Comlete Varance Decomoston Methods Cédrc J. allaberry
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
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
Lmt on tradtonal methods: Conjont Inluence y +. y + +. uch relaton wll not be catured wth tradtonal samlng-based senstvty analyss 4
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 0 + + j, j + +,,...,,,, j> However, ths decomoston s NOT unque 5
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
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
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
obol varance decomoston / V d V ntegraton o the square o on the whole range o. 9
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
0 0 0 0 0 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
Proertes o obol Varance Decomoston / 0., 0 0 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
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
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
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
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 4 4 4 Hghly nonlnear and non-monotonc uncton Involvng comle nteracton between U and V sngulartes or V/4, V/4 and V/4 6
Eamle Tradtonal enstvty Results CC RCC PCC PRCC RC RRC U 0.05 0.0 0.6 0.7 0.05 0.0 V 0.467 0.400 0.494 0.47 0.47 0.40 R equal to ~ 0.9 9% o the varance elaned 7
Eamle obol varance decomoston Parameter j Tj U ~ j 8.50-4 0.686 V ~ j 0.95 0.979 Almost 98% o the varance s elaned 8
Eamle FAT Parameter j Tj U ~ j.8 0-0.700 V ~ j 0.5 0.97 9% to 98% o the varance s elaned 9
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
Reerences OBOL obol', I.M., enstvty Estmates or Nonlnear Mathematcal Models. Mathematcal Modelng & Comutatonal Eerment, 99. 4:. 407-44. FAT Cuker, R.I., H.B. Levne, and K.E. huler, Nonlnear enstvty Analyss o Multarameter Model ystems. Journal o Comutatonal Physcs, 978. 6:. -4 altell, A.,. Tarantola, and K.P.-. Chan, A Quanttatve Model-Indeendent Method or Global enstvty Analyss o Model Outut. Technometrcs, 999. 4:. 9-56. 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-85000. Revew rovded at NL by Rob Rechard and Kathryn Knowles.