Session 4 Part 1 Paper 2.doc

8/10/2019 Session 4 Part 1 Paper 2.doc

1/18

ADVANCES IN METHODS FOR UNCERTAINTY AND SENSITIVITY ANALYSIS

Nicolas DEVICTOR CEA/Cadarache

DEN/CAD/DER/SESI/LCFR

Buildi ! "#"#$#%& S'()aul(Le*(Dura ce Cede+

icolas,de-ic'or.cea, r

i co(o0era'io s 1i'h2 Nadia )EROT3 Michel MAR4UES a d Ber'ra d IOOSS 5CEA DEN/CAD/DER/SESI/LCFR637ulie 7AC4UES 5INRIA Rh8 e(Al0es3 )hD s'ude '63Chris'ia LAVER9NE 50ro essor a' Mo '0ellier " U i-ersi': a d INRIA Rh8 e(Al0es6,

AbstractA lo' o ;e'hods e+is' 'o s'ud: 'he i lue ce o u cer'ai 'ies o 'he resul's o se-ere accide 'sco;0u'er codes i use or Le-el " )SA, The i'e; a>ili': 'o e+ceed a 'hreshold,These ;e'hods are o 'e o' sui'a>le3 ro; a 'heore'ical 0oi ' o -ie13 1he 'he 0he o;e a 'ha' are;odelled >: 'he co;0u'er code are disco 'i uous i 'he -aria'io ra !e o i lue ' 0ara;e'ers3 or so;e i 0u' -aria>les are s'a'is'icall: de0e de ',A 'er a o-er-ie1 o s'a'is'ical a d 0ro>a>ilis'ic ;e'hods 'o s'ud: 'he i lue ce o u cer'ai 'ies o i 0u' -aria>les o 'he code res0o ses3 'he 0ur0ose o 'he 0a0er is 'o !i-e a descri0'io o so;e;a'he;a'ical ;e'hods 'ha' are i 'eres'i ! i 'he ra;e1or? o se-ere accide ' s'udies a d Le-el ")SA 2( res0o se sur aces3 a d s0eci icall: 'he sui'a>le ;e'hods or 'heir -alida'io , The use o a res0o sesur ace i 'roduces a addi'io al error o 'he resul's o 'he u cer'ai ': a d se si'i-i': a al:sis, Thees'i;a'io o 'ha' error is o' eas: 'o co;0u'e i ;os' cases, @e 1ill !i-e a o-er-ie1 o 'his

0ro>le;a'ic i case o 'he co;0u'a'io o 'he -aria ce o 'he res0o se,( clus'eri ! ;e'hods3 'ha' could >e use ul 1he 1e 1a ' a00l: s'a'is'ical ;e'hods >ased o Mo 'e(Carlo si;ula'io ,I 'he case o de0e de ' i 0u' -aria>les3 a e1 se si'i-i': i dice has >ee de-elo0ed 1i'h 'he ai; 'oo>'ai use ul a d co;0rehe si>le se si'i-i': i dices,)rac'ical i 'eres' o 'hese < e1= ;e'hods should >e co ir;ed3 >: a00lica'io s o real 0ro>le;s,

1 Introduction

A lo' o ;e'hods e+is' 'o s'ud: 'he i lue ce o u cer'ai 'ies o 'he resul's o se-ere accide 'co;0u'er codes i use or Le-el " )SA, The i'e; a>ili': 'ha' a res0o se e+ceeds a'hreshold,A lo' o 'hese ;e'hods could o' >e sui'a>le3 ro; a 'heore'ical 0oi ' o -ie13 1he 'he 0he o;e a'ha' are ;odelled >: 'he co;0u'er code are disco 'i uous i 'he -aria'io ra !e o i lue ' 0ara;e'ers3or so;e i 0u' -aria>les are s'a'is'icall: de0e de ', The 0ur0ose o 'he 0a0er is 'o !i-e a o-er-ie1 o so;e ;a'he;a'ical ;e'hods li?e o li ear res0o se sur aces a d clus'eri ! ;e'hods3 'ha' ca >euse ul 1he 1e 1a ' 'o a00l: s'a'is'ical ;e'hods >ased o Mo 'e(Carlo si;ula'io , For 'he 0ro>le;o clus'eri !3 a e+a;0le >ased o 'he direc' co 'ai ;e ' hea'i ! 0he o;e o is 0ro0osed,The use o a res0o se sur ace i 'roduces a addi'io al error o 'he resul's o 'he u cer'ai ': a dse si'i-i': a al:sis, The es'i;a'io o 'ha' error is o' eas: 'o co;0u'e i ;os' cases, @e !i-e ao-er-ie1 o 'his 0ro>le;a'ic i case o 'he -aria ce o 'he res0o se,I 'he case o correla'ed i 0u' -aria>les3 a e1 se si'i-i': i dice has >ee de-elo0ed 1i'h 'he ai; 'oo>'ai use ul a d co;0rehe si>le se si'i-i': i dices,

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mailto:[email protected]:[email protected]


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2 Overview of methods for uncertainty analysis

O ce 'he di ere ' sources o u cer'ai 'ies ha-e >ee ide 'i ied a d ;odelled3 'he 0ro>le; is ho1 'o 0ro0a!a'e 'hese u cer'ai 'ies 'hrou!h 'he code a d ho1 'o assess 'he 0ro>a>ili': 'ha' a res0o se o 'hecode e+ceeds a 'hreshold, This sec'io 0rese 's di ere ' ;e'hods used 1i'h 'heir ad-a 'a!es a d

dra1>ac?s, A lo' o re0or's e+is' o 'he su> ec'3 a d 1e ca see 'he >i>lio!ra0h: o # ,I 'he s'udies o u cer'ai ': 0ro0a!a'io 3 1e are i 'eres'ed 1i'h 'he u cer'ai ': e-alua'io o ares0o se Y accordi ! 'o u cer'ai 'ies o 'he i 0u' da'a X = (X i , i=1,..,p) a d o 'he u c'io al rela'io

f co ec'i ! Y 'o X , Y c is a 'hreshold or a cri'ical -alue i ecessar:,

2.1 Methods for propagating the uncertainties

I order 'o !e' i or;a'io a>ou' 'he u cer'ai ': o Y 3 a u;>er o code ru s ha-e 'o >e 0er or;ed,For each o 'hese calcula'io ru s3 all ide 'i ied u cer'ai 0ara;e'ers are -aried si;ul'a eousl:,Accordi ! 'o 'he e+0loi'a'io o 'he resul' o 'hese s'udies3 'he u cer'ai ': o 'he res0o se ca >ee-alua'ed ei'her i 'he or; o a u cer'ai ': ra !e or i 'he or; o a 0ro>a>ili': dis'ri>u'io u c'io5 pdf 6,

2.1.1 Uncertainty rangeA '1o(sided 'olera ce i 'er-al [m,M] o a res0o se Y 3 or a rac'ile a d a co ide ce le-el is !i-e >:2

( ){ } M Y m P P Such a rela'io ;ea s 'ha' o e ca a ir;3 1i'h a' 'he ;os' (1- ) 0erce ' o cha ces o error3 'ha' a'leas' 0erce 's -alues o 'he res0o se Y lies >e'1ee 'he -alues m a d M ,To calcula'e 'he li;i's m a d M 3 'he 'ech i ue usuall: used is a ;e'hod o si;ula'io co;>i ed 1i'h'he or;ula o @il?s, The or;ula o @il?s de'er;i es 'he ;i i;al si*e N o a sa;0le 'o >e !e era'edra do;l: accordi ! 'o 'he -alues o a d 5c , " a d $ or e+a;0les6For a '1o(sided s'a'is'ical 'olera ce i 'er-als3 'he or;ula is2

( ) 1 N N 1 N 1The ;i i;u; u;>er o calcula'io s ca >e ou d i Ta>le #,

O e(sided s'a'is'ical 'olera ce li;i' T1o(sided s'a'is'ical 'olera ce li;i'

/ %, % %, %, %, % %, %,

%, % "" G "$% $& $&&

%, " " G $ G $

%, GG % G G #$% "

Table 1 : Minimum number of calculations N for one-sided and two-sided statistical tolerancelimits

The >ou ds m a d M o 'he co ide ce i 'er-al o Y are o>'ai ed >: re'ai i ! 'he ;i i;al a d;a+i;al -alues o 'he sa;0le {Y j , j = 1,,N}.This ;e'hod su00oses 'ha' 'he u c'io g is co 'i uous a d 'ha' all u cer'ai 'ies o 'he i 0u' da'a X iare dis'ri>u'ed accordi ! 'o co 'i uous la1s,The ad-a 'a!e o usi ! 'his 'ech i ue is 'ha' 'he u;>er o code calcula'io eeded is i de0e de ' o 'he u;>er o u cer'ai 0ara;e'ers3 >u' 1e ca o>'ai lar!e co ide ce i 'er-als o 'he >ou ds,

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2.1.2 Density of probabilityThe u cer'ai ': e-alua'io i 'he or; o a pdf !i-es richer i or;a'io 'ha a co ide ce i 'er-al, Bu''he de'er;i a'io o 'his dis'ri>u'io ca >e e+0e si-e i co;0u'i ! 'i;es, The ollo1i ! 0ara!ra0hsdescri>e 'he -arious ;e'hods a-aila>le or 'his e-alua'io ,

2.1.2.1 Methods of Monte-CarloThe ;e'hod o Mo 'e(Carlo is used 'o >uild pdf 3 >u' also 'o assess 'he relia>ili': o co;0o e 's or s'ruc'ures or 'o e-alua'e 'he se si'i-i': o 0ara;e'ers, Mo 'e Carlo si;ula'io co sis's o dra1i !sa;0les o 'he >asic -aria>les accordi ! 'o 'heir 0ro>a>ilis'ic charac'eris'ics a d 'he eedi ! 'he;i 'o 'he 0er or;a ce u c'io , I 'his 1a:3 a sa;0le o res0o se {Y j , j = 1,..,N} is o>'ai ed,The pdf is o>'ai ed >: i''i ! a la1 o 'he sa;0le {Y j , j = 1,..,N} , This i''i ! is a 1ell ? o1 0ro>le;a d ;a : 'es's e+is' a d are ada0'ed 'o 'he la1 'es'ed 5Chi(s uare3 Jol;o!oro-(S;ir o-3 A derso (Darli !,,, 6, De!rees o co ide ce ca >e associa'ed 'o 'he i''i !,I' is o>-ious 'ha' 'he uali': o 'he i''i ! de0e ds o 'he u;>er o si;ula'io s carried ou' a d o 'he!ood re0ar'i'io o 'hese si;ula'io s i 'he ra do; s0ace3 es0eciall: i 'he 'ails o dis'ri>u'io s are o eo 'he i 'eres's o 'he s'ud:, I' is ecessar: 'o o'ice 'ha' o rule e+is's3 1he 'here is o a 0riori? o1led!e o 'he ':0e o pdf 3 'o de'er;i e 'he u;>er o si;ula'io s ecessar: 'o o>'ai 'hisdis'ri>u'io 1i'h co ide ce,I' is ecessar: 'o selec' 'he 0oi 's3 1hich >ri ! 'he ;a+i;al i or;a'io 3 >u' 'he de'er;i a'io o 'hese 0oi 's re;ai s a o0e ues'io ,The 0ri ci0al ad-a 'a!e o 'he ;e'hod o Mo 'e(Carlo3 is 'ha' 'his ;e'hod is -alid or s'a'ic3 >u' also

or d: a;ic ;odels a d or 0ro>a>ilis'ic ;odel 1i'h co 'i uous or discre'e -aria>les, The ;aidra1>ac? o 'his ;e'hod is 'ha' i' re uires a lar!e u;>er o calcula'io s a d ca >e 0rohi>i'i-e 1heeach calcula'io i -ol-es a lo ! a d o erous co;0u'er 'i;e,

2.1.2.2 Method of moments

A o'her ;e'hod 'o o>'ai 'he de si': o 0ro>a>ili': o 'he res0o se is 'o calcula'e 'he irs' our ;o;e 's >: usi ! 'he 9auss i 'e!ra'io ;e'hod a d 'he 'o i' a dis'ri>u'io o 0ro>a>ili': 'o 'hese

;o;e 's >: usi ! 'he )earso or 7oh so ;e'hods 5c , G a d 6,The irs' o> ec'i-e is 'o e-alua'e 'he irs' ;o;e 's o 'he ra do; res0o se Y , The e+0ec'a'io o Y ca >e calcula'ed >:32

( ) ( )( ) ( ) ( ) == p1 X dxdx x f x g X g E Y E 1here f x is 'he oi ' de si': dis'ri>u'io o K,This e ua'io ca >e e-alua'ed >: a s uari ! ;e'hod o 9auss, This ;e'hod allo1s 'he i 'e!ra'io o a co 'i uous u c'io 1i'h 'he desired 0recisio , I' co sis's i 'he discre'isa'io o 'he i 'er-al o i 'e!ra'io i a u;>er o X (coordi a'es xi 'o 1hich a 1ei!h' wi is associa'ed, The u;>er o X (coordi a'es is a u c'io o 'he desired 0recisio , For a co 'i uous u c'io !5+63 1e o>'ai 2

( ) ( ) =

N

1i

ii

b

a

x g dx x g ) x(

)rac'icall:3 a se' o order or'ho!o al 0ol: o;ial 0 5+6 %3#3",,, are associa'ed 'o 'he 1ei!h' u c'io@5+6, These 0ol: o;ials -eri : 'he ollo1i ! rela'io s2

( )

( ) 1 x p ) x(

ji !i"dx ) x( p x p ) x(

b

a

#i

b

a ji

=

=

The N K(coordi a'es o a s uari ! or;ula 1i'h a 1ei!h' u c'io (x) are 'he *eros o 'he

0ol: o;ial p N (x)3 1hich has e+ac'l: N *eros i 'he i 'er-al a3 > , These 0ol: o;ials are !e erall:de i ed >: rela'io s o recurre ce, The 1ei!h's are calcula'ed >: sol-i ! 'he s:s'e; o li ear e ua'io s2

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( ) ( ) N 1i x p ) x( x pb

ai

N

1 j ji

== =

The 'he a-era!e is e-alua'ed ro;2( ) ( )( ) ( )

==

N

1iii $ g X g E Y E

a d 'he ;o;e ' o order % ro;2

( ) ( )[ ] ( ) ( )[ ] =

= N

1i

% Y X ,i X ,i X

% Y Y $ g dx x f x g % M

Fro; 'he irs' our ;o;e 's ? o1led!e3 i' is 0ossi>le 'o de'er;i e 'he associa'ed dis'ri>u'io o )earso ,)earso a d al, 5 $ 6 sho1 'ha' o e ca de i e i a a00ro+i;a'e 1a: a de si': o 0ro>a>ili': ro; 'hea-era!e3 'he s'a dard de-ia'io a d '1o addi'io al coe icie 's called coe icie 's o Fisher2

The coe icie ' o s:;;e'r:2&&

11 ==

The coe icie ' o la' ess2 ''

##

==

@here 1 is 'he S?1e ess a d # 'heJur'osis,A !rea' u;>er o co 'i uous dis'ri>u'io s ca >e 1ri''e i 'he ollo1i ! or;2

#1 p#

p1" ) xa( )a x )( x( f ) x( f =

1here 'he 0ara;e'ers a 13 a #3 p1 a d p# ca >e real or i;a!i ar: a d f(x" ) is de i ed >: 'he co di'io

= 1dx ) x( f ,

The dis'ri>u'io la13 1hich de0e ds o G 0ara;e'ers3 ca >e e+0ressed accordi ! 'o m3 "# 33 , These' o 'hese dis'ri>u'io la1s is called 'he a;il: o )earso , The cur-es ca ha-e se-eral sha0es 5>ell(sha0ed cur-e3 cur-ed i 73 cur-ed i U6,This ;e'hod is e icie ' 'o es'i;a'e a pdf i 'he u;>er o ra do; -aria>les is s;all,

2.2 Sensitivity analysis

Se si'i-i': ;easures o 'he i;0or'a ce o i 0u's u cer'ai 'ies o 'he u cer'ai ': o 'he res0o se is ai;0or'a ' i or;a'io 'ha' 0ro-ides !uida ce as 'o 1here 'o i;0ro-e 'he s'a'e o ? o1led!e i order 'oreduce 'he ou'0u' u cer'ai 'ies ;os' e ec'i-el:3 or >e''er u ders'a d 'he ;odelli !, I e+0eri;e 'alresul's are a-aila>le 'o co;0are 1i'h calcula'io s3 se si'i-i': ;easures 0ro-ide !uida ce 1here 'o

i;0ro-e 'he ;odels o 'he co;0u'er code, A s'a'e o 'he ar' has >ee carried ou' a d is 0rese 'ed i ,@e ca dis'i !uish '1o ?i ds o se si'i-i': a al:sis2( local se si'i-i': a al:sis >ased o di ere 'ial a al:sis a d o 0ro>a>ilis'ic 'ool( !lo>al se si'i-i': a al:sis 1i'h 'he ai; o ra ?i ! 'he 0ara;e'ers accordi ! 'o 'heir co 'ri>u'io

o 'he code res0o se -aria ce3 >ased o 'he -aria ce o co di'io al e+0ec'a'io ,

2.2.1 Sensitivity indices fTo a00or'io 'he -aria'io i 'he ou'0u' 'o 'he di ere ' 0ara;e'ers3 ;a : 'ech i ues could >e used5see or ;ore de'ails63 each :ieldi ! di ere ' ;easures o se si'i-i':,A usual a00roach is 'o >ase 'he se si'i-i': a al:sis o a li ear re!ressio ;e'hod3 1hich is >ased o'he h:0o'hesis o a li ear rela'io >e'1ee res0o se a d i 0u' 0ara;e'ers, This3 i case o se-ereaccide ' is o 'e res'ric'i-e, Ho1e-er3 'he ;e'hod is si;0le a d uic?3 a d 0ro-ides use ul i si!h's i

case o a res'ric'ed u;>er o sa;0li !, Three di ere ' se si'i-i': coe icie 's ha-e >ee co sidered3each o e 0ro-idi ! a sli!h'l: di ere ' i or;a'io o 'he rele-a ce o a 0ara;e'er2 S'a dardi*edRe!ressio Coe icie 's 5SRC63 )ar'ial Correla'io Coe icie 's 5)CC6 a d Correla'io Coe icie 's

G


5/18


6/18

K#3 P3 K i are i de0e de ' i 0u's3 a d X i #3 P3 X i l are l !rou0s o i 'ra(de0e de ' or i 'ra(correla'edi 0u's,@e 1ro'e ;o odi;e sio al o i de0e de ' -aria>les 5K #3 P 3 K 06 li?e ;ul'idi;e sio ali de0e de ' -aria>les 5K #3 PKi3 X i #3 X i "3P3 X i l6,Thus 1e de i e irs' order se si'i-i': i dices2

( )Y * Y E *

+ j

j X /

= # i l

To co ec' 'his 'o ;o odi;e sio al -aria>les3 i # i 3 1e ha-e2

( ) ( )Y * X Y E *

Y *

Y E * +

j j j

//==

X

a d i i i l 3 or e+a;0le i i "2

{ } [ ]( )( )Y * X X Y E * + + % i% i% i% i j "##"3,,,3## 33/

++++++ ==

Li?e i classical a al:sis3 1e ca also de i e hi!her order i dices a d 'o'al se si'i-i': i dices,I' is i;0or'a ' 'o o'e 'ha' i all i 0u' -aria>les are i de0e de '3 'hose se si'i-i': i dices are clearl:'he sa;e 'ha 5#6, A d so3 ;ul'idi;e sio al se si'i-i': i dices ca 1ell >e i 'er0re'ed li?e a!e eralisa'io o usual se si'i-i': i dices 5#6,I 0rac'ice3 'he ;ul'idi;e sio al se si'i-i': i dices ca >e assess ro; Mo 'e(Carlo ;e'hods li?e iSo>olQ or McJa:Q ;e'hods 5c , 6, The assess;e ' is o 'e 'i;e co su;i !, So;e co;0u'a'io ali;0ro-e;e 's3 >ased o 'he idea o di;e sio reduc'io 3 are i 0ro!ress a d -er: 0ro;isi !,Co clusio s o 'his 0oi ' are e+0ec'ed a' 'he e d o 'he :ear,

2.3 Methods for assessing the probability

The 0ro>a>ili': ) 'o e+ceed a 'hreshold accordi ! 'o a s0eci ied 0er o;a ce cri'erio or ailurecri'erio is !i-e >:2

M = p f ma/c c i0 i / gi2 / c i0 i / 3imi0 = g(X 1 , X # ,,X / )The 0er or;a ce u c'io 3 also a;ed 'he li;i' s'a'e u c'io 3 is !i-e >: M = " , The ailure e-e ' isde i ed as 'he s0ace 1here M 4 " 3 a d 'he success e-e ' is de i ed as 'he s0ace 1here M 5 " , Thus a

0ro>a>ili': o ailure ca >e e-alua'ed >: 'he ollo1i ! i 'e!ral2

= /#1/#1 dx ,...,dx ,dx ) x ,..., x , x( X f f ... P 5"61here f X is 'he oi ' de si': u c'io o X 1 ,X # ,, X / 3 a d 'he i 'e!ra'io is 0er or;ed o-er 'he re!io

1here M 4 " , Because each o 'he >asic ra do; -aria>les has a u i ue dis'ri>u'io a d 'he: i 'erac'3'he i 'e!ral 5"6 ca o' >e easil: e-alua'ed, T1o ':0es o ;e'hods ca >e used 'o es'i;a'e 'he 0ro>a>ili': o ailure2 Mo 'e Carlo si;ula'io 1i'h or 1i'hou' -aria ce reduc'io 'ech i ues a d 'hea00ro+i;a'e ;e'hods 5FORM/SORM6, More de'ails are a-aila>le i & 3 a d #% ,

2.3.1 Monte Carlo SimulationDirec' Mo 'e Carlo si;ula'io 'ech i ues ca >e used 'o es'i;a'e 'he 0ro>a>ili': ) de i ed i E s,5"6, Mo 'e Carlo si;ula'io 5 Fi!ure # 6 co sis's o dra1i ! sa;0les o 'he >asic -aria>les accordi ! 'o'heir 0ro>a>ilis'ic charac'eris'ics a d 'he eedi ! 'he i 'o 'he 0er or;a ce u c'io , A es'i;a'e o 'he 0ro>a>ili': o ailure P f ca >e ou d >:2

N

N P f f =


7/18

1here N f is 'he u;>er o si;ula'io c:cles i 1hich g(.) 4 " 3 a d N 'he 'o'al u;>er o si;ula'io

c:cles, As N a00roaches i i i':3 f P a00roaches 'he 'rue 0ro>a>ili': o ailure, The accurac: o 'hees'i;a'io ca >e e-alua'ed i 'er;s o i's -aria ce co;0u'ed a00ro+i;a'el: as

( ) N

P P 1 ) P ( *a f f f

I' is reco;;e ded 'o ;easure 'he s'a'is'ical accurac: o 'he es'i;a'ed 0ro>a>ili': o ailure >:co;0u'i ! i's coe icie ' o -aria'io as

( )

f

f f

f P

N

P P 1

) P ( 67*

5$6

The s;aller 'he coe icie ' o -aria'io 3 'he >e''er 'he accurac: o 'he es'i;a'ed 0ro>a>ili': o ailure,

I' is e-ide ' ro; 5$6 'ha' as N a00roaches i i i':3 ) P ( *a f a d ( ) f P 67* a00roach *ero,

For a s;all 0ro>a>ili': o ailure a d a s;all u;>er o si;ula'io c:cles3 'he -aria ce o f P ca >eui'e lar!e, Co se ue 'l:3 i' ;a: 'a?e a lar!e u;>er o si;ula'io c:cles 'o achie-e a s0eci ic

accurac:,The a;ou ' o co;0u'er 'i;e eeded or 'he direc' Mo 'e Carlo ;e'hod is lar!e3 s0eciall: i our case1here each si;ula'io c:cle i -ol-es a lo ! calcula'io 0er or;ed >: a se-ere accide ' co;0u'er code,

Figure 1 : Reliability assessment by Monte-Carlo simulation

More e icie ' Mo 'e(Carlo ;e'hods ha-e >ee de-elo0ed a d de i e 'he a;il: o u' i' is o' e icie ' !e erall: or 'he assess;e ' o s;all 0ro>a>ili':,

2.3.2 Approximated methods (FORM/SORM)The irs'( a d seco d(order relia>ili': ;e'hods 5FORM/SORM6 co sis' o G s'e0s 5 Fi!ure " 62

Failure do;ai

Sa e do;ai


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'he 'ra s or;a'io o 'he s0ace o 'he >asic ra do; -aria>les X 1 , X # ,,X / i 'o a s0ace o s'a dard or;al -aria>les3

'he research3 i 'his 'ra s or;ed s0ace3 o 'he 0oi ' o ;i i;u; dis'a ce ro; 'he ori!i o'he li;i' s'a'e sur ace 5'his 0oi ' is called 'he desi! 0oi '63

a a00ro+i;a'io o 'he ailure sur ace ear 'he desi! 0oi '3

a co;0u'a'io o 'he ailure 0ro>a>ili': corres0o di ! 'o 'he a00ro+i;a'i ! ailure sur ace,FORM a d SORM a00l: 'o 0ro>le;s 1here 'he se' o >asic -aria>les are co 'i uous,

Figure 2 : Reliability assessment with F RM!" RM methods

Transformation of spaceThe choice o 'he 'ra s or;a'io 'o >e used de0e ds o 'he charac'eris'ics o 'he oi ' de si': o ra do; i 0u' -ec'or X , The ;os' curre ' 'ra s or;a'io s are 'he 'ra s or;a'io o Rose >la'' 1he'he oi ' de si': is ? o1 3 a d 'he 'ra s or;a'io o Na'a 3 1he 'he 0ro>a>ilis'ic ;odel is o l: ;adeu0 o 'he ;ar!i al de si'ies a d o 'he ;a'ri+ o co-aria ce,I 'he 9aussia s0ace3 'he sur ace o ailure is de i ed >: 8($) = g[9 -1($)].

Design point researchThe Haso er(Li d i dice HL is de i ed as 'he ;i i;al dis'a ce >e'1ee 'he ailure sur ace a d 'heori!i o 'he 9aussia S0ace, The calcula'io o HL co sis's i sol-i ! 'he ollo1i ! 0ro>le; o o0'i;isa'io u der co s'rai '2

$$mi/ 0

" )$( 8 =

The 0oi ' associa'ed 1i'h 'his ;i i;al dis'a ce is o 'e called 'he desi! 0oi ', I' corres0o ds 'o 'he 0oi ' o ;a+i;u; o 0ro>a>ili': o ailure a d is o'ed $ , I deed3 i 'he 9aussia S0ace3 'he oi 'de si': o 'he -ec'or : is s:;;e'rical i ro'a'io 1i'h res0ec' 'o 'he ori!i a d 'hus i -ol-es 'ha' 'hedesi! 0oi ' coi cides 1i'h 'he ;os' 0ro>a>le 0oi ' o 'he ailure,The 0ro>a>ili': o ailure ca >e calcula'ed >:2

( )( )

( )( )

( ) $d $ xd x f ": 8 P P ": 8 N

" x g x f

===

1here N is 'he ;ul'i( or;al s'a dard dis'ri>u'io o di;e sio N,

FORM method

&

%

Failure

do;ai

U#

U"

U

95U6 %

HLSa edo;ai


9/18

A00ro+i;a'e ;e'hod FORM 5Firs' Order Relia>ili': Me'hod6 co sis's i a00roachi ! 'he sur ace o ailure >: a h:0er 0la e 'a !e ' 'o 'he sur ace o ailure a' 'he desi! 0oi ', The a es'i;a'e o 'he

0ro>a>ili': o ailure is o>'ai ed >:2

P f = (- ;< )

1here is rela'ed 'o cu;ula'i-e dis'ri>u'io o 'he s'a dard or;al la1, The 0recisio o 'hisa00ro+i;a'io de0e ds o 'he o (li eari': o 'he ailure sur ace,The ? o1led!e o 'his desi! 0oi ' ;a?es i' 0ossi>le 'o de'er;i e 'he ;os' i lue 'ial -aria>les orelia>ili':, B: su00osi ! 'ha' 'here is a si !le desi! 0oi ' a d 'ha' 'he i de+ o relia>ili': de i ed >:Haso er a d Li d HL is 0osi'i-e3 'he -ec'or direc'io al cosi e u i' is de i ed >: =$ ;< ,The co;0o e 's i are also called ac'ors o se si'i-i': a d 'he ac'ors i are i 'er0re'ed li?e ac'orso i;0or'a ce associa'ed 'o 'he -aria>les U i, A -aria>le associa'ed 1i'h o e si! i ica ' i is re!ardedas ha-i ! a si! i ica ' i lue ce o 'he 0ro>a>ili': o ailure,

I 'he li ear a00ro+i;a'io is o' sa'is ac'or:3 ;ore 0recise e-alua'io s ca >e o>'ai ed 'o ro;a00ro+i;a'io s 'o hi!her orders o 'he ailure sur ace a' 'he desi! 0oi ', #% descri>es so;ei;0ro-e;e 's o seco d order called SORM Me'hods3 >ased o a a00ro+i;a'io o 'he ailuresur ace >: a uadra'ic sur ace a' 'he desi! 0oi ',A o'her idea 'o co 'ri>u'e 'o 'he -alida'io o a resul' FORM or 'o i;0ro-e 'he 0recisio o 'he resul'is 'o use a ;e'hod o si;ula'io i 'he -ici i': o 'he desi! 0oi ' $ . The ;e'hods o i;0or'a cesa;0li ! a d direc'io al si;ula'io are 'he ;os' used,For a i;0or'a ce sa;0li ! 5 Fi!ure $ 6 arou d 'he desi! 0oi '3 'he 0ro>a>ili': o ailure is es'i;a'ed

>:2( )

( ) ( ) ( )

( )( ) H i

i

i)

hh d

hF i=

=

uu

u u uu

uu

## %

#5 6

1here >($) is 'he de si': o i;0or'a ce de i ed >: a ;ul'i or;al dis'ri>u'io ce 'red o 'he 0oi ' o desi! ,

Figure # : Conditional im$ortance "am$ling

2.3.3 Comparison of Monte-Carlo methods and FORM/SORMThe Mo 'e Carlo si;ula'io ;e'hods are co;0le'el: !e eral3 a d a00l: 'o a : dis'ri>u'io o 'he >asicra do; -aria>les3 i cludi ! discre'e ra do; -aria>les, Fur'her;ore3 'here is o re uire;e 's o 'he

ailure u c'io s o l: 'he si! o 'he ailure u c'io is >ei ! used,

% U

"

#

U

)

9 5 u 6 %

Failuredo;ai


10/18

FORM a d SORM are a al:'ical a d a00ro+i;a'e ;e'hods3 a d 'heir accurac: is !e erall: !ood or s;all 0ro>a>ili'ies 5#% ($(#%(&6, The a al:'ical 0ro0er'ies e a>le 'he ;e'hods 'o :ield rela'i-el:i e+0e si-e se si'i-i': ac'ors, The >asic ra do; -aria>les ;us' >e co 'i uous3 a d 'he ailure

u c'io ;us' >e co 'i uous, @i'h 'he o0'i;isa'io 0rocedures 0rese 'l: used i ;os' cases3 'heailure u c'io s should >e s;oo'h,

For s;all order 0ro>a>ili'ies FORM/SORM are e+'re;el: e icie ' as co;0ared 'o si;ula'io;e'hods3 i 'he u;>er o ra do; -aria>les is o' 'oo hi!h, The C)U('i;e is or FORMa00ro+i;a'el: li ear i 'he u;>er o >asic -aria>les / 3 a d 'he addi'io al C)U('i;e or a SORMco;0u'a'io !ro1s a00ro+i;a'el: 1i'h / #, The a>solu'e co;0u'a'io 'i;e de0e ds o 'he 'i;e

ecessar: 'o e-alua'e 'he ailure u c'io , This 'i;e ;a: i ac' de0e d o 'he ac'ual -alues o 'he >asic -aria>les, E+'re;e -alues ;a: 'a?e lo !er due 'o i creased o (li eari'ies i 'he 0ro>le;, TheC)U('i;e is i de0e de ' o 'he 0ro>a>ili': le-el3 assu;i ! a co s'a ' 'i;e or e-alua'io o 'he

ailure u c'io ,The ollo1i ! 'a>le su;;ari*es 'he ad-a 'a!es a d dra1>ac?s o Mo 'e(Carlo si;ula'io a dFORM/SORM ;e'hods,

Table 2 : Com$arison of the characteristics of reliability methods

3 Response surface methods

3.1 Principle

To a-oid 'he 0ro>le; o lo ! co;0u'er 'i;e i 'he ;e'hod o Mo 'e(Carlo3 i' ca >e i 'eres'i ! 'o >uild a a00ro+i;a'e ;a'he;a'ical ;odel called res0o se sur ace or surro!a'e ;odel or ;e'a(;odel,The ai; o 'he ;e'hod is 'o a00ro+i;a'e f(X) >: a si;0le ;a'he;a'ical ;odel3 such as a 'h order

0ol: o;ial ) X ( g ? 1i'h u de'er;i ed coe icie 's3 ro; a da'a>ase o co;0u'a'io s 5see or e+a;0le## 6, Di ere ' ':0es o res0o se sur ace ca >e used2 0ol: o;ial3 'hi 0la'e s0li es3 eural e'1or?s3

!e eralised li ear ;odel 59LM63 )LS 5)ar'ial Leas' S uares6 re!ressio P For so;e a;il: o

res0o se sur aces3 e+0eri;e 'al desi! 'ools could >e use ul 'o o0'i;ise 'he si*e o 'he da'a>ase,I 0u' da'a i a res0o se sur ace ;e'hod 5RSM6 are2

#%

"imulations F RM!" RM

RESULTS

FA%&'R( )R *A*%&%T+

(RR R N T,( ("T%MAT% N

)R *A*%&%T+ -%"TR%*'T% N F T,(R(") N"(

ASSUMPTIONS

N A""'M)T% N N T,( RAN- M.AR%A*&(" /-%"CR(T(0 C NT%N '"0-()(N-ANC+12

N A""'M)T% N N T,( &%M%T "TAT(F'NCT% N

DRA !A"#S

C M)'TAT% N C "T"/de$ends on the $robability le3el2

RESULTS

FA%&'R( )R *A*%&%T+

M "T %NF&'(NT%A& .AR%A*&("/)R *A*%&%T+2

(FF%C%(NC+/de$ends on the number of random 3ariables2

ASSUMPTIONS

C NT%N '" RAN- M .AR%A*&("

C NT%N '" &%M%T "TAT( F'NCT% N

DRA !A"#S

N (RR R N T,( ("T%MAT% N

4& *A& M%N%M'M


11/18

a sa;0le D o 0oi 's 5 55i63 *5i663 1here5 is 'he ra do; -ec'or 5+ #3 P3 + 063 * f 556 a d )5K3 6 'he 0ro>a>ili': la1 o 'he ra do; -ec'or 5K3 6 5u ? o1 i 0rac'ice6 a a;il: F o res0o se sur ace u c'io 5+3c63 1here c is ei'her a 0ara;e'er -ec'or or a i de+ -ec'or 'ha' ide 'i ies 'he di ere ' ele;e 's o F,The ai; o a RSM is3 i co sidera'io o 'he sa;0le 3 'o de'er;i e 'he u c'io % i F 'ha' has 'he

>eha-iour 'he eares' o f , The >es' u c'io i 'he a;il: F is 'he 'he u c'io % 'ha' ;i i;i*ed 'heris? u c'io 2

I assu;0'io s o or;al dis'ri>u'io s a d co s'a ' s'a dard de-ia'io s are do e3 'he loss u c'io isde i ed3 >:

L(z, f( x , c )) = (x ,f)= [z( x ) - f(x , c )]

I 0rac'ice a e;0irical ris? u c'io is used2

Be ore a : use o res0o se sur ace3 i' is ecessar: 'o uali : i' or a oresee u'ilisa'io , Thisuali ica'io ?ee0s a 0ar' o su> ec'i-i':, The charac'eris'ic W !ood a00ro+i;a'io X is su> ec'i-e a d

de0e ds o 'he use o 'he res0o se sur ace, The use could i 'roduce addi'io al co s'rai 's, For e+a;0le3 1e ca eed co s'rai 's li?e co ser-a'is;3 a >ou d o 'he re;ai der3 a >e''er accurac: i ai 'eres' area 5dis'ri>u'io 'ailP6P For relia>ili': s'udies3 a !ood re0rese 'a'io o 'he do;ai o ;a+i;u; o ailure 0ro>a>ili': is o 'e su icie ' a d i' is o' ecessar: 'o see? a !ood uali': o a00ro+i;a'io i 'he e 'ire ield o -aria'io o 'he i 0u' 0ara;e'ers, I 'he res0o se sur ace is used ia 0ro>le; 1here 'he ? o1led!e o u cer'ai 'ies is i;0recise3 i' is o' udicious 'o see? res0o sesur aces e+0lai i ! 3 o 'he -aria>ili':,

3.2 Validation of a response su rfaceI a : case3 1e e+0ec' ro; a res0o se sur ace uali'ies o a00ro+i;a'io a d 0redic'io 2

#, 'he uali': o a00ro+i;a'io is !i-e >: s'a'is'ical a al:ses carried ou' o 'he >ases o 0oi 'used 'o >uild 'he sur ace 5'his se' o 0oi 's is called here


12/18

Figure 6 : &inear relationshi$ between the out$uts of the function g and the res$onse surface

3.3 Feedback on the response surface

E+0lici' res0o se sur ace u c'io are ;ore u ders'a da>le or 0h:sicis', I' is 'he case or 'he di ere 'a;ilies3 a0ar' eural e'1or?s 5NNRS6, NNRS a d 9LM are ;ore sui'a>le or co 'i uous a d

co;0le+ ;odels, Bu' 'heir 0rac'ical 1or? 1i'h eural e'1or?s a d 9LM ;odels assu;e ae+0erie ce 2

or eural e'1or? 2 u;>er o la:er3 choice o ac'i-a'io u c'io P or 9LM3 choice o 'he res0o se 0ro>a>ili': dis'ri>u'io 3 li ? u c'io P

The 0rac'ical 0ro>le;s e cou 'ered >: 'he use o 'he res0o se sur ace ;e'hod are i 2( 'he a al:sis o s'ro !l: o (li ear 0he o;e a 1here i' is o' o>-ious 'o i d a a;il: o

ade ua'e u c'io s 5;ul'i(la:er eural e'1or?s ca >ri ! a solu'io 6

( 'he a al:sis o disco 'i uous 0he o;e a2 a solu'io is 'o >uild a 0reli;i ar: u c'io o classi ica'io a d 'o calcula'e res0o se sur aces or each >ra ch o 'he >i urca'io 5see sec'ioG6,

O e ues'io o 'e e cou 'ered is 2 is 'he si*e a d 'he uali': o 'he da'a>ase su icie ' [A si;0le a al:se co sis's i s'ud:i ! 'he co -er!e ce o so;e s'a'is'ics 5;ea 3 -aria ce or o'her61he 'he da'a>ase is 'ru ca'ed, The Fi!ure !i-es a e+a;0le, Co -er!e ce o 'he -aria ce a des'i;a'io o 'he >ias could >e assess >: use o Boo's'ra0 ;e'hod, #" !i-es a 'heore'ical e+a;0le,

#"

Values co;0u'ed >: 'he u c'io !

Values co;0u'ed >: a 0ol: o;ial res0o se

sur ace

3 3 3" 3 3& &3# &3G3&

3#

3G

3

&

&3$


13/18

Figure 7 : Con3ergence of statistics with the si8e of the database

3.4 About the impact of response surface error

The use o a res0o se sur ace or a surro!a'e ;odel i a u cer'ai ': a d se si'i-i': a al:sis i;0lies!e erall: a >ias or a error o 'he resul's o 'he u cer'ai ': a d se si'i-i': a al:sis3 >ecause 'hedi ere ce >e'1ee 'he '1o ;odels is e-er a ra do; oise,Usual ues'io s are2

( @ha' is 'he i;0ac' o 'his e'1ee 'he res0o sesur ace SR a d 'he s'udied u c'io 2

5+#3 P3 + 06 SR5+#3 P3 + 06 ( 5+#3 P3 + 06

Assu;e 'ha' 1e ha-e a !ood a00ro+i;a'io o 'he residual u c'io , The (SR a00ears as a !ooda00ro+i;a'io o 'he 'rue u c'io ,Assu;e 'ha' all K i are i de0e de '3 a d se si'i-i': a al:sis ha-e >ee do e o 'he '1o u c'io SR a d 3 a d 1e o'e S SR3i a d S 3i 'he co;0u'ed se si'i-i': a al:sis,The assess;e ' o V5E5 5K #3 P3 K 06/Ki66 ro; S SR3i a d S 3i is o>'ai ed ro;2

( )( ) ( )( )( )[ ] ( )[ ]( )

( )( ) pi pi p

p

pi

p

pi+(i f

X X f *

X X X E X X X +( E

X X f *

X X * +

X X f *

X X +(* + +

33

33333co-"

33

33

33

33

#

##

#

#3

#

#33

+

+=

#$

0,0000

0,2000

0,4000

0,6000

0,8000

1,0000

1,2000

1,4 000

30 40 50 60 70 80 90 Database size

Mean

S'a dardde-ia'io

Minimum

Maximum


14/18

The co;0u'a'io o 'he co-aria ce 'er; re uires a Mo 'e(Carlo si;ula'io , The i' is !e erall:i;0ossi>le 'o deduce resul's o 'he le i de0e de ' o K #3 P3 K 0 a d 5K#3 P3 K 06,The S 3i V5E5 5K#3 P3 K 06/ Ki66 / V5Y6 is es'i;a'ed >:SSR3i / 5V55+#3 P3 + 066 V5SR5+#3 P3 + 0666

I 1e are i 'eres'ed >: 'he i;0ac' o 'he residual u c'io o l: o a ua 'ile3 use ul resul's3 >asedo l: o di ere 'ial a al:sis3 are a-aila>le i FORM/SORM ;e'hods are used 'o co;0u'e aa00ro+i;a'io o a ua 'ile 5see #$ 6,

4 Discontinuous model

4.1 Principles

For disco 'i uous u c'io 3 o usual res0o se sur ace a;il: is sui'a>le, I 0rac'ice3 disco 'i uous >eha-iour ;ea s !e erall: 'ha' ;ore 'ha o e 0h:sical 0he o;e o is i;0le;e 'ed i 'he u c'io f ,The 3 'o a-oid ;isleadi ! i i 'er0re'a'io o resul's o u cer'ai ': a d se si'i-i': a al:sis3discri;i a ' a al:sis should >e used 'o de i e areas 1here 'he u c'io is co 'i uous, S'a'is'icala al:sis are led o each co 'i uous area,

Di ere ' ;e'hods are a-aila>le i s'a'is'ical 'ools 5li?e R or @EJA6 or ;a'he;a'ical so '1ares2( eural e'1or?s 1i'h si!;oid ac'i-a'io u c'io 5 #G 63( 9LM ;odels 1i'h a lo!i' li ? or lo!is'ic re!ressio 5 # 63( Vec'or su00or' ;achi e 5a e+'e sio o eural e'1or? 5 # 3 # 63( Decisio 'ree a d -aria 's li?e ra do; ores'P 5 #& 6

Free1are 'ool>o+es e+is's a d are -er: eas: 'o use,)rac'ical 0ro>le;s are o 'e e cou 'ered i 'he sa;0le is W li earl: se0ara>le X, The Fi!ure sho1s ae+a;0le, I 1e ha-e a irs' se' o da'a 5see >lac? lac? li es6 'ha' e+0lai all 'he se', I e1 calcula'io s are do e3 a d 1e o>'ai '1o e1 lue 0o'a'oes63 'he 'he 0redic'io error could >e o e!li!i>le 1i'h 'he 0re-ious se0ara'or

u c'io s, @or?s are i 0ro!ress 'o de-elo0 ;e'hods 'ha' 0er;i' 'o o>'ai a ro>us' se0ara'or u c'ioli?e 'he red li e, Su00or' -ec'or ;achi es a d ;e'hods >ased o Decisio Trees are -er: 0ro;isi !

or 'ha' case,

#G


15/18

Figure 9 : )roblem of robustness in linear clustering

4.2 Example

The i 'eres' o 'hese discri;i a ' ;e'hods has >ee s'udied o a e+a;0le3 i 'he ra;e1or? o aco 'rac' 1i'h 'he )SA Le-el " 0ro ec' a' IRSN,I 'his e+a;0le3 1e s'ud: 'he direc' co 'ai ;e ' hea'i ! 5DCH6 0he o;e o , The res0o se is 'he

0rese ce o coriu; i 'he co 'ai ;e ' ou'side 'he reac'or 0i' i' is a discre'e res0o se 1i'h -alue % 5 o

coriu;6 or # 50rese ce6, Ni e i 0u' -aria>les3 s'a'is'icall: i de0e de '3 are 'a?e i 'o accou '2 MCOR 2 ;ass o coriu;3 u i or;l: dis'ri>u'ed >e'1ee "% %%% a d &% %%% ?! F RO 2 rac'io o o+:ded r3 u i or;l: dis'ri>u'ed >e'1ee %3 a d #3 )VES 2 0ri;ar: 0ressure3 u i or;l: dis'ri>u'ed >e'1ee # a d # >ars3 DIAM 2 >rea? si*e3 u i or;l: dis'ri>u'ed >e'1ee # c; a d # ;3 ACAV 2 sec'io de 0assa!e da s le 0ui's de cu-e 5-arie e 're & a d "" ; " 6 FRAC 2 rac'io o coriu; e 'rai \e direc'l: e ec'ed i 'he co 'ai ;e '3 u i or;l: dis'ri>u'ed

>e'1ee % a d #3 CDIS 2 dischar!e coe icie ' a' 'he >rea?3 u i or;l: dis'ri>u'ed >e'1ee %3# a d %3 3 JFIT 2 ad us';e ' 0ara;e'er3 u i or;l: dis'ri>u'ed >e'1ee %3# a d %3$3 H@AT 2 1a'er hei!h' i 'he reac'or 0i'3 discre'e ra do; -aria>le 5% or $ ;e'er6

The calcula'io s ha-e >ee 0er or;ed 1i'h 'he RU)UICUV ;odule o Escadre Mod #," i "%%%, The;odule has >ee si! i ica 'l: i;0ro-ed si ce "%%% 'he ollo1i ! resul's ha-e i 0rac'ice o'

0h:sicall: ;ea i !,A da'a>ase o $%% calcula'io s is a-aila>le, The i 0u's -ec'ors or 'hese calcula'io s ha-e >ee!e era'ed ra do;l: i 'he -aria'io do;ai ,The da'a>ase has >ee s0li''ed i '1o da'a>ases 2

( A 'rai i ! se' 1i'h 'he irs' "%% calcula'io s3( a 'es' >ase 1i'h 'he o'her #%% calcula'io s,

S'a'is'ical charac'eris'ics o 'hese se's are !i-e i Ta>les $ a d G,

Mi i;u; Ma+i;u; Mea S'a dard De-ia'ioMCOR ",%% e %%G , &G e %%G ,%&%&e %%G #, $e %%G

#

#s'W co 'i uous Xse'

" dW co 'i uous Xse'


16/18

F RO %, %" %, & %, G %,#$&&)VES #,#% % # ,"#GG & ,% # G ,G GDIAM %,%# %, %, % %,"& #ACAV &,%#G& "#, % #G& $ G,% "FRAC %,%%"$ %, %,G %," G

CDIS %,$%## %,& $ %, & %,#&%JFIT %,#%#" %," $ %,"%#$ %,%

Table # : "tatistical characteristics for the training set

Mi i;u; Ma+i;u; Mea S'a dard De-ia'ioMCOR ",%"% e %%G ,&$ e %%G G, # &e %%G #, e %%GF RO %, % # %, $ %, %,#G#&)VES $,G&& # G, ### &$,G & G ,GG#$DIAM %,%#"$ %, & % %,G$ %,"&ACAV &,%""G "#, %# #G,& # G,%FRAC %,%%$G %, %, $& %,$#%#CDIS %,$% G %,& &$ %, %# %,# "JFIT %,#%%$ %," % %,# %,%

Table 6 : "tatistical characteristics for the test base

The irs' 'ool 'ha' 1e ha-e 'ried is 'he !e erali*ed li ear ;odel 1i'h a 3 gi0 li ?, For a se' o all 'hecalcula'io s3 or or each su>(sa;>le3 i' is 0ossi>le 'o i d a ;odel 'ha' e+0lai s #%% o 'he dis0ersioo 'he resul's or 'he 'rai i ! se' i' is ecessar: 'o i 'roduce i 'erac'io 'er;s a d so;e'ra s or;a'io s 5lo!ari'h;ic or 0o1er6, The 3 1e ha-e a li earl: se0ara>le 0ro>le;, Bu' 'here is so;edra1>ac?s 2

( 'he lis' o 'he 'er;s 'ha' are s'a'is'icall: si! i ica ' -aries s'ro !l: 1i'h 'he 'rai i ! se'( 'he 0redic'io error is arou d "% ,

The use o eural e'1or?s sho1s si;ilar 0ro>le;s3 >ecause 'he u c'io al s'ruc'ural is si;ilar 'o alo!is'ic re!ressio ,@e ha-e 'he 'ried o'her ;e'hods, The Ta>le sho1s a s: 'hesis o 'he resul's o>'ai ed 1i'h di ere ';e'hods 5 #& 6, The @EJA so '1are 5111,cs,1ai?a'o,ac, */;l/1e?a6 has >ee used or 'he s'ud:,The 0ara;e'ers or each ;e'hod are !i-e , For 'he '1o se's3 1e !i-e 'he 0erce 'a!e o 0oi 's 'ha' are

>adl: es'i;a'ed >: 'he ;odel, For E+a;0le Ra do; Fores' ;e'hod e+0lai 0er ec'l: 'he 'rai i ! se',A ;ore !lo>al i dica'or o 'he uali': o 'he ;odel is o>'ai ed >: cross -alida'io ;e'hod3 a00lied o'he 1hole da'a>ase,

@e o'ice 'ha' 'he 'es' error o 'he 'es' >ase could >e lo1er 'ha i's o>'ai ed >: cross(-alida'io 5c ,7G& a d Ra do; Fores'6, I 1e de i e 'he 0redic'io error o l: ro; a 'es' >ase3 'his ac' is o 'e

o>ser-ed3 1he 1e used o l: a 'es' >ase3 a d 'his error is !e erall: >iased,The ;os' e icie ' ;e'hod or 'ha' e+a;0le is 'he Ra do; Fores' ;e'hod 5 o'e 2 o'hers ;e'hods ha-e >ee 'ried3 >u' 'he resul's are o' !i-e here6,I 0rac'ice3 1e ha-e o'ice 'ha' 'he ;e'hods 7G& a d Ra do; Fores' are as'er 'ha 'he al!ori'h;s

>ased o o0'i;isa'io s'e0 5li?e Na]-e Ba:es3 SVM3 Neural Ne'1or?P6, The 0ri ci0le o decisio'rees a d ra do; ores' is si;0le a d >ased o 'he >uildi ! o a se' o lo!ical co;>i a'io o decisiorules, The: are o 'e -er: reada>le3 a d ha-e -er: 0redic'io ca0a>ili'ies 5li?e sho1 >: 'he e+a;0le6,

Nai-e Ba:es SVM SMO 7G& Ra do; Fores'Descri0'io o 'he;e'hod

O0'i;isa'io o 'he classi ica'io 0ro>a>ili': i a

Su00or' -ec'or ;achi e ;e'hod

1i'h SMO

Decisio Tree;e'hod also

a;ed CG,

Al!ori'h; >asedo a se' o

decisio 'rees

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Ba:esiara;e1or? u der

9aussiaassu;0'io s

al!ori'h; or 'heo0'i;isa'io s'e0

al!ori'h;, 0redic'or,

Me'hod 0ara;e'ers (D (C"(E" (S (IG%(9%,%# (C%," (J%

(A#%%%%%$ M" (S#(T%,%%# (A()#% (#%

(N% (R (M(V(# (@ #

Training setBadl: classi ied , &, %Error ;a'ri+ #% ## #" # # #% " %

# & " # # % # $ % # $Test setBadl: classi ied

Error ;a'ri+ ## $ ## $ #% G &% " &G G &" # &

Cross 3alidation/# calculationsBadl: classi ied &, ,$$ , ,$$Error ;a'ri+ " # #& "$ "# "% " #

## "G& " G " % " $

Table 7 : Results with different methods

5 Conclusions

This 0a0er has !i-e a o-er-ie1 o s'a'is'ical a d 0ro>a>ilis'ic ;e'hods 'o s'ud: 'he i lue ce o u cer'ai 'ies o i 0u's -aria>le o 'he code res0o ses,I 'he use o 'hese ;e'hods3 so;e 0ro>le;s are e cou 'ered due 'o correla'ed ra do; -aria>les or disco 'i uous ;e'hods, The 0a0er descri>es e1 resul's a d ideas 'o o-erco;e 'hese 0ro>le;s,)rac'ical i 'eres' o 'hese < e1= ;e'hods should >e co ir;ed3 >: a00lica'io o real 0ro>le;s,

6 References

# NEA/CSNI/R5 6$" H, 9laeser, U cer'ai ': e-alua'io o 'her;al(h:draulic code resul'es, B/0 /a0i /a3 m 0i/g

/ CD !0-E!0ima0 M 0> d! i/ N$c3 a B/!0a33a0i / +af 0@ F/a3@!i! (DE-#"""). @ashi !'o 3DC3 No-e;>er "%%%,

$ @il?s S,S, De'er;i a'io o sa;0le si*es or se''i ! 'olera ce li;i's F//. Ma0>. +0a0i!0 ,3 #" 3 00, #( 3 # G#,

G )earso E,S, a d Tu?e: M, Dis'ri>u'io s 1hose irs' our ;o;e 's are ? o1 , Di m 0 i%a3 3 00 #$$(#$ 3 #

Balde1ec? H,3 MG0> d ! d ! 3 m /0! fi/i! !0 c>a!0iH$ !. Fpp3ica0i /! I 3a gG !0a0i!0iH$ 0 I3a mGca/iH$ d 3a $p0$ , )hD U i-ersi': E-r:(Val dQEsso e, #

Sal'elli A,3 Cha J, e' Sco'' E,M, 5Eds6, + /!i0i2i0@ F/a3@!i!, 7, @ile:3 @ile: Series i)ro>a>ili': a d S'a'is'ics, "%%%

7ac ues 7,3 La-er! e C, a d De-ic'or N, Se si'i-i': a al:sis i 0rese ce o ;odel u cer'ai ':a d correla'ed i 0u's, P c di/g! f +FM7 #""' , "%%G

& Ru>i s'ei R,Y, +im$3a0i /! a/d M /0 -6a 3 M 0> d , @ile: Series i )ro>a>ili': a dMa'he;a'ical S'a'is'ics3 7, @ile: ^ So s, #

#


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De-ic'or N, Jiabi3i0G 0 mGca/iH$ K mG0> d ! J7 M +7 M 0 c $p3ag ! a2 c d ! c d !dLG3Gm /0! fi/i! pa !$ fac ! d Gp /! adap0a0i2, Th_se3 U i-ersi'\ Blaise )ascal3Cler;o '(Ferra d, #

#% Melchers R,E,3+0 $c0$ a3 3iabi3i0@ a/a3@!i! a/d p dic0i /3 7,@ile: ^ So s, ### Bo+ 9,E,) a d Dra0er N,R, Empi ica3 M d 3 D$i3di/g a/d !p /! +$ fac , 7, @ile: ^ So s3

@ile: Series i )ro>a>ili': a d Ma'he;a'ical s'a'is'ics, ##" De-ic'or N, a d Mar'i e* 7(M, No li ear re!ressio ;e'hods i u cer'ai ': a d se si'i-i': s'udies

a d relia>ili': co;0u'a'io s, P c di/g! f E+ E< #""" , "%%%#$ )e dola M3 Jiabi3i0G d ! !0 $c0$ ! / c /0 x0 dLi/c 0i0$d !, !0a0i!0iH$ ! 0 dLGca 0! d

m dG3i!a0i /, )hD 'hesis3 U i-ersi'\ Blaise )ascal3 Cler;o '(Ferra d, "%%%#G Dre: us 9, e' al, H\raul', G! a$x d / $ / ! - MG0> d 3 gi 0 app3ica0i /!, E:rolles, "%%"# McCulla!h ), a d Nelder 7, 8 / a3iA d 3i/ a m d 3!, " d Edi'io , Cha0;a a d Hall, # Su:?e s 7, e' al, < a!0 !H$a ! K +$pp 0 * c0 Mac>i/ !, @orld Scie 'i ic, "%%"# Be (Hur A,3 Hor D,3 Sie!el;a H, a d V, Va0 i? V, Su00or' -ec'or clus'eri !, $ /a3 f

Mac>i/ < a /i/g ! a c> 3 " 2#" `#$ , "%%##& @ri''e H, a d Fra ? E, a0a Mi/i/g K P ac0ica3 Mac>i/ < a /i/g 9 3! a/d 9 c>/iH$ !

wi0> a2a Bmp3 m /0a0i /!, Mor!a Jau ;a , #

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