7 th World Cogresses of Structural ad Multdscplary Optmzato COE Seoul, May 5 May 7, Korea New Trasformato of Depedet Iput Varables Usg Copula for RBDO BUoojeog NohU, K.K. Cho*, ad Lu Du Departmet of Mechacal & Idustral Egeerg College of Egeerg The Uversty of Iowa Iowa Cty, IA 54, U.S.A. Emal: HUoh@egeerg.uowa.eduUH HUkkcho@egeerg.uowa.eduUH HUludu@egeerg.uowa.eduU. Abstract For the performace measure approach (PMA of RBDO, a trasformato betwee the put radom varables ad the stadard ormal radom varables s requred to carry out the verse relablty aalyss. Sce the trasformato uses the jot cumulatve desty fucto (CDF of put varables, the jot CDF should be kow before carryg out RBDO. I may dustral RBDO problems, eve though the put radom varables are correlated, they are ofte assumed to be depedet because oly margal dstrbuto ad covarace are practcally obtaed ad the jot CDF s very dffcult to obta. Wth the assumpto of depedet put varables, t s easy to costruct the jot CDF, ad Roseblatt trasformato, whch trasforms the codtoal CDF of put varables to the stadard ormal dstrbuto, has bee used for RBDO. However, whe put varables are correlated, Roseblatt trasformato caot be drectly used because t s hard to obta the jot CDF of correlated varables. O the other had, Nataf trasformato ca be used for correlated put varables because t oly requres margal dstrbuto ad covarace. However, sce Nataf trasformato uses aussa copula, whch jos multvarate ormal ad margal dstrbutos, t caot be used for put varables wth o-aussa jot dstrbuto. I ths paper, a ew trasformato that uses a o-aussa copula, such as Clayto copula, as the jot CDF of correlated put varables, whch s the followed by Roseblatt trasformato, s proposed for o-aussa correlated varables. I addto, t s show that the correlato coeffcet betwee put varables sgfcatly affect RBDO results ad dfferet trasformatos such as Nataf trasformato usg aussa copula ad the ew trasformato usg o-aussa copula (Clayto copula provde dfferet RBDO results.. Keywords: Relablty-based desg optmzato (RBDO, verse relablty aalyss, Roseblatt trasformato, Nataf trasformato, copula famly. 3. Itroducto The RBDO process requres two optmzato procedures: the desg optmzato the put radom varable space ad the verse relablty aalyss the stadard ormal radom varable space []. Thus, a trasformato betwee these two varables s ecessary for the verse relablty aalyss RBDO. Roseblatt ad Nataf trasformatos are commoly used for the relablty aalyss. Roseblatt trasformato [] s a mathematcally exact method ad requres complete formato of the put varables such as jot CDF or codtoal CDF [3, 4]. O the other had, Nataf trasformato s a approxmate method that oly requres the covarace matrx ad margal CDF [3-6] ad used to costruct a jot CDF (Nataf model whch s detfed as aussa copula. I fact, Nataf trasformato s a combato of aussa copula ad Roseblatt trasformato. Hece, f the put varables of the RBDO problem are depedet, ether method ca be used because two methods have the same trasformato formulato ths case. However, f ot, they may yeld dfferet RBDO results depedg o the put formato. I may RBDO problems, the put radom varables such as the materal propertes are correlated. To solve the RBDO problems wth the correlated put varables, a jot PDF or cumulatve dstrbuto fucto (CDF of put varables should be avalable. However, ofte dustral applcatos, oly lmted formato such as the margal dstrbutos ad covarace are practcally avalable, whereas the put jot probablty dstrbuto fuctos (PDF are very dffcult to obta [4, 6]. Thus, the lterature, most RBDO studes have assumed all put radom varables are depedet ad mostly the Roseblatt trasformato method has bee used.
