Reliability Engineering and System Safety

Transcription

1 Relablty Egeerg ad System Safety 96 (20) Cotets lsts avalable at SceceDrect Relablty Egeerg ad System Safety joural homepage: The detfcato of model effectve dmesos usg global sestvty aalyss Serge Kuchereko a,, Balazs Fel b, lay Shah a, Wolfgag Mautz c a CPSE, Imperal College Lodo, South Kesgto Campus, Lodo SW7 2A, UK b Departmet of Process Egeerg, Uversty of Paoa, Veszprem, Hugary c Lehrstuhl für Alagesteuerugstechk, Fachberech Chemetechk, Uverstät Dortmud, Germay artcle fo Artcle hstory: Receved 23 Jauary 200 Receved revsed form 9 ovember 200 Accepted 8 ovember 200 Avalable ole 25 ovember 200 Keywords: Mote Carlo tegrato Quas-Mote Carlo Global sestvty aalyss Sobol sestvty dces Effectve dmesos abstract It s show that the effectve dmesos ca be estmated at reasoable computatoal costs usg varace based global sestvty aalyss. amely, the effectve dmeso the trucato sese ca be foud by usg the Sobol sestvty dces for subsets of varables. The effectve dmeso the superposto sese ca be estmated by usg the frst order effects ad the total Sobol sestvty dces. The classfcato of some mportat classes of tegrable fuctos based o ther effectve dmeso s proposed. It s show that t ca be used for the predcto of the QMC effcecy. Results of umercal tests verfy the predcto of the developed techques. & 200 Elsever Ltd. All rghts reserved.. Itroducto Moder mathematcal models of real systems physcs, chemstry, bology, ecoomcs ad other areas ofte have hgh complexty wth hudreds or eve thousads of varables. Straghtforward modellg usg such models ca be computatoally costly or eve mpossble. There s a demad for complexty reducto techques whch are ot oly geeral ad applcable to ay complex o-lear model but also rgorous that ther applcato provdes estmates of the approxmato errors. Varace based global sestvty aalyss allows to develop such complexty reducto techques. Recetly a ew class of measures was troduced by Borgoovo [,2]. These measures are kow as momet-depedet. They are based o the etre dstrbuto of the output wthout a specfc referece to ts momets. Potetally, momet-depedet measures ca also be used for complexty reducto. For modellg ad complexty reducto purposes t s mportat to dstgush betwee the model omal dmeso ad ts effectve dmeso. The otos of the effectve dmeso the trucato ad superposto sese was troduced by Caflsch et al. [3]. Qute ofte complex mathematcal models have effectve dmesos much lower tha ther omal dmesos. The kowledge of model effectve dmesos s very mportat as t allows to apply varous complexty reducto techques. Correspodg author. Tel.: ; fax: E-mal address: s.kuchereko@c.ac.uk (S. Kuchereko). The effectve dmeso the trucato sese d T loosely speakg s equal to the umber of mportat factors the model. Idetfcato of mportat ad ot mportat varables allows to fx ot mportat varables at ther omal values. The resultat model would have lower complexty wth dmesoalty reduced from to d T. A codto d T 5 ofte occurs practcal problems. Aother type of complexty reducto s assocated wth the effectve dmeso the superposto sese d S : the fucto has the effectve dmeso the superposto sese d S f t s almost a sum of s-dmesoal fucto compoets the AOVA decomposto. For some problems such as path-depedet opto prcg mathematcal face chagg the order whch put varables are sampled ca dramatcally decrease d T. Such techques are kow as dmeso reducto. Most results o dmeso reducto are emprcal ad qualtatve (see for example [3]). A straghtforward evaluato of the effectve dmesos from ther deftos s ot practcal the geeral. Owe troduced the dmeso dstrbuto for a square tegrable fucto [4]. The effectve dmeso ca be defed through a quatle of the dmeso dstrbuto. He showed that for some classes of fuctos quatles of the dmeso dstrbuto ca be explctly calculated but they are dffcult to estmate a geeral case. I ths paper we show that global sestvty aalyss based o the Sobol sestvty dces (SI) allows to estmate the effectve dmesos at reasoable computatoal costs. Evaluato of the Sobol SI ecesstates the computato of hghdmesoal tegrals. The classcal grd methods become computatoally mpractcal whe the umber of dmesos creases /$ - see frot matter & 200 Elsever Ltd. All rghts reserved. do:0.06/j.ress

2 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) becauseof thecurseofdmesoalty.thecovergecerateof Mote Carlo (MC) tegrato methods does ot deped o the umber of dmesos. However, the rate of covergece Oð =2 Þ,where s the umber of sampled pots, attaed by MC s rather slow. A hgher rate of covergece ca be obtaed by usg quas-mote Carlo (QMC) methods based o uformly dstrbuted sequeces stead of pseudo-radom umbers. Asymptotcally, QMC ca provde the rate of covergece O( ). For suffcetly large, QMC should always outperform MC. However, practce such sample szes qute ofte are feasble, especally whe hgh-dmesoal problems are cocered. May umercal expermets demostrated that the advatages of QMC ca dsappear for hgh-dmesoal problems. There were clams that the degradato performace of QMC occurs at 2 [5].I cotrast, other papers reported the superorty of QMC over MC for some tegrads wth ¼ 360 [6]. Some explaatos for such cosstet results were gve usg the oto of the effectve dmeso [3].I[7] t was show how the AOVA compoets are lked to the effectveess of QMC tegrato methods. Sloa ad Wozakowsk [8] studed the effcecy of the quas-mote Carlo algorthms for hgh-dmesoal tegrals. They detfed classes of fuctos for whch the effect of the dmeso s eglgble. These are the so-called weghted classes whch the behavor the successve dmesos s moderated by a sequece of weghts. There s o computatoally feasble techque that would predct the effcecy of QMC hgh dmesos. I ths paper we use Sobol SI as a quattatve measure of the QMC effcecy. Ths paper s orgazed as follows. Secto 2 brefly descrbes MC ad QMC tegrato algorthms ad ssues cocerg the possble degradato of QMC effcecy hgher dmesos. Secto 3 gves a descrpto of the Sobol SI. Secto 4 presets mproved formulas for evaluato of the Sobol SI. The oto of the effectve dmeso s troduced Secto 5. The classfcato of fuctos based o Sobol SI s suggested Secto 6. It s show how ths classfcato ca be used for the predcto of the QMC effcecy. Test examples ad umercal results are cosdered Secto 7. Fally, coclusos are gve Secto MC ad QMC algorthms Cosder the evaluato of a tegral I½ f Š¼ f ðxþ dx, H where the fucto f ðxþ s tegrable the -dmesoal ut hypercube H ad suffcetly regular. The Mote Carlo quadrature formula s based o the probablstc terpretato of a tegral. A approxmato to ths expectato s I ½f Š X f ðx Þ, where fx g s a sequece of radom pots H of legth. The approxmato I [f] coverges to I[f] wth probablty. Cosder a tegrato error e defed as e ¼ I½ f Š I ½ f Š : The expectato of e 2 s Eðe 2 Þ¼ s2 ðf Þ, where s 2 ðf Þ s the varace. The root mea square error of the MC method s e MC ¼ðEðe 2 ÞÞ =2 ¼ sðf Þ =2 : I cotrast to grd methods, the covergece rate of MC methods does ot deped o the umber of varables although t s rather slow. The effcecy of MC methods s determed by the propertes of the radom umbers. Radom umber samplg s proe to clusterg. As ew pots are added radomly, they do ot ecessarly fll the gaps betwee already sampled pots. I cotrast, lowdscrepacy sequeces (LDS) are specfcally desged to place sample pots as uformly as possble. The dscrepacy s the measure of devato from uformty. Cosder a umber of pots from a sequece fx g a -dmesoal rectagle Q whose sdes are parallel to the coordate axes, Q AH. The, the dscrepacy s defed as D ¼ sup Q Q A H mðqþ, where m(q) s a volume of Q ad Q s the umber of pots of the sequece fx g that are cotaed Q. The Koksma Hlawka equalty gves a upper boud for the QMC tegrato error: e QMC rvðf ÞD : ðþ Here, V(f) s the varato of f ðxþ the sese of Hardy ad Krause [9]. For a oe-dmesoal fucto wth a cotuous frst dervatve t s smply Vðf Þ¼ jdf ðxþ=dxj dx: ð2þ H I hgher dmesos, the Hardy-Krause varato may be defed terms of the tegral of partal dervatves. Further t s assumed that f ðxþ s a fucto of bouded varato. For radom umbers, the expected dscrepacy s D ¼ OððllÞ= =2 Þ, whle the dscrepacy of LDS s of the order D ¼ O log ðþ : ð3þ There are a few well-kow ad commoly used LDSs. Dfferet prcples were used for ther costructo by Halto, Faure, Sobol, ederreter ad others. The LDS developed by ederreter has the best theoretcal asymptotc propertes [9]. However, may practcal studes have prove that the Sobol LDS s may aspects superor to other LDS [6,0]. The Sobol LDS was costructed by followg the three ma requremets []:. Best uformty of dstrbuto as Good dstrbuto for farly small tal sets. 3. A very fast computatoal algorthm. Pots geerated by the Sobol LDS produce a very uform fllg of the space eve for a rather small umber of pots, whch s a very mportat pot practce. The boud o the tegrato error () s a weak oe ad s ot partcularly meagful practce. It was show expermetally that the QMC tegrato error s determed by the varace ad ot by the varato of the tegrad [2]. It s geerally accepted that the rate of the dscrepacy determes the expected rate of the accuracy, so oe ca use a estmate of the QMC covergece rate e QMC ¼ O log ðþ : ð4þ Asymptotcally, ths rate of covergece s O( ). umerous computatoal studes showed that QMC methods ca provde sgfcat mprovemet over MC. The aalyss of (4) shows that e QMC s a creasg fucto of up to some threshold value of, expðþ. The accelerated covergece rate O( ) sets at

3 442 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) For hgh-dmesoal problems such a large umber of sample pots s feasble. Ths s oe of the reasos why practce the advatages of usg QMC ca dsappear at hgh ad eve moderate values of. The study of some test problems [5] led ts authors coclude that eve for problems of moderate dmesoalty (42) QMC offers o practcal advatage over MC. Other authors [3,4] also reported the degradato of performace of QMC hgh dmesos ad for a dscotuous fucto eve low dmesos. I cotrast, [6] t was foud that for some hgh-dmesoal tegrads ( ¼ 360), QMC sgfcatly outperformed MC. These results were later cofrmed other papers (e.g. [3,4,5]). Sobol oted that a error boud () wth D gve by (3) was obtaed wth the assumpto that the fucto f ðxþ depeds equally o all varables [6]. I practcal applcatos may fuctos qute ofte strogly deped oly o a small subset of varables: x,x 2,...,x s,r o 2 o o s,so whle the depedece o other varables ca be weak. I ths case, ca be substtuted by s (4). Ths cosderato s based o a very mportat property of LDS: the projecto of the -dmesoal LDS o the s-dmesoal subspace forms the s-dmesoal LDS. That meas partcular that the accelerated covergece rate of the QMC tegrato ca set at lower values of 4, expðsþ.it s mportat to ote that practce low-dmesoal projectos have good uform dstrbutos, whle hgh dmesos LDS are ot partcularly well equdstrbuted for feasble. The effects o the covergece of certa propertes of tegrads cludg varace, varato, smoothess ad dmeso were studed [4]. It was foud that the varato does ot affect the covergece, whle the varace provdes a rough upper boud, but t does ot accurately predct the performace. Caflsch et al. [3] troduced the oto of a effectve dmeso. It was suggested that QMC s superor to MC f the effectve dmeso of a tegrad s ot too large. The oto s based o the Aalyss Of VAraces (AOVA). I [7] t was show how the AOVA compoets are lked to the effectveess of QMC tegrato methods. Owe [4] troduced the dmeso dstrbuto for square tegrable fuctos ad showed how t s lked wth Sobol SI [7]. Further detals are gve Secto Global sestvty dces May practcal problems deal wth fuctos of a very complex structure. Global sestvty aalyss (SA) ca provde formato o the geeral structure of a fucto by quatfyg the varato the output varables to the varato of the put varables. The method of global SA s superor to the local SA methods such as regresso aalyss, rak trasformato, etc. as t s geeral ad ca be appled to both lear ad hghly o-lear fuctos [8]. Oe of the most effcet global SA techques s based o the Sobol SI [7]. Ths techque provdes a uambguous formato o the mportace of dfferet subsets of put varables to the output varace. Cosder a tegrable fucto f ðxþ defed the ut hypercube H. It ca be expaded the followg form: f ðxþ¼f 0 þ X X s s ¼ o o s f...s ðx,...,x s Þ: Ths expaso s a sum of 2 compoets. It ca also be preseted as f ðxþ¼f 0 þ X f ðx Þþ X o j f j ðx,x j Þþþf 2... ðx,x 2,...,x Þ: Each of the compoets f ::s ðx,...,x s Þ s a fucto of a uque subset of varables from x. The compoets f (x ) are called frst order terms, f j (x,x j ) the secod order terms ad so o. ð5þ It ca be prove [7] that the expaso (5) s uque f f ðx H ::s,...,x s Þ dx k ¼ 0, rkrs, ð6þ whch case t s called a decomposto to summads of dfferet dmesos [9]. Ths decomposto was troduced [20,9]. Later t became kow as the AOVA decomposto. The AOVA decomposto s orthogoal,.e. for ay two subsets uaw a er product f u ðxþf w ðxþ dx ¼ 0: ð7þ H It follows from (5) ad (6) that f ðxþ dx ¼ f 0, H f ðxþ Y dx k ¼ f 0 þf ðx Þ, H k a f ðxþ Y dx k ¼ f 0 þf ðx Þþf j ðx j Þþf,j ðx,x j Þ ð8þ H k a ð,jþ ad so o. For square tegrable fuctos, the varaces of the terms the AOVA decomposto add up to the total varace of the fucto s 2 ¼ X X s ¼ o o s s 2 ::s, where s 2 ::s ¼ R H f 2 ::s ðx,...,x s Þ dx...dx s. Sobol defed the global SI as the ratos S ¼ s2 ::s ::s s : 2 All S are o-egatve ad add up to oe ::s X X s ¼ o o s S ::s S ::s ¼ : ca be vewed as a atural sestvty measure of a set of varables x,...,x s. It correspods to a fracto of the total varace gve by f ðx ::s,...,x s Þ. For example, S s the ma effect of avarablex, S 2 s a measure of teractos betwee x ad x 2 (.e. that part of the total varace due to parameters x ad x 2 whch caot be explaed by the sum of the effects of parameters x ay x 2 ) ad so o. For fuctos of a addtve structure, oly the low-order SI are mportat. I a extreme case whch there s o teracto amog the put varables, f ðxþ¼f 0 þ X f ðx Þ all hgher order SI are equal to zero. Thus, X S : Ths case s very mportat for the uderstadg of the performace of QMC tegrato. It wll be cosdered the followg secto. I the geeral case, all SI ca be mportat for SA. Ther straghtforward calculato usg the AOVA decomposto would result 2 tegral evaluatos of the summads f ðx ::s,...,x s Þ usg (8) ad 2 tegral evaluatos for calculatos of s 2 (9). For ::s hgh-dmesoal problems such a approach s mpractcal. For ths reaso Sobol troduced the SI for subsets of varables. Cosder two complemetary subsets of varables y ad z: x ¼ðy,zÞ: ð9þ

4 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) Let y ¼ðx,...,x m Þ,r o o m r,k ¼ð,..., m Þ.Thevarace correspodg to y s defed as s 2 y ¼ Xm s 2 y X s ¼ ð o o sþ A K s 2,...,s : cludes all partal varaces s2,s 2 2,...,s 2 such that ther ::s subsets of dces ð,..., s ÞAK. The varace s 2 z s defed smlarly. The total varace ðs tot y Þ2 s defed as ðs tot y Þ2 ¼ s 2 s 2 z : ðs tot y Þ2 cossts of all s 2 such that at least oe dex ð pþak whle ::s the remag dces ca belog to the complmetary set K.The correspodg global SI are defed as S y ¼ s2 y s 2, y ¼ ðstot y Þ2 : s 2 Obvously y ¼ S z. y S y accouts for all teractos betwee y ad z. The total sestvty dces were troduced by Homma ad Saltell [2]. The mportat dces practce are S ad S tot. Ther kowledge most cases provdes suffcet formato to determe the sestvty of the aalyzed fucto to dvdual put varables. The use S ad S tot reduces the umber of dex calculatos from O(2 ) to just O(2). Extreme cases are ¼ 0 meas that f ðxþ does ot deped o x ( ths case S s also equal to 0); S meas that f ðxþ depeds oly o x ( ths case s also equal to ); S ¼ correspods to the absece of teractos betwee varable x ad other varables. It wll be show the ext secto how ths case relates to the effcecy of QMC tegrato. 4. Orgal ad mproved formulas for evaluato of SI Oe of the most mportat results obtaed by Sobol s a effectve way of computg SI. Gve x ad xu beg two depedet sample pots, where x ¼ðy,zÞ ad xu ¼ðyu,zuÞ, S y ad y are calculated usg the followg formulae [22]: R 0 S y ¼ f ðxþf ðy,zuþ dx dzu f 0 2 R 0 f, ð0þ 2 ðxþ dx f0 2 y ¼ 2 R H ½f ðxþ f ðyu,zþš 2 dx dyu R 0 f : ðþ 2 ðxþ dx f0 2 I the geeral multdmesoal case, the tegrals (0) ad () are evaluated usg MC or QMC methods. Formulae (0), () are based o geeratg two depedet sample pots x ¼ðy,zÞ, xu ¼ðyu,zuÞ ad evaluatg the three fuctos f ðxþ,f ðy,zuþ,f ðyu,zþ. I ths case a Mote Carlo algorthm for (0) has a form P h f ðy,zþf ðy,zuþ P 2 f ðy,zþ S y P h f 2 ðy,zþ P 2 : ð2þ f ðy,zþ The exteded verso of the Sobol method preseted by Saltell [23]. It has a addtoal advatage of the reduced cost of evaluatg S y ad S y tot. amely, for calculato of all oe-dmesoal dces t uses (+2) model evaluato rather tha (2+) for the orgal Sobol formulas. Moreover, t was show [23] that these (+2) model evaluatos ca be used for computg all twodmesoal dces. The exteded verso s based o usg a dfferet set of fucto values, amely f ðxþ,f ðxuþ,f ðy,zuþ. Oe ca otce that for less mportat varables values of the terms omator of (2) ca be very close. It ca result the sgfcat loss of accuracy. Stuato ca be mproved usg modfed formula for S y. It s easy to see that f0 2 ¼ðR H f ðxþ dxþ 2 ¼ ð R H f ðxþ dxþð R H f ðxuþ dxuþ. Hece, formula (0) ca be rewrtte as R H f ðxþf ðy,zuþ dx dzu ð R S y H f ðxþ dxþð R H f ðxuþ dxuþ R : ð3þ H f 2 ðxþ dx f 2 0 Ths expresso ca be reformulated as R H f ðxþ½f ðy,zuþ f ðxuþš dx dxu S y R : ð4þ H f 2 ðxþ dx f 2 0 The correspodet Mote Carlo algorthm has a form S y P f ðy,zþ½ f ðy,zþ f ðyu,zuþš h 2 : ð5þ P f 2 ðy,zþ P f ðy,zþ The mproved formula (5), whch was suggested by Kuchereko [24], s based o the same set of fucto values as the exteded verso suggested by Saltell [23]. A comparso of the orgal ad mproved formulas preseted [24,25] shows that for small value dces the mproved formula produces a few orders of magtude more accurate results. A comprehesve dscusso of computato of S y tot ca be foud [36]. 5. Effectve dmesos The AOVA decomposto was used for the troducto of a oto of the effectve dmeso [3]. Let L ¼ f,2,...,g ad jyj be a cardalty of a set ydl. Defto. The effectve dmeso of f the superposto sese s the smallest teger d S such that X 0 o jyj o d S S y p, ð6þ where p s the threshold, 0opo. Codto (6) meas that the fucto f s almost a sum of d S -dmesoal fuctos. The effectve dmeso d S s the order of the hghest effect oe eeds to clude the sum P 0 o jyj o d S S y order to reach the target p. Aother oto of the effectve dmeso was mplctly troduced [6]. I[3] t was called the effectve dmeso the trucato sese. Defto 2. The effectve dmeso of f the trucato sese s the smallest teger d T such that X S y p: ð7þ 0 o y D f,2,...,dt g I other words, the effectve dmeso d T s the hghest umber of varables, whch eed to be cluded the sum P 0 o y D f,2,...,dt g S y order to reach the target p. The value d S does ot deped o the order whch the put varables are sampled, whle d T does. For the same pd S rd T. It was suggested that the effcecy of QMC methods o hghdmesoal problems ca be attrbuted to the low effectve dmeso of the tegrad ( oe or both of the seses), although o formal proof was gve [3]. By reducg the effectve dmeso, a hgher effcecy of QMC tegrato ca be acheved. Oe example of such a approach s a smulato drve by Browa moto. It was show that by chagg the order whch the

5 444 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) varables are sampled from the LDS the effectve dmeso ca be reduced ad thus the accuracy ca be sgfcatly mproved [3]. A straghtforward evaluato of the effectve dmeso from ts deftos (6), (7) s ot practcal the geeral case as t would requre the calculato of all 2 compoets s 2 y ðf yþ. A quasregresso method suggested [7] s less computatoally expesve. A trucated orthogoal decomposto based o orthogoal polyomals s used for a drect estmato of s 2 y ðf yþ. The method allows the separato of hgh ad low order subcompoets of f y. The lower frequecy or smoother parts of the AOVA compoets of a tegrad f are kow to be related to the accuracy of tegrato rules appled to f. Ths method s stll dffcult to use for the predcto of QMC effcecy ad there are some uresolved umercal ssues such as the possblty of the egatve varace estmates. Owe troduced a probablty measure mðyþ o o-empty subsets ydf,...,sg, whch mðyþ s proportoal to the varace cotrbuto to f of the subset u of put varables of f [4]. IfU s a radom m dstrbuted subset, the ts cardalty, deoted juj, sa radom varable. The dstrbuto ðþ of the radom varable juj s the dmeso dstrbuto of f. The effectve dmeso ca be defed through a quatle of the dmeso dstrbuto ðþ. Although such quatles are hard to estmate, Owe cosdered several cases of addtve ad multplcatve test fuctos for whch such quatles ca be explctly calculated. Owe also troduced the oto of mea dmesos. A smlar cocept was also suggested ad used [26]. Oe of the advatages of usg mea dmesos s that they do o deped o the arbtrary threshold level p. The mea dmeso was computed for some commoly cosdered test fuctos. It was show that may of these fuctos are sums or products of uvarate fuctos ad have very low effectve dmeso. To aalyze a class of sotropc test fuctos troduced by Capstck ad Kester [27], Owe lked Sobol SI wth the dmeso dstrbuto. It allowed hm to show umercally that the fucto classes uder cosderato are fact very early a superposto of fuctos of 3 or fewer varables. Owe also otced that low effectve dmeso s ot suffcet to state that QMC wll be more effcet tha MC for dscotuous fuctos or fuctos wth spkes such as some of Gez s fuctos. Ths observato s accordace wth earler fdgs made [4]. The set of varables z ca be regarded as ot mportat f z 5. I ths case t s possble to fx a value of z at some omal pot z 0 ad to use f ðy,z 0 Þ as a approxmato to f ðxþ. The approxmato error depeds o the choce of z 0 : dðz 0 Þ¼ ½f ðxþ f ðy,z s 0 ÞŠ 2 dx: ð8þ The followg theorem shows that dðz 0 Þ s of the same order as z. Theorem. For a arbtrary z 0 the error dðz 0 Þ z. If z 0 s assumed to be radom ad uformly dstrbuted, the the expected value s Edðz 0 Þ¼2 z : ð9þ Proof of ths theorem ca be foud [25]. A corollary of the theorem s the followg asserto from [7]: for a arbtrary e40 wth probablty exceedg e dðz 0 Þo þ S e tot z : ð20þ Cosder set y ¼ðx,...,x d Þ,rdr ad a complmetary set z ¼ðx d þ,...,x Þ. Usg equalty z ¼ S y ad (7) for d T ¼ d t s easy to see that z r p, ð2þ hece Edðz 0 Þr2ð pþ: ð22þ 6. Classfcato of fuctos based o Sobol SI Fuctos wth respect to ther depedece o varables ca broadly be dvded to two categores: fuctos wth ot equally mportat varables ad fuctos wth equally mportat varables. Fuctos wth equally mportat varables accordg to the relatoshp betwee the values of S ad S tot ca be further dvded to two subgroups. Altogether, three dfferet types of fuctos ca be dstgushed: Type A: Fuctos wth ot equally mportat varables. Such fuctos are characterzed by the small effectve dmeso d T (ad small d S because of the codto: d S rd T ). I terms of Sobol SI, ths case ca be wrtte as y y b Stot z : z ð23þ Here y s a group of leadg varables, z s a group of complmetary varables, y, z are the umber of varables groups y ad z correspodgly, z ¼ y. Type B: Fuctos wth domat low-order terms. Such fuctos are characterzed by the small effectve dmeso d S 5. I a extreme case of d S ¼ S ¼, rr: ð24þ As a result X S ad S =. Type C: Fuctos wth domat hgh-order teracto terms. Such fuctos are characterzed by the hgh effectve dmeso d S. For such fuctos S o, rr: ð25þ Ths codto ca also be wrtte as X S o: Ths classfcato s summarzed Table. Type A fuctos are probably the most commo type of fuctos ecoutered practce. For ths case QMC ca atta the rate of covergece Oð a Þ wth a, although the presece of hgh-order teracto terms ca somewhat decrease the covergece rate. Table Classfcato of fuctos based o Sobol sestvty dces. Fucto type Descrpto Relatoshp betwee tot S ad S d T d S QMC s more effcet tha MC A A few domat varables y =y bstot z = z 5 5 Yes B o umportat subsets; mportat low-order teracto terms S S j,8,j S =,8 5 Yes C o umportat subsets; mportat hgh-order teracto terms S S j,8,j S = 5,8 o

6 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) I the AOVA decomposto of type B fuctos, the effectve dmeso d S s small. I the extreme case t s equal to, ad a fucto f ðxþ ca be preseted as a sum of oe-dmesoal fuctos The expresso for F has the geeral form F ¼ X a 2 þ X X ¼ 0 ¼ 0 j o a j j, ð28þ f ðxþ¼ X f ðx Þ: QMC would always outperform MC for type B fuctos rrespectve of the omal dmeso. Although addtve or early addtve tegrads are ot very commo, there are mportat applcato areas such as facal mathematcs where such tegrads are typcal. For type C fuctos both of the effectve dmesos are equal or early equal to a omal dmeso. For ths type of fuctos QMC wll lose ts advatage over MC hgh dmesos because of the mportace of hgh-order terms the AOVA decomposto. The evaluato of all ma ad total effects Sobol sestvty dces for type B ad C fuctos requres (+2) fucto calculatos. Computatoal costs ca be further reduced by usg the RS/QRS-HDMR method whch case the umber of fucto evaluatos s equal to. The detfcato of the effectve dmeso d T for type A fuctos may requre a few teratos before a set of omportat varables z satsfyg codto (2) s foud. 7. umercal results 7.. Path tegrals Cosder the Weer path tegral I ¼ F½xðtÞŠ d x x, C ð26þ where C s the space of all fuctos x(t) cotuous the terval 0rt rt wth a boudary codto xð0þ¼x 0. The tegral (26) ca be regarded as a expectato wth respect to the Weer measure o C, so that I ¼ EðF½xðtÞŠÞ. Here xðtþ s a radom Weer processes (also kow as a Browa moto). A Mote Carlo approach cossts of costructg may radom paths xðtþ, computg F½xðtÞŠ ad averagg the results. We cosder two dscretzato algorthms for radom paths xðtþ geerato. The frst oe s kow as the stadard dscretzato algorthm. It follows drectly from the defto of xðtþ. The secod oe s the alteratve dscretzato algorthm also kow as the Browa brdge. It s based o the use of codtoal dstrbutos. Both algorthms were descrbed [28,29]. The alteratve dscretzato algorthm was later aalyzed [30] wth the framework of the quas-mote Carlo approach. Both algorthms have the same varace, hece ther Mote Carlo accuraces are also the same but the correspodg quas-mote Carlo algorthms have dfferet effceces wth the Browa brdge havg the much hgher covergece rate (although there are fuctoals F[x(t)] for whch the Browa brdge does ot offer a cosstet advatage quas-mote Carlo tegrato [3]). Cosder a fuctoal F½xðtÞŠ ¼ T 0 x 2 ðtþ dt: ð27þ Ths tegral ca be evaluated aalytcally. We assume that the dffuso costat the defto of Weer s measure s 2 ad that boudary value xðtþ¼x 0 s fxed. The terval 0rt rt s dvded to equal parts. It s assumed that ¼ 2 l, l s a teger umber l40. Radom values of the process at the momets of tme t ¼ð=ÞT, rrt are sampled by usg depedet ormal (0;) varable. A cotuous path x(t) s replaced wth a polygoal approxmato x (t), detals ca be foud elsewhere [29]. where are depedet ormal radom varables, a ad a j are coeffcet values whch deped o the type of approxmato for x (t). Applyg global sestvty aalyss t s easy to show that the frst ad secod order SI are gve by a2 S ¼ 2 s 2 ðf Þ, S j ¼ a2 j s 2 ðf Þ, ð29þ whle all hgher order SI are equal to zero. Here s 2 ðf Þ s the varace s 2 ðf Þ¼2 X a 2 þx o ja 2 j : ð30þ s 2 ðf Þ has the same value for both algorthms, so they are equvalet as far as the Mote Carlo method s cocered. The results of the aalytcal evaluato of coeffcets a show for the stadard dscretzato coeffcets a learly decrease wth the dex umber. For the Browa brdge dscretzato sestvty dces of the frst few varables are much larger tha those of the subsequet varables. They also decrease more rapdly tha sestvty dces for the stadard dscretzato. For the Browa brdge the frst two sestvty dces are cosderably larger tha oes for the stadard method. It results partcular the much hgher value of the sum of the frst order sestvty dces P S for the Browa brdge dscretzato tha that for the stadard dscretzato (Table 2). The results show for the stadard approxmato P S decreases wth the crease of the umber of dscretzato pots approxmately as 2=. As a result the mportace of the secod order teractos grows wth. They become domat at 44. I cotrast, for the Browa Brdge approxmato P S s much hgher tha that of the secod order dces P S j ad t s practcally depedet of the umber of dscretzato pots. Table 2 also shows the effectve dmesos. The effectve dmeso the superposto sese s equal to 2 for both approxmatos rrespectve of. The effectve dmeso the trucato sese s estmated usg relatoshp (22) ad values of z.itable 2 the followg otato s used: z (d T ) s a value of z for a set z ¼ðx dt þ,...,x Þ. For the Browa Brdge approxmato d T ¼ 2 for ay ad t belogs to the type A fucto. For the stadard approxmato d T s close to ðd T 3 4 Þ, however because of the small effectve dmeso the superposto sese t belogs to the type B fuctos. Table 2 Sestvty dces for the stadard ad Browa Brdge approxmatos. Idex Fucto Measure umercal values ¼8 ¼32 3A Browa S P Brdge S P approxmato Sj RH x2 ðtþdt z ðd T Þ d T 2 2 d S 2 2 4B Stadard S P approxmato S R P H x2 ðtþdt Sj z ðd T Þ d T 6 22 d S 2 2

7 446 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) It s well kow that the tal low-dmesoal coordates of the low dscrepacy sequeces (LDSs) are more uformly dstrbuted tha the later hgh-dmesoal coordates [9,]. The Browa brdge costructo uses the best coordates from each -dmesoal vector pot to determe most of the structure of a path ad reserves the later coordates to fll fe detals. I other words, the most mportat varables are determed wth the best dmesos of LDSs. It results a sgfcatly mproved accuracy of quas-mote Carlo tegrato. I cotrast, the stadard costructo does ot accout for the specfcs of LDSs dstrbuto propertes. umercal results for the covergece rates preseted [32] cofrm that for the stadard Mote Carlo method there s o dfferece betwee the two dscretzatos. O the other had the Browa brdge dscretzato method wth the Sobol sequece provdes sgfcatly more accurate results tha the stadard dscretzato Test problems commoly used quadrature To test the classfcato preseted above, QMC ad MC tegrato methods were compared cosderg seve dfferet test fuctos preseted Tables 3 5. All fuctos are defed H. Ther tegral values are equal to. Most of the fuctos are kow test fuctos used prevously [33,5,34] ad some other papers for testg QMC tegrato methods. Fuctos 2A, 3B ad 3C were used [35,0] as test fuctos for global sestvty aalyss. The measures S,,S = ad P S were calculated aalytcally. Aalytcal ad umercal results for selected dmesos ( ¼ 2, 0, 00) are preseted Tables 3 5. For each of the cosdered fuctos, the root mea square error e ¼ K X K k ¼ ði½ f Š I k ½ f ŠÞ 2! =2 averaged over 50 rus (K ¼ 50) s preseted Fgs. 3 as a fucto of. For the MC method all rus were statstcally depedet. For QMC tegrato for each ru a dfferet part of the Sobol LDS was used. For practcal purposes, MC ad QMC tegrato errors ca be approxmated as c a : ð3þ The expoets for the expoetal decay a (3) for QMC ad MC tegratos were extracted from the tred les. The tred les ad correspodg values for ð aþ are preseted Fgs. 3. The AOVA decomposto for fucto A has the followg form (for smplcty, a three-dmesoal case s cosdered): fx ð,x 2,x 3 Þ¼ x þx x 2 x x 2 x 3 ¼ f 0 þf ðx Þþf 2 ðx 2 Þþf 3 ðx 3 Þþf,2 ðx,x 2 Þþf,3 ðx,x 3 Þ þf 2,3 ðx 2,x 3 Þþf,2,3 ðx,x 2,x 3 Þ Table 3 Sestvty dces for type A fuctos. Idex Fucto f ðxþ Ref. Measure Aalytcal values umercal values ¼2 ¼0 ¼00 A P ð Þ Q [5] S 2 ð ð x j 2 Þ Þ j ¼ ð 2 Þ þ 3 P 0 ð 3 Þ S [ ] A Q j4x 2jþa, þa a ¼ a 2 ¼ 0, a 3 ¼¼a 00 ¼ 6:52 [35] S P S ðþdþ ð2 Þ ðþcþ 2C þð 2ÞD ðþcþ 2 ðþdþ ð 2Þ C ¼ 3ða þþ,d ¼ 2 3ða 3 þþ Table 4 Sestvty dces for type B fuctos. Idex Fucto f ðxþ Ref. Measure Aalytcal values umercal values ¼2 ¼0 ¼00 B Q x 0:5 [34] S P S A þ 2ð 0:5Þ þ 2ð 2 Þ2 2B þ Q ffffffff [34] S p x P S þ 2 þ h ð 2 þ2þ þ 2 þ 2 3B Q j4x 2jþa, þa a ¼ 6:52 [35] S ð Þ þ 3ða þ Þ 2 P S ða þ Þ 2 þ 3ða þ Þ 2

8 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) Table 5 Sestvty dces for type C fuctos. Idex Fucto f ðxþ Ref. Measure Aalytcal values umercal values ¼2 ¼0 ¼00 C Q j4x 2j [33] S P S ð 4 3 Þ ðð 4 3 Þ Þ C ð2þ Q [-] S x P S ð 3 4 Þð Þ ðð 4 3 Þ Þ ¼ 3 8 þ 3 4 x þ 3 þ 8 8 þ 4 x 2 þ 8 4 x 3 þ 4 x 4 x 2 þ 2 x x 2 þ þ 8 4 x þ 4 x 3 2 x x 3 8 þ 4 x 2 þ 4 x 3 2 x 2x 3 þ 8 4 x 4 x 2 4 x 3 þ 2 x x 2 þ 2 x x 3 þ 2 x 2x 3 x x 2 x 3 þ : 8 Oe ca see by comparg f (x ), f 2 (x 2 ) ad f 3 (x 3 ) that the varable x s more mportat terms of ts varace tha x 2 ad x 3. It s also mportat to otce that frst order terms are more mportat tha teracto oes: the rato S / s close to oe both P low ad hgh dmesos. S s also close to oe (the aalytc values for sestvty dces for arbtrary are gve [36]). To check codto (23),,2 ad 3,4,...,200 were calculated for ¼ 200. Ther values are,2 ¼ 0:94 ad Stot 3,4,...,200 ¼ 0:, hece d T ¼ 2 assumg that p ¼ 0:9. These results cofrm that codto (24) s satsfed, whch case QMC should be more effcet tha MC rrespectve of dmesoalty. Ideed, the results of umercal tegrato cofrm ths predcto (Fg. a): for a hgh-dmesoal problem wth ¼360, the expoet for algebrac decay a QMC ¼ 0:94 (3) s oly margally smaller tha theoretcally predcted asymptotcal value a QMC ¼ :0. The costat c s lower for the QMC method. Fucto 2A Table 3 was wdely used papers o global sestvty aalyss, where t was called g-fucto [35,0]. It ca be see that, as the value of a creases, the mportace of the correspodg varable decreases. By varyg values of a t s possble to chage the type of the g-fucto. Three dfferet sets of fa g were egeered such a way that all three types of fuctos were cosdered. For fucto 2A at ¼ 2,2 ¼ Stot 3,4,...,00 ¼ 0:64, so codto (23) s satsfed ad d T s close to 2. At the same tme the teracto terms are domat: P S 0. The effcecy of QMC s stll hgher tha that of MC at ¼ 00, although a QMC s oly equal to 0.7 (Fg. b) ad the costats c (3) are almost equal for both methods. All cosdered test fuctos wth equally mportat varables are fact symmetrcal wth regard to ther varables f ð...x,...,x j,...þ¼fð...x j,...,x,...þ, 8f,jg,aj: Type B fuctos B ad 2B (see Table 4) have very smlar values ad P S (both beg very close to oe). Fg. 2a ad b of S = cofrm that the tegrato errors for both fuctos exhbt a smlar behavor wth QMC outperformg MC by several orders of magtude at ¼ 360. Wth all a beg equal to 6.52, the g-fucto becomes a type B fucto (fucto 3B Table 4). The aalyss of the global SI shows that for ths fucto the teracto terms (although ot beg domat) become more mportat at hgh (S = decreases from Fg.. The tegrato error e vs. the umber of sampled pots. (a) Fucto A ( ¼ 360), (b) fucto 2A ( ¼ 00) at ¼2 to 0.55 at ¼00). The values of the tegrato errors for the QMC ad MC methods are very smlar up to 2 6.At 4 QMC becomes more effcet tha MC (Fg. 2c). For fuctos C ad 2C the rato of S /S tot rapdly decreases to 0 wth, whch meas that the hgher order terms become domat. The effectve dmesos for such fuctos are equal to ther omal values. I ths case, QMC loses ts advatage over MC hgh dmeso. I partcular, a QMC a MC. The results preseted Fg. 3 cofrm ths predcto.

9 448 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) Fg. 3. The tegrato error e vs. the umber of sampled pots. (a) Fucto C ( ¼ 20), (b) fucto 2C ( ¼ 20). Fg. 2. The tegrato error e vs. the umber of sampled pots. (a) Fucto B ( ¼ 360), (b) fucto 2B ( ¼ 360), (c) fucto 3B ( ¼ 00). 8. Coclusos It has bee show that global sestvty aalyss allows the estmato of the effectve dmesos at reasoable computatoal costs. amely, d T ca be foud by calculato of the Sobol sestvty dces for subsets of varables. d S ca be estmated by ether usg calculatg the frst order effects ad the total Sobol SI or by usg the RS/QMC-HDMR method. Global sestvty aalyss ca also be used to predct the effcecy of the QMC method. Fuctos wth respect to ther depedece o the put varables ca be loosely dvded to three categores: fuctos wth ot equally mportat varables (type A) for whch d T 5; fuctos wth equally mