs) SVM regression on the toy model SVM regression mM Toy measurement

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 15 / 29

slide-21
SLIDE 21

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 5 10 Variable (ex : energy) Measurements (ex : Xs) SVM regression on the toy model SVM regression mM Toy measurement r=1 pseudo−measurement

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 15 / 29

slide-22
SLIDE 22

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 2 4 6 8

Real matrix Estimation

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 15 / 29

slide-23
SLIDE 23

Validation

55 25Mn: (n,2n) cross section

Values extracted from [MTN11] N=11, r=1, ΣF is a 11x11 matrix

12 13 14 15 16 17 18 19 20 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Energy (eV) Cross section (mb) SVM model Experimental cross sections Pseudo−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 0.2 0.4 0.6 0.8

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 16 / 29

slide-24
SLIDE 24

Validation

55 25Mn: total cross section

Values extracted from EXFOR [EXF] N=399, r=1, ΣF is a 399x399 matrix

5 10 15 x 10

6

2.5 3 3.5 4 Energy (eV) Cross section (mb) SVM model Experimental data Pseudo−measurements

50 100 150 200 250 300 350 50 100 150 200 250 300 350 −0.01 −0.005 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

slide-25
SLIDE 25

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

slide-26
SLIDE 26

Quality measure of the obtained estimation

Estimator quality

Criterion choice: quality of ΣF and/or quality of ΣF

−1?

⇒ distance criterion

1

Distance between ΣF and ΣF: Frobenius norm I E( ΣF − ΣFfro) = I E     

N

  • i=1

N

  • j=1

| Cij − Cij|2  

1 2

  

2

Distance between ΣF

−1 and Σ−1 F : Kullback-Leibler distance [LRZ08]

KL( ΣF, ΣF) = trace(( ΣF)−1.ΣF) − log(det( ΣF

−1.ΣF)) − N

Criterion estimation → parametric bootstrap

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 19 / 29

slide-27
SLIDE 27

Quality measure of the obtained estimation

Bootstrap principle

F : θF, ΣF: unknown (F(1), ..., F(n)) : h, ΣF: sample measurements (F′(1), ..., F′(n))1 , ..., (F′(1), ..., F′(n))B resample measurements

  • ΣF

1

  • ΣF

B

Resampling allows to make statistics on ΣF:

  • I

E( ΣF), Var( ΣF), Bias( ΣF),... Resample strategy: classical: draw with replacement in the initial sample (F(1), ..., F(n)) parametric: simulations with a fixed law (here N(h, ΣF))

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 20 / 29

slide-28
SLIDE 28

Quality measure of the obtained estimation

Estimation of IE( ΣF − ΣFfro) and KL( ΣF, ΣF)

1st case: with r = 0, ΣF invertible Usual parametric bootstrap: (F′(1), ..., F′(n))1, ..., (F′(1), ..., F′(n))B where F′(i) ֒ → N(H, ΣF) with H = (h(E1), ..., h(EN))t.

c c ΣF

b

term at the ith line and jth column, for i and j ∈ {1, ..., N} and for b = 1, ..., B: Cb

ij =σiσj

n

n

X

k=1

(F′(k)b

i

− h(Ei))(F′(k)b

j

− h(Ej)) q b V′

i

q b V′

j

and Cb

ii = σ2 i

I E( ΣF − ΣF) ≈ 1 B

B

  • b=1
  • ΣF

b

− ΣF KL( ΣF, ΣF) ≈ 1 B

B

  • b=1

KL(

  • ΣF

b

, ΣF)

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 21 / 29

slide-29
SLIDE 29

Quality measure of the obtained estimation

Estimation of IE( ΣF − ΣFfro) and KL( ΣF, ΣF)

2nd case: with r = 0, ΣF non invertible 2 simultaneous parametric bootstraps:

(F′(1), ..., F′(n))1, ..., (F′(1), ..., F′(n))B where F′(i) ֒ → N(H, c ΣF) et (S′(1), ..., S′(r))1, ..., (S′(1), ..., S′(r))B where S′(i) ֒ → N(H, diag(c ΣF))

c c ΣF

b

term at the ith line and jth column, for i and j ∈ {1, ..., N} and for b = 1, ..., B: Cb

ij = σiσj

n + r

n

X

k=1

(F′(k)b

i

− h(Ei))(F′(k)b

j

− h(Ej)) q b V′

i

q b V′

j

+ σiσj n + r

r

X

k=1

(S′(k)b

i

− h(Ei))(S′(k)b

j

− h(Ej)) q b V′

i

q b V′

j

and Cb

ii = σ2 i

I E( ΣF − ΣF) ≈ 1 B

B

  • b=1
  • ΣF

b

− ΣF KL( ΣF, ΣF) ≈ 1 B

B

  • b=1

KL(

  • ΣF

b

, ΣF)