I ths paper, t s foud that Roseblatt trasformato s mpractcal for problems wth correlated put varables due to dffculty of costructg a jot PDF from the margal dstrbutos ad covarace. O the other had, Nataf trasformato ca be used to costruct a jot CDF, whch s called Nataf model or aussa copula the copula famly [7], from the lmted formato such as margal dstrbutos ad covarace. Sce Nataf trasformato uses copula whch s a jot fucto of margal dstrbutos, eve f put varables are mxed wth dfferet types of dstrbutos, the jot PDF or CDF ca be easly costructed. Further, Nataf trasformato s applcable for the ormal ad logormal dstrbutos wth postve correlato that cover majorty of practcal dustral applcatos, ad thus, t s applcable to a broad class of RBDO problems [8-]. However, sce Nataf trasformato s orgated from aussa copula, ad t may ot be applcable to o-aussa dstrbutos except the ormal ad logormal dstrbutos. Eve though the ormal ad logormal dstrbutos cover majorty of practcal dustral applcatos, there are stll some applcatos wth o-aussa varables, whch wll be dscussed Secto 6.. I ths paper, a ew trasformato that combes the advatages of Roseblatt trasformato ad copulas s developed for RBDO problems wth o-aussa correlated radom put varables. Usg a umercal example, t s show that the correlated radom put varables do sgfcatly affect the RBDO result ad the proposed trasformato s applcable to RBDO problems wth correlated put varables wth o-aussa jot CDF whch has a o-aussa dstrbuto. 4. Roseblatt Trasformato Roseblatt trasformato s a well-kow trasformato method that maps the orgal varables oto the stadard ormal varables. It s defed as the followg successve codtog Φ ( u = F ( x Φ ( u = F ( x x ( Φ u = F x x, x,, x ( ( where s umber of put varables, (,,..., F x x x x s the CDF of codtoal o = x, = x,, = x, ad Φ ( s the verse CDF of the stadard ormal varables. Based o Eq. (, whe the multvarate jot PDF or codtoal CDFs are kow, the probablty of falure ca be exactly estmated usg Roseblatt trasformato. For depedet put varables, the probablty of falure ca be obtaed from the jot PDF, whch s smply multplcato of the margal PDFs. However, eve though Roseblatt trasformato has advatages, t may ot be wdely applcable to practcal egeerg problems due to followg reasos. Frst, the jot PDF or codtoal CDFs should be avalable for all varables to estmate the probablty of falure, whch s ofte too expesve or dffcult to obta dustral applcatos where the margal CDF ad covarace are commoly avalable. Also, whe the dstrbuto types of put varables are mxed,.e., some of the varables are logormal ad others are expoetal or Webull, t s ot possble to express the jot PDF a mathematcal formulato. Thus, Roseblatt trasformato ca be used oly for lmted cases where all put varables have ormal dstrbuto ad a jot PDF or codtoal CDFs are provded. However, f a copula, whch s a jot CDF of margal dstrbutos, s used stead of the multvarate jot CDF, Roseblatt trasformato ca be used because copulas oly requre margal dstrbutos ad correlato parameters that ca be practcally obtaed. It s dscussed Secto 7 detal. 5. Copulas The copula s orgated from a Lat word for lk or te that coects dfferet thgs. I statstcs, the copulas are fuctos that jo multvarate dstrbuto fuctos to ther oe dmesoal margal dstrbuto fuctos. That s, copulas are multvarate dstrbuto fuctos whose margal dstrbutos are uform o the terval o [,]. Accordg to Sklar s theorem [7], f the radom varables have margal dstrbutos, the there exsts a -dmesoal copula C such that F ( ( ( (,..., x,..., x,..., = C F x F x ( If margal dstrbutos are all cotuous, the C s uque. Coversely, f C s a -dmesoal copula ad F ( x (,..., ad F x are margal CDFs, the the jot dstrbuto s a -dmesoal fucto wth margal dstrbutos [7]. By takg the dervatve of Eq. (, the jot PDF s defed as f ( x,, x = c( F ( x ( (,, F x Π f x (3 where cu (,, u (,, u wth u = F( x ad ( C u = u u = f x s margal PDF for =,,.
5.. Fréchet-Hoeffdg Bouds Ay copula Cu (,, u les betwee the Fréchet-Hoeffdg lower ad upper bouds for every (,, u u I ad the bouds are themselves copulas whch are gve as [7] max( u + + u +, C( u,, u m( u,, u (4 where I = I I I I =,. ( [ ] For the two-dmesoal case, Eq. (4 ca be wrtte as max( u+ u, C( u, u m( u, u (5 W u, u = max( u+ u,, M ( u, u = m( u, u, ad cosder a depedet copula ( u, u = u u. W u, u, M ( u, u, ad ( u, u are graphcally show Fg. u - u space. Let ( These copulas ( M(u,u.8.6.4. W(u,u.8.6.4. Π(u,u.8.6.4..5 u..4 u.6.8.5 u..4 u.6.8.5 u..4 u.6.8 (a (b (c Fgure. raph of the Copulas: (a Fréchet-Hoeffdg Upper Boud Copula W (b Fréchet-Hoeffdg Lower Boud Copula M (c Idepedet Copula Three copulas ca be easly compared by drawg these copulas alog the dagoal drecto u = u as show Fg.. The graph of ay copula s a cotuous surface wth I 3 (Fg., ad alog the horzotal, vertcal, ad dagoal drectos, all copulas are odecreasg fuctos. Moreover, all copulas are uformly cotuous o I [7]..9.8.7 W(u,u M(u,u Π(u,u C(u,u.6.5.4.3....4.6.8 u =u Fgure. raph of Copulas W, M, ad alog the Dagoal Drecto 5.. Depedece Measures To measure depedece betwee two radom varables, several types of measures such as Pearso s rho, Spearma s rho ad Kedall s tau are used. Pearso s rho, whch s also called a product momet correlato coeffcet, s frst dscovered by Bravas 846 [] ad developed by Pearso 896 []. Pearso s rho dcates the degree of lear relatoshp betwee two radom varables as follows. Cov(, ρ = (6 σ σ where σ ad σ are stadard devatos of ad, respectvely ad Cov(, s the covarace betwee ad
. Sce Pearso s rho oly dcates the lear relatoshp betwee two radom varables, t s ot a good measure for olear relatoshp betwee two radom varables, whch ofte occurs practcal egeerg applcatos. If gve data follows o-aussa jot dstrbuto, aother measure eeds to be troduced to estmate depedecy betwee radom varables. Therefore, Pearso s rho s oly vald whe the jot PDF s multvarate ormal dstrbuto. Ulke Pearso s rho, the followg two measures, Spearma s rho ad Kedall s tau, do ot requre the assumpto that the relatoshp betwee two radom varables s lear. Spearma s rho ad Kedalls tau measures the correspodece of two rakgs betwee radom varables a dfferet way. Sce these measures estmate the relatoshp betwee two rakgs of radom varables, they are called rak correlato. The rak correlato coeffcet s frst troduced by Spearma, who s psychologst, 94 [3]. I psychology, more tha ay other scece, t s hard to fd a measure that estmates correlato betwee two varables because there are some cases where correlato caot be measured quattatvely. For example, the depedece betwee heredtary qualtes of brothers caot be quattatvely measured f Pearso s rho that requres specfc values of two varables s used. O the other had, f chldre of a school are dvded to coscetous ad o-coscetous group, the correlato ca be measured by coutg how much brothers ted to be the same dvso. Thus, that case, comparso (rakg of two groups s a better way rather tha measurg Pearso s rho. The rak does ot chage uder strctly creasg fucto; hece t ca be expressed as copulas because copulas are varat uder strctly creasg trasformato of ts margs. As prevously metoed, sce the rak correlato coeffcet measures the degree of correspodece betwee two varables, the correspodece should be mathematcally defed. Cocordace s oe way of expresso for correspodece. If large values of oe ted to be assocated wth large values of the other ad small values of oe wth small values of the other, two radom varables are sad to be cocordat. Lkewse, f large values of oe ted to be assocated wth small values of the other, two radom varables are called dscordat. Sce the copulas play a mportat role cocordace ad depedece measures are also assocated wth cocordace, a cocordace fucto Q eeds to be troduced. The cocordace s the dfferece betwee the probablty of cocordace ad the probablty of,, ad defed as dscordace for a par of radom vectors ( ad ( Q = P ( ( > P ( ( < where (, ad (, H( xy, = C( F( x, ( y ad H = C ( F( x, ( y wth same margs u = F( x ad v ( y be expressed as copulas (7 are depedet vectors of cotuous radom varables wth jot dstrbuto ( ( ( I =. Equato (7 ca Q = Q C, C = 4 C u, v dc u, v (8 5.. Spearma s Rho Spearma s rho s frst troduced by Spearma 94 [3]. Spearma s rho s defed to be proportoal to the probablty of cocordace mus the probablty of dscordace betwee two radom vectors (, ad (, 3 wth same margs u = F( x ad v = ( y, but wth dfferet copulas, H ( xy, = C F( x, ( y of (, ad ( ( =Π ( = ( ( of (, 3. The populato verso of Spearma s rho s defed as ρ s = 3( P ( ( 3 > P ( ( 3 < H xy, xy, F x y (9 Here, the multplcato of 3 s to make Spearma s rho to have rages to. Equato (9 ca be rewrtte terms of a copula as ρ s = 3 Q ( C, Π = uvdc ( u, v 3 = C ( u, v dudv 3 I ( I The sample verso of Spearma s rho s 6 rs = d = ( where d s the dfferece of two rakgs ad s the umber of samples. Usg Eq. (, Spearma s rho ca be estmated from a gve copula, or coversely usg Eq. (, the correlato parameter betwee margal dstrbutos a copula ca be estmated provded that the copula type s defed. For may copulas, sce there s a explct relatoshp betwee correlato parameter ad Spearma s rho, the correlato parameter ca be easly obtaed. 5.. Kedall s Tau Kedall s tau s frst troduced by Kedall 938 [4]. Kedall s tau s the probablty of cocordace mus the (
= ad. The populato verso of Kedall s tau s expressed usg cocordace fucto as τ = QCC (, = 4 C( uv, dc( uv, ( I The sample verso of Kedall s tau s c d t = = ( cd/ (3 c+ d where c s the umber of cocordat pars ad d s the umber of dscordat pars ad s the umber of samples. Usg Eq. (, Kedall s tau ca be calculated from a gve copula or f Kedall s tau s kow, the the correlato parameter of copulas also ca be calculated. Lke Spearma s rho, for may copulas, sce there are explct formulatos whch defe the relatoshp betwee correlato parameter ad Kedall s tau, the correlato parameter ca be easly obtaed provded the copula type s determed. More detaled formato o Spearma s rho ad Kedall s tau s preseted Ref. 5. probablty of dscordace betwee two radom vectors (, ad (, wth same margs u F( x v = ( y, but wth a commo copula, H ( xy, = C( F( x, ( y of (, ad (, 5.3. Commoly Used Copulas I ths secto, two commoly used copulas: the ellptcal copulas ad Archmedea copulas are troduced. These copulas are most popular copulas because they ca be exteded to multvarate dstrbutos ad easy to hadle because of ther tractable characterstcs. For example, ellptcal copulas provde a lear trasformato from orgal varables to stadard ormal varables, ad thus t s easy to trasform from oe to aother. I case of Archmedea copulas, each copula has a uque geerator fucto ad the geerator s used to calculate the depedece measure, Kedall s tau. Thus, wthout usg Eq. ( that requre double tegrato o I, Kedall s tau ca easly obtaed as wll be explaed Secto 5.3.. 5.3. Ellptcal Copulas Ellptcal copulas are copulas of ellptcal dstrbutos. A radom varable has a ellptcal dstrbuto, f ad oly f there exst S, R, ad A that satsfy =μ + R AS = μ +A (4 k k where S s k-dmesoal radom vector that are uformly dstrbuted o the ut sphere δ = { s : s's = } ; R s d k a o-egatve radom varable, depedet of S ; A wth AA' = Σ where Σ s covarace matrx, ad d ad k are the umber of radom varables ad S, respectvely [6]. Whe R has a ch-square dstrbuto wth d degree-of-freedom, has a stadard ormal dstrbuto N (, ad accordgly has a ormal dstrbuto N ( μ, Σ for d-dmesoal varables. If t dstrbuto T ( ν,, ad has a t dstrbuto wth (, R / d has F dstrbuto wth d ad ν degrees-of-freedom, has a stadard T ν μ, Σ for d -dmesoal varables. The t dstrbuto ad ormal dstrbuto provde t copula ad ormal (aussa copula, respectvely [7]. aussa copula s defed as multvarate ormal dstrbuto of stadard ormal varables C( u (,, u =Φ Φ u,, Φ ( u (5 aussa copula s used Nataf trasformato to trasform orgal varables x to correlated stadard ormal varables y wth Φ(,P' where P' s the reduced covarace matrx. It s dscussed Secto 6 detal. The t copula ca be costructed usg t dstrbuto lke aussa s costructed usg ormal dstrbuto. The t copula s defed as multvarate t dstrbuto of stadard t varables wth TP ', ν (,P', ν where ν s the degree of freedom. C( u (,, u = T P', ν Ft u,, Ft ( u ν ν (6 5.3. Archmedea Copula Archmedea copula s costructed completely dfferet way from the ellptcal copula that uses multvarate dstrbuto. A mportat compoet of costructg Archmedea copulas s a geerator fucto ϕ whch s a complete mootoe decreasg fucto. The fucto ϕ ( t s completely mootoc o a terval I = [,] f t s cotuous ad has dervatves of all orders that alterate sg k d ( ( t, k d ϕ k =,, (7
ϕ = ad ϕ ( = ad the verse,, the a -copula, whch s called Archmedea copula, for all ca be If ϕ s a cotuous ad strctly decreasg fucto from I to [, ] such that ( ϕ s completely mootoc o [ defed as ( = ϕ ϕ( + + ϕ( C u,, u u u (8 Each Archmedea copula has the correspodg uque geerator fucto ad the geerator fucto provdes copulas as see Eq. (8. As stated the begg of ths secto, oce the geerator s provded, Kedall s tau ca be easly obtaed as ϕ ( t τ = + 4 dt (9 ϕ '( t Ulke Kedall s tau, there s o explct formulato that estmates Spearma s rho usg the geerator. 6. Nataf Trasformato Nataf trasformato s used to costruct a ormal CDF, whch s detfed as aussa copula (Nataf model, by trasformg orgal varables to stadard ormal varables. Nataf trasformato cotas two steps: trasformato from orgal varables to correlated stadard ormal varables ad trasformato from correlated ormal varables to depedet stadard ormal varables U. The frst step approxmates the jot CDF usg aussa copula ad the secod step requres a lear trasformato whch s same wth Roseblatt trasformato for ormal varables [8]. 6.. Nataf Trasformato Usg aussa Copula Usg Sklar s theorem, aussa copula, whch s also called Nataf model, ca be obtaed as F ( ( ( (,... x,..., x,..., =Φ Φ F x Φ F x where F s the margal CDF of x. Cosder the trasformato from to as y =Φ F ( x,,..., = The, the jot PDF s defed by F y y F f x f x f (,..., x,..., x = = = x x x x y y φ( y φ( y y f ( x F where =, = φ ( y,p '. If the margal PDFs f ( x for,, x φ y y y ( ( ( φ ( y,p' ( ( ( = are avalable, the the reduced covarace matrx P' = { ρ ' j} ca be estmated. Thus, s a -dmesoal stadard ormal varable wth a jot PDF ( φ y,p'. Therefore, the jot PDF ca be calculated usg the margal PDFs of the put varables ad the reduced covarace matrx. The reduced correlato coeffcet betwee two varables ca be estmated by a teratve process usg the double tegral as ' ρ = E ZZ = zzφ y, y; ρ dydy ( (3 j j j j j j where Z = ( μ / σ. However, sce the teratve process s very tedous ad ukows are wth the double tegral, Eq. (3 s approxmated by ' ρj = Rj ρj (4 to obta the reduced coeffcet. I Eq. (4, R j s approxmated by a polyomal as Rj = a+ bv + cv + dρj + eρj + f ρjv + gvj + hvj + kρjvj + lvv j (5 where V ad V j are the coeffcet of varato ( V = σ / μ for each varable ad the coeffcets deped o the types of put varables. For dfferet types of put varables, the correspodg coeffcets are gve Refs. 3, 4 ad 6. The maxmum error of the estmated correlato coeffcet obtaed from Eq. (4 s ormally much below %, ad eve f the expoetal dstrbuto or egatve correlato s volved, the maxmum error the correlato coeffcet s at most up to % [4, 6]. Therefore, the approxmato provdes adequate accuracy wth less computatoal effort. The verse relablty aalyss s carred out usg the trasformed stadard ucorrelated ormal varables U. Sce the relatoshp betwee the orgal correlated varables ad the correlated stadard ormal varables s gve
Eq. (, the ext step s to trasform the correlated stadard ormal varables to the ucorrelated stadard ormal varables U usg a lear trasformato. Cosder the followg lear equato. =A+BU (6 where ~ N(,I has the reduced correlato matrx Σ = P' ad U ~ N(,I has the covarace matrx Σ U =I. The mea of ca be calculated as E [ ] =EA+BU [ ] =A+BEU [ ] =A= (7 I the same way, the covarace matrx of ca be calculated as T T P' = Σ = Var[ A+BU ] = Var [ BU ] =BΣUB =BB (8 Sce the covarace matrx of s postve defte, P' ca be decomposed to the lower ad upper tragular T matrx, B ad B usg Cholesky factorzato. Each elemet b j B ca be calculated as bk, = j k = bj = (9 j ρj bb k kj / b, > j k = Usg Eqs. ( ad (6, the relatoshp (Nataf trasformato betwee orgal varables ad depedet stadard varables U ca be obtaed as x = F ( Φ( b u x = F ( Φ ( b u+ bu (3 x = F Φ b u + b u + + b u ( ( Sce the orgal varables ca be expressed terms of the depedet stadard ormal varables, the relablty aalyss ca be carred out by usg Eq. (3. If oly the covarace matrx ad margal dstrbuto are avalable, Nataf trasformato s oly possble way to costruct the jot CDF of the put radom varables whle Roseblatt trasformato s ot. Further, sce Nataf trasformato ca accurately approxmate jot PDF for logormal varables wth postve correlato as well as for ormal varables ad for combed ormal ad logormal varable, whch cover majorty of egeerg applcatos, t ca be wdely used []. For multvarate ormal dstrbuto, Roseblatt trasformato s same wth Nataf trasformato, ad thus, Nataf trasformato s a combato of aussa copula ad Roseblatt trasformato. 6.. Lmtato of Nataf Trasformato Nataf trasformato s applcable to ormal ad logormal varables, but t may ot be applcable to other dstrbutos whch have o-symmetrc assocato betwee two radom varables. I practcal egeerg applcatos, there are cases wth o-symmetrc assocato betwee two radom varables. For stace, the followg example shows 3 pars of correlated data from a exhaust mafold used o a Chrysler 5.L ege. Two radom varables ad are show Fg. 3 ad the data are collected from a mache capablty study performed o the mache bulder s floor Chrysler Corporato [9]. Fgure 3. Exhaust Fold [9]
Fgure 4 (a shows the scatter plot of the data. Usg the data, the mea values ad stadard devatos of ad are calculated as μ = 8.945, μ =.87, σ =.569E 3, σ =.54E 3 ; ad Pearso s rho ad Kedall s tau are.44 ad.4, respectvely. As see the Fg. 4, the jot dstrbuto does ot ft the bvarate ormal (aussa copula because the data are ot symmetrcally dstrbuted. The data are desely dstrbuted the left-lower ed, but they are wdely spread out the rght-upper ed. As see Fg. 4 (b ad (c, the PDF cotours usg Clayto copula are dese ear the area where data are clustered. O the other had, the PDF cotours usg aussa copula evely fall apart from each other regardless of the desty of the data. Therefore, Clayto copula better descrbes the data tha aussa copula..89.89.88.88.87.87.86.86.85.85.84 8.94 8.943 8.944 8.945 8.946 8.947 8.948 8.949 8.95 (a (b (c.84 8.94 8.943 8.944 8.945 8.946 8.947 8.948 8.949 8.95 Fgure 4. (a Scatter Plot of Data (b PDF Cotour Usg aussa Copula (c PDF Cotour Usg Clayto Copula I ths example, sce the true jot dstrbuto may ot be kow ad the data has lack of formato, t s ot clear to fgure out whch copula descrbes the data better. I fact, case of the data, how may samples are requred to select a approprate copula type for a gve data s a mportat ssue, whch has ot bee vestgated ths paper. I ths paper, t s assumed that we have eough data whose dstrbuted shape s clearly show ad t ca be fgured out whch type of copulas s approprate to descrbe the gve data. Suppose that we have a data set wth 5 samples as show Fg. 5, whch are geerated from oe of Archmedea copulas (true model whch s gve as θt θt C( u, v = l( exp( u + exp( v e θt where the geerator s ϕ θ ( t = exp( t e, ad t t / θt (3 θ s the correlato parameter of the true model, whch s.4. The the correspodg Kedall s tau ca be calculated as.36 for gve θ t =.4 usg Eq. (9. Two radom varables ad are supposed to have stadard ormal dstrbuto u =Φ ( x ad v =Φ ( x. Whe 5 pars of data are used, Kedall s tau s calculated as τ =.35 ad the mea values of ad are.85 ad.45, stadard devatos of ad are estmated as.8 ad.7, respectvely. 4 3 x - - -3-4 -4-3 - - 3 4 x Fgure 5. 5 Pars of Data
To characterze the put varables, we eed to select a copula whch best descrbes the data gve Fg. 5. Of course, f the Archmedea copula Eq. (3 s selected, the the true copula s selected. However, for ths gve data set, we could select Clayto copula, whch s oe of Archmedea copulas, because the PDF cotour usg Clayto copula s smlar wth the data set as show Fg. 5. Of course, t s clear that aussa copula caot descrbe the data set Fg. 5. Thus, suppose we have selected the followg Clayto copula as a jot PDF, θ / c c θ θ c C( u, v = u + v for θc (3 where the geerator s ( t ( t θ c ϕ θ = ad θ c s the correlato parameter of Clayto copula. I Clayto copula, there θ s a relatoshp betwee θ c ad Kedall s tau as τ θc = τ (33 If the data are fte, the correct correlato parameter for Clayto copulaθ c s.968, whch s obtaed from Eq. (33 usg the true Kedall s tau τ =.36. If Kedall s tau s calculated from the 5 pars of data,.e., τ =.35, the the correlato parameter of Clayto copula s θ c =.996 from Eq. (33. Usg the estmated correlato parameter θ c, Clayto copula ca be obtaed. As stated Secto 6., Nataf trasformato, aussa copula s used to descrbe the data. Hece, t s terestg to verfy how accurately aussa copula descrbe the data wth o-symmetrc assocato by comparg wth other o-aussa copulas. aussa copula s defed as ( ( ( ( ( ( Φ u Φ v s ρst+ t C u, v = N Φ u, Φ v = exp dsdt π ρ ( ρ (34 I aussa copula, Pearso s rho ρ s calculated from the data usg Eq. (6. If the data are fte, the correct Pearso s rho would have bee.497. However, whe 5 pars of data are used, Pearso s rho s calculated as ρ =.56. Usg the estmated Pearso s rho, aussa copula s obtaed. Fgure 6 shows that the PDF cotour usg true model; ad Clayto copula ad aussa copula, wth fte data ad 5 data, respectvely. As see Fg. 6, PDF cotour usg Clayto copula s smlar wth the true model, whch fts well wth the data Fg. 5. O the other had, the jot PDF cotour usg aussa copula s qute dfferet from the true copula. The dfferece ca be clearly show by comparg CDFs usg each copula as see Fg. 7 (a ad (b. As expected, the CDF usg Clayto copula s close to the oe usg true model whereas the CDF usg aussa copula s dfferet from oe usg true model especally at the left tal whch s a mportat rego to estmate falure rate. The dffereces of CDFs from these copulas ca be clearly show f CDFs are magfed the rego x = 3. to. as show Fgs. 8 (a ad (b. Cosder a specfc CDF value F( x, x =.35% (.e., β t = 3. or 3-σ desg, whch s dcated by the horzotal les Fgs. 8 (a ad (b. Whe 5 data sets are used, Clayto copula ad aussa copula detfy x =.89 ad x =.4, respectvely for the CDF value of.35%. Whe these values are compared wth the true value x =.96, the result from Clayto copula s deed better. Whe data sets are fte, Clayto copula ad aussa copula detfy x =.78 ad x =.3 respectvely for the CDF value of.35%. Thus, the result of Clayto copula s better. From ths observato, f we used Clayto copula to characterze the put dstrbuto, we wll fd the optmum pot that s ear the true optmum desg RBDO whle aussa copula fals. The RBDO results usg dfferet copulas are show Secto 7. 3 True model 3 Clayto copula 3 aussa copula x x x - - - - - - -3-3 -3-3 - - 3-3 - - 3-3 - - 3 x x x (a True Copula (b Clayto Copula Usg Ifte Data (c aussa Copula Usg Ifte Data
(d aussa Copula Usg 5 Data (e Clayto Copula Usg 5 Data Fgure 6. PDF Cotours of Copulas (a (b Fgure 7. CDFs Usg Dfferet Copulas: (a Estmated Margal ad Correlato Parameters (5 Samples ad (b Exact Margal ad Correlato Parameters 3 x -3.5 CDF usg true model CDF usg Clayto copula CDF usg aussa copula 3 x -3.5 CDF usg true model CDF usg Clayto copula CDF usg aussa copula F(x,x.5 F(x,x.5.5.5-3. -3 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -. -. - -3. -3 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -. -. - x =x (a x =x (b Fgure 8. CDFs Usg Dfferet Copulas for the Rage from x = 3. to. Usg (a 5 Data ad (b Ifte Data 7. New Trasformato As stated Secto 6, f the data follow o-aussa copula, Nataf trasformato that uses aussa copula may geerate correct jot PDF, whch could lead to a correct RBDO result. For correlated put varables wth o-aussa jot dstrbuto, a o-aussa copula whch provdes best ft to the data eeds to be selected.
7.. New Trasformato Usg Copulas Nataf trasformato trasforms the orgal varables to the correlated stadard ormal varables usg aussa copula, ad the trasforms correlated stadard ormal varables to depedet stadard ormal varables usg lear trasformato that s the same as Roseblatt trasformato for correlated ormal varables. Lkewse, for o-aussa copula, oce a copula whch captures the data s selected, ad the Roseblatt trasformato ca be used to trasform to the ormalzed aussa jot dstrbuto. I relablty aalyss, sce the sestvtes of the costrats wth respect to the stadard ormal varables s requred, we eed to take dervatves of Eq. (. The sestvty of the costrat wth respect to u j s xk = = J kj (35 uj k= xk uj k= xk x f ( x,, x k k φ ( uk = for k = j uk f ( x,, xk where J kj = f ( x,, x k x F( xk x,, x k k k x = for k > j uj f ( x,, xk = j x uj ad each compoet of Jacoba matrx J kj ca be calculated usg dervatves of Eq. (. Thus, oce the explct formulato of jot PDF ad CDF (.e., copula s gve, Eq. (35 ca be used to carry out relablty aalyss. I the followg example, the depedet coupla, aussa coupla, ad Clayto copula are used for comparso RBDO. 7. Numercal Example Cosder the RBDO formulato of a two dmesoal mathematcal example, m. cost(d = d+ d s.t. P( ( Φ( βt, =,,3 d, d, βt = 3. ( = / ( = ( + 5 / 3 ( / ( = 8/ + 8 + 5 ( 3 Assume that the true put model s Archmedea copula gve Eq. (3 wth θ t =.4 (τ =.36. As PDF cotours ad CDFs usg dfferet copulas are show Fgs. 6, 7, ad 8 for 5 sample data ad fte data, RBDO results usg these copulas are compared. I ths example, sce put varables has ormal dstrbuto,, ~ N ( 5,.3,the data geerated from stadard ormal dstrbutos Secto 6. ca be learly trasformed to the data wth ormal dstrbutos. Ths s to use same CDFs ad PDF cotours for a gve data because CDFs ad PDF cotours are ot chaged durg lear trasformato. Thus, from the ew data, Kedall s tau s τ =.35 ; Pearso s rho s ρ =.56 ; ad mea values ad stadard devatos of ad are μ = 4.975, μ = 4.987, σ =.3, ad σ =.3 respectvely. As show Fgs. 6, 7, ad 8, sce PDF cotours ad CDFs are smlar for 5 data ad fte data, RBDO results are close as show Fg. 9. Fgures 9 (a ad (b shows the optmal desg pots ad target cotours ( u = βt of the depedet copula, aussa copula, Clayto copula, ad true model, for 5 data ad fte data, respectvely. As ca be see Fgs. 9 (a ad (b, the optmum desg obtaed usg Clayto copula s close to the optmum desg obtaed usg the true put model, whch s also cofrmed Tables ad. O the other had, f aussa copula s used, the RBDO result s far from the optmum desg obtaed usg the true copula. Moreover, whe depedet copula s used,.e., whe t s assumed that two put varables are depedet, the RBDO results are very dfferet from the optmum desg obtaed usg the true copula. Thus, the put correlato, as well as the copula types, sgfcatly affects the RBDO results. (36
(a (b Fgure 9. RBDO Results Usg Dfferet Copulas wth (a 5 Data ad (b Ifte Data Copula types Table. RBDO Results Usg Dfferet Copulas wth 5 Data Optmum desg pot Cost Actve costrats Idepedet copula 3.443 3.95 6.738, aussa copula 3.69 3.93 6.885, Clayto copula 3.89 3.6 6.998, True copula 3.933 3.7 7.3, Copula types Table. RBDO Results Usg Dfferet Copulas wth Ifte Data Optmum desg pot Cost Actve costrats Idepedet copula 3.439 3.87 6.76, aussa copula 3.67 3.9 6.863, Clayto copula 3.868 3.6 6.984, True copula 3.933 3.7 7.3, 8. Dscussos ad Cocluso I ths paper, a RBDO method that deals wth the correlato of put varables usg aussa copula ad o-aussa copula s proposed. For ths, two represetatve trasformato methods, Roseblatt trasformato combed wth o-aussa copula such as Clayto copula ad Nataf trasformato usg aussa copula, are vestgated for applcablty to RBDO problems wth correlated put varables. Roseblatt trasformato s a mathematcally exact trasformato method, but t has lmted applcatos sce the trasformato of orgal radom varables to stadard ormal varables ca be carred out oly whe a jot CDF or codtoal CDFs are avalable. Thus, t s ot applcable to practcal applcatos where oly the covarace matrx ad margal dstrbuto are avalable. O the other had, Nataf trasformato, whch uses aussa copula the copula famly, s foud to be practcally applcable where oly covarace matrx ad margal dstrbutos are vald. Nataf trasformato ca costruct a exact jot CDF whe the put varables are ormal or whe the ormal ad logormal varables are combed. Further, whe the put varables are logormal, t ca accurately costruct a jot CDF for postve or eve some egatve correlatos. However, Nataf trasformato s ot applcable for o-aussa correlated varables. I ths paper, a ew trasformato method that combes Roseblatt trasformato wth o-aussa copula s proposed for RBDO of problems wth o-aussa
correlated put varables to demostrate that the correlato ad jot dstrbuto types of correlated put varables sgfcatly fluece the optmum results of RBDO. 9. Ackowledgemet Research s supported by the Automotve Research Ceter that s sposored by the U.S. Army TARDEC.. Refereces. ou, B.D., Cho, K.K., ad Du, L., Adaptve Probablty Aalyss Usg A Ehaced Hybrd Mea Value Method, Structural ad Multdscplary Optmzato, Vol. 9, No., pp. 34-48, 5.. Roseblatt, M., Remarks o a Multvarate Trasformato, The Aals of Mathematcal Statstcs, Vol. 3, pp. 47-47, 95. 3. Lu, P-L. ad Der Kuregha, A., Multvarate Dstrbuto Models wth Prescrbed Margals ad Covaraces, Probablstc Egeerg Mechacs, Vol., No., pp. 5-, 986. 4. Melchers, R.E., Structural Relablty Aalyss ad Predcto, d edto, J. Wley & Sos, New ork, 999. 5. Nataf, A., Détermato des dstrbutos de probabltés dot les marges sot doées, Comptes Redus Hebdomadares des Séaces de l Académe des Sceces, Vol. 55, pp. 4-43, 96. 6. Dtlevse, O. ad Madse, H.O., Structural Relablty Methods, J. Wley & Sos, New ork, 996. 7. Roser B.N., A Itroducto to Copulas, Sprger, New ork, 999. 8. Tobas, P.A. ad Trdade, D.C., Appled Relablty, d edto, CRC Press, 995. 9. Hah,.J. ad Shapro, S.S., Statstcal Models Egeerg, Joh Wley & Sos, 994.. Cho, K.K., Noh,., Du, L., Relablty Based Desg Optmzato wth Correlated Iput Varables, 7 SAE World Cogress, Aprl 6-9, 7, Detrot, MI.. Bravas, A., Aalyse Mathematque sur les Probabltes des Erreurs de Stuato d'u Pot, Memores par dvers Sava, 9, 55-33, 846.. Pearso, K., Mathematcal cotrbutos to the theory of evoluto. III. Regresso, heredty ad pamxa, Phlos. Tras. Royal Soc. Lodo Ser. A, Vol. 87, pp. 53 38, 896. 3. Spearma, The proof ad measuremet of assocato betwee two thgs, The Amerca Joural of Psychology, Vol. 5, pp. 7-, 94. 4. Kedall, M., A New Measure of Rak Correlato, Bometrka, Vol. 3, pp. 8-89, 938. 5. Kruskal, W. H., Ordal Measures of Assocatos, Amercal Statstcal Assocato Joural, Vol. 53, No. 84, pp. 84-86, 958. 6. Joe, H., Multvarate Models ad Depedece Cocepts, Chapma & Hall, Norwell, 997. 7. Hoag Pham, Sprger hadbook of egeerg statstcs, Sprger, Lodo, 6. 8. Madse, H.O., Krek, S, ad Ld N.C., Methods of Structural Safety, Pretce-Hall, New Jersey, 986. 9. Hoag Pham, Recet Advaces Relablty ad Qualty Egeerg, World Scetfc, Sgapore,.