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

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 25-28 S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 1 / 29

Motivations: experimental cross-sections 25 55 Mn n2n cross section 1.1 Available informations: 1 0.9 the measurements n2n cross section 0.8 their uncertainty 0.7 0.6 EXFOR data 0.5 0.4 12 13 14 15 16 17 18 19 20 Energy (MeV) → covariance matrix estimation and its inverse Example Evaluated cross sections uncertainty: generalized χ 2 ([VDUVB11]) S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 2 / 29

Motivations: experimental cross-sections covariances HOW? Empirical estimator: only one measurement per energy � n cov ( F i , F j ) ≈ 1 ( F ( k ) − ¯ F i )( F ( k ) − ¯ F j ) i j n k = 1 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 energy Conventionnal approach: propagation error formula [IBC04],[Kes08] � ∂ F i .∂ F j cov ( F i , F j ) ≈ . cov ( q k , q l ) ∂ q k ∂ q l param. exp. ( q k , q l ) 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 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

Outline Notations 1 Experimental covariances estimation: new method 2 Validation 3 Quality measure of the obtained estimation 4 Conclusion 5 S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 4 / 29

Notations Outline Notations 1 Experimental covariances estimation: new method 2 Validation 3 Quality measure of the obtained estimation 4 Conclusion 5 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

Experimental covariances estimation: new method Outline Notations 1 Experimental covariances estimation: new method 2 Validation 3 Quality measure of the obtained estimation 4 Conclusion 5 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 their uncertainty) The idea: Repeat the available information in order to artificially increase the number of measurements The method: Construction of a regression model h (SVM, polynomial,...) 1 Generation of r pseudo-measurements ( S ( 1 ) , ..., S ( r ) ) : gaussian 2 noise centered on h : N (( h ( E 1 ) , ..., h ( E N )) t , diag ( σ 1 , ..., σ N )) � Σ F : empirical estimator of Σ F 3 Diagonal terms are imposed 4 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 their uncertainty) The idea: Repeat the available information in order to artificially increase the number of measurements The method: Construction of a regression model h (SVM, polynomial,...) 1 Generation of r pseudo-measurements ( S ( 1 ) , ..., S ( r ) ) : gaussian 2 noise centered on h : N (( h ( E 1 ) , ..., h ( E N )) t , diag ( σ 1 , ..., σ N )) � Σ F : empirical estimator of Σ F 3 Diagonal terms are imposed 4 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 F i = h ( E i ) = w . E i + b with E i ∈ R s and w ∈ R s (here s = 1 ). The svm regression model is a solution of the constraint optimisation problem: 1 2 � w � 2 flat function: minimize � F i − ( w . E i + b ) ≤ ε errors lower than ε : subject to ( w . E i + b ) − F i ≤ 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

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: Construction of a regression model h (SVM, polynomial,...) 1 Generation of r pseudo-measurements ( S ( 1 ) , ..., S ( r ) ) : gaussian 2 noise centered on h : N (( h ( E 1 ) , ..., h ( E N )) t , diag ( σ 1 , ..., σ N )) � Σ F : empirical estimator of Σ F 3 Diagonal terms are imposed 4 S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 10 / 29

Experimental covariances estimation: new method Pseudo-measurements method Σ F term at the i th line and j th column, for i and j ∈ { 1 , ..., N } : � ( F ( k ) − h ( E i ))( F ( k ) n � − h ( E j )) C ij = σ i σ j i j � � � n + r � V i V j k = 1 ( S ( k ) − h ( E i ))( S ( k ) � r − h ( E j )) + σ i σ j i j � � � n + r � V i V j k = 1 and C ii = σ 2 i where n = 1 . S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 11 / 29

Experimental covariances estimation: new method Number r of pseudo-measurements? Constraint on r 0 r too large: r too small: � � Σ F diagonal matrix Σ F non invertible r = 0 � � Σ F invertible Σ F non invertible r such as � no pseudo-measurements Σ F invertible S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 12 / 29

Validation Outline Notations 1 Experimental covariances estimation: new method 2 Validation 3 Quality measure of the obtained estimation 4 Conclusion 5 S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 13 / 29

Validation Toy model Goal: Compare � Σ F with Σ F : real case: Σ F is unknown ⇒ we build a toy model with Σ F fixed ⇒ numerical validation   Toy Model: M 1   . M := .  ( ≡ F ) ֒ → N N ( m M , Σ M )  . M N number of M realisations n = 1 , N = 5 0 1 0 1 1 . 25 9 . 49 − 1 . 20 2 . 39 − 0 . 03 0 . 07 B C B C − 3 . 49 − 1 . 20 3 . 64 − 3 . 21 0 . 03 − 0 . 09 B C B C B C B C θ = − 0 . 67 Σ F = 2 . 39 − 3 . 21 5 . 27 0 . 48 0 . 10 B C B C @ A @ A − 7 . 41 − 0 . 03 0 . 03 0 . 48 6 . 79 0 . 01 − 2 . 29 0 . 07 − 0 . 09 0 . 10 0 . 01 5 . 72 S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 14 / 29

Validation Pseudo-measurements with the ’Toy Model’ SVM regression on the toy model 10 5 Measurements (ex : Xs) 0 −5 −10 m M −15 −20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Variable (ex : energy) S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29

Validation Pseudo-measurements with the ’Toy Model’ SVM regression on the toy model 10 5 Measurement (ex : Xs) 0 −5 −10 m M Toy measurement −15 −20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Variable (ex : energy) S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29

Validation Pseudo-measurements with the ’Toy Model’ SVM regression on the toy model 10 5 Measurement (ex : Xs) 0 −5 SVM regression −10 m M −15 Toy measurement −20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Variable (ex : energy) S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29

Validation Pseudo-measurements with the ’Toy Model’ SVM regression on the toy model 10 5 Measurements (ex : Xs) 0 −5 SVM regression −10 m M Toy measurement −15 r=1 pseudo−measurement −20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Variable (ex : energy) 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 1 8 6 2 2 4 3 3 2 4 4 0 −2 5 5 1 2 3 4 5 1 2 3 4 5 Real matrix Estimation S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 15 / 29

Validation 55 25 Mn : (n,2n) cross section Values extracted from [MTN11] N=11, r=1, � Σ F is a 11x11 matrix 1.1 SVM model Experimental cross sections 1 1 Pseudo−measurements 0.8 2 0.9 0.6 3 Cross section (mb) 4 0.4 0.8 5 6 0.2 0.7 7 0 8 9 0.6 −0.2 10 −0.4 11 0.5 1 2 3 4 5 6 7 8 9 10 11 0.4 12 13 14 15 16 17 18 19 20 Energy (eV) S. Varet ( CEA-DAM-DIF, ENS Cachan ) Experimental covariances WONDER-2012 16 / 29