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 22 / 29

slide-30
SLIDE 30

Quality measure of the obtained estimation

Bootstrap on the ’Toy Model’

N = 5, n = 1, B = 30, ΣF = 15.6680, r = 1,

2 4 6 8 10 12 10 12 14 16 18 20 Index of the 1−sample of measures Empirical mean and variance of ||C2F − ΣF|| its bootstrap estimation Empirical mean and variance of ||C2F − ΣF|| its bootstrap estimation (B=30). ||C2F − ΣF|| Σb=1

B

||C2F

b − C2F||/B

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 23 / 29

slide-31
SLIDE 31

Quality measure of the obtained estimation

Bootstrap on the ’Toy Model’

N = 5, n = 1, B = 30, ΣF = 15.6680, r = 1,

2 4 6 8 10 12 10 12 14 16 18 20 Index of the 1−sample of measures Empirical mean and variance of ||C2F − ΣF|| its bootstrap estimation Empirical mean and variance of ||C2F − ΣF|| its bootstrap estimation (B=30). ||C2F − ΣF|| Σb=1

B

||C2F

b − C2F||/B

2 4 6 8 10 12 2 3 4 5 6 7 Index of the 1−sample of measures Empirical mean and variance

  • f KL and its bootstrap estimation

Empirical mean and variance of KL and its bootstrap estimation (B=30). KL KL boot

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 23 / 29

slide-32
SLIDE 32

Quality measure of the obtained estimation

55 25Mn: (n,2n) cross section [MTN11]

12 13 14 15 16 17 18 19 20 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Energy (eV) Cross section (mb) SVM model Experimental cross sections Pseudo−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 0.2 0.4 0.6 0.8

I E( ΣF − ΣF) ≈ 0.019 I E(KL( ΣF)) ≈ 6.834 Var( ΣF − ΣF) ≈ 1.2 10−5 Var(KL( ΣF)) ≈ 2.351

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 24 / 29

slide-33
SLIDE 33

Quality measure of the obtained estimation

55 25Mn: total cross section

5 10 15 x 10

6

2.5 3 3.5 4 Energy (eV) Cross section (mb) SVM model Experimental data Pseudo−measurements

50 100 150 200 250 300 350 50 100 150 200 250 300 350 −0.01 −0.005 0.005 0.01 0.015 0.02 0.025

I E( ΣF − ΣF) ≈ 2.385 I E(KL( ΣF)) ≈ 386.36 Var( ΣF − ΣF) ≈ 4.3 10−5 Var(KL( ΣF)) ≈ 6.19

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 25 / 29

slide-34
SLIDE 34

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

slide-35
SLIDE 35

Conclusion

Conclusion

ΣF estimation

1

Classical approaches

  • ne measurement per energy

experimental informations rarely available quite unrealistic assumptions

2

Our approach:

  • nly measurements are needed

quality measure of the estimation Estimator quality Numerical validation Application

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 27 / 29

slide-36
SLIDE 36

Conclusion

Conclusion

ΣF estimation Estimator quality Estimation of b ΣF − ΣF with Bootstrap Estimation of KL(b ΣF) with Bootstrap Numerical validation Application

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 27 / 29

slide-37
SLIDE 37

Conclusion

Conclusion

ΣF estimation Estimator quality Numerical validation Toy model Application

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 27 / 29

slide-38
SLIDE 38

Conclusion

Conclusion

ΣF estimation Estimator quality Numerical validation Application Application to 55

25Mn

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 27 / 29

slide-39
SLIDE 39

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

slide-40
SLIDE 40

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 uncertainty

analysis of large-scale systems −i: Deterministic methods, Nuclear Science and Engineering 147 (2004), 189–203. Grégoire Kessedjian, Mesures de sections efficaces d’actinides mineurs d’intérêt pour la transmutation, Ph.D. thesis, Bordeaux 1, 2008.

  • E. Levina, A. Rothman, and J. Zhu, Sparse estimation of large covariance matrices via a

nested 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 empire

code 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 and Computing 14 (2004), 199–222.

  • S. Varet, P

. Dossantos-Uzarralde, N. Vayatis, and E. Bauge, Experimental covariances contribution 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 Processing Systems 9 (1997), 281–287.

  • S. Varet ( CEA-DAM-DIF, ENS Cachan )

Experimental covariances WONDER-2012 29 / 29