Complete Variance Decomposition Methods. Cédric J. Sallaberry

Transcription

1 Comlete Varance Decomoston Methods Cédrc J. allaberry

2 enstvty Analyss y y [,,, ] [ y, y,, ] y ny s a vector o uncertan nuts s a vector o results s a comle uncton successon o derent codes, systems o de, ode Queston: What art o the uncertanty n y can be elaned by the uncertanty n each element o? Tradtonal amlng-based enstvty Method Cature lnear relatonsh between one nut and one outut CC, PCC, RC Cature monotonc relatonsh between one nut and one outut RCC, PRCC, RRC

3 Lmt on tradtonal methods: Non-monotonc nluence y 0.5 y 0.75 < else 0.5 Tradtonal methods wll al to cature ths knd o relatonsh between and y

4 Lmt on tradtonal methods: Conjont Inluence y +. y + +. uch relaton wll not be catured wth tradtonal samlng-based senstvty analyss 4

5 Hgh dmensonal Model reresentaton / We would lke to nd a method that : cature any knd o relatonsh between nut and outut cature conjont nluence Man Idea: Decomose the uncton nto unctons deendng on any ossble combnatons o nuts y j, j + +,,...,,,, j> However, ths decomoston s NOT unque 5

6 Hgh dmensonal Model reresentaton / I all the arameters are orthogonal and then the decomoston s unque 0 E [ y] y E j> [ y] + j, j + +,,...,,,, nce all terms are orthogonal, the cross roducts are all equal to zero y,,,..., j> V + V j + V V + Decomoston o the varance o y wth V V,,..., d,,...,,..., d...d One mortant consequence s that we have to consder ndeendence between nut arameters 6

7 Hgh dmensonal Model reresentaton / Dvdng by V y wth j y,,,..., j> V + V j + V V + + +, j +,,..., j> contrbuton o to the varance o contrbuton o the nteracton o to the varance o y,,..., enstvty ndces y and contrbuton o the nteracton o all arameters to the varance o y j 7

8 obol Varance Decomoston /4 We calculate the average o the uncton or a gven value o E y E y Derence between the mean I we know and the mean I we don t know t. j, j y Monte Carlo Aroach Two samles o sze n are created ame set o value or Derent set o values or all other j, j ame oeraton done or each,,, 8

9 obol varance decomoston / V d V ntegraton o the square o on the whole range o. 9

10 obol varance decomoston / Hgher Order By ng the value o and j j the conjont nluence o and j can be calculated.,j, reresentng the nluence o the sole nteracton o and j, s dened by ntegratng, E y, E y E y E, j j j j + y calculated n the revous ste Hgher order, u to V,,, can be dened the same way Total Order By ng the value o all varables but one can calculate the nluence o all nuts wth ther nteractons, ecet wth -. The derence T - reresents the nluence o solely and all ts nteracton wth the others nuts. Ths nde s called total senstvty nde o. 0

11 d d d d d d d E Proertes o obol Varance Decomoston / Eamles n dmenson : Eected value o equal to zero Indeed, we have and 0 s constant relatvely to nce the uncton s ntegrated on, we can relace wth d Ŝ

12 Proertes o obol Varance Decomoston / 0., d d d dv 0, d d d dv Eamles n dmenson : Two roos o orthogonalty dv h g h g 0, Denton: Two unctons g and h dened on are sad to be orthogonal ther nner or dot roduct s equal to 0 : 0 as shown n revous slde

13 Fourer Amltude enstvty Test / Basc Idea In the moments calculaton, convertng the outut rom a uncton o varables.e., the elements o to a uncton o one varable.e., s lead to convert the multdmensonal ntegral to a mono-dmensonal ntegral. Ω r d Ω π r, π π r,, d d...d [ G snω s, G snω s,, G snω s ] Each nut s assocated wth a unque requency ω The unctons G are used to rovde a better coverage o the doman

14 Fourer Amltude enstvty Test / mall requences Good aromaton o search curve wth a small number o onts Large requences Good coverage o the doman But Bad coverage o the doman But Need large number o onts or aromatng the search uncton 4

15 Fourer Amltude enstvty Test / V y π Ak + Bk k π π r [ G snω s, G snω s,, G snω s ] Fourer eres Reresentaton where A B k k π π π π π π [ G snω s, G snω s,, G snω s ] [ G snω s, G snω s,, G snω s ] y k V A + B kω kω cos ksds sn ksds 5

16 Eamle Functon or llustraton U, V U + V + U + V + g V cos U wth [ π.5 ] g V mn ma + +, 0, 0 V V V Hghly nonlnear and non-monotonc uncton Involvng comle nteracton between U and V sngulartes or V/4, V/4 and V/4 6

17 Eamle Tradtonal enstvty Results CC RCC PCC PRCC RC RRC U V R equal to ~ 0.9 9% o the varance elaned 7

18 Eamle obol varance decomoston Parameter j Tj U ~ j V ~ j Almost 98% o the varance s elaned 8

19 Eamle FAT Parameter j Tj U ~ j V ~ j % to 98% o the varance s elaned 9

20 Concluson trong Ponts o Varance Decomoston Methods Cature nonlnear and nonmonotonc relatonsh between nut and outut Allows calculaton o conjont nluence o two or more nuts Weak Ponts o Varance Decomoston Methods Non neglgble cost n number o smulatons requred uose nut arameters are ndeendent to each other 0

21 Reerences OBOL obol', I.M., enstvty Estmates or Nonlnear Mathematcal Models. Mathematcal Modelng & Comutatonal Eerment, 99. 4: FAT Cuker, R.I., H.B. Levne, and K.E. huler, Nonlnear enstvty Analyss o Multarameter Model ystems. Journal o Comutatonal Physcs, :. -4 altell, A.,. Tarantola, and K.P.-. Chan, A Quanttatve Model-Indeendent Method or Global enstvty Analyss o Model Outut. Technometrcs, : see the related aer or many more Calculaton done wth the sotware mlab, avalable at htt://webarm.jrc.cec.eu.nt/uasa/ Ths work has been erormed at anda Natonal Laboratores NL, whch s a multrogram laboratory oerated by anda Cororaton, a Lockheed Martn comany, or the Unted tates Deartement o Energy s Natonal Nuclear ecurty Admnstraton under contract DE-AC04-94AL Revew rovded at NL by Rob Rechard and Kathryn Knowles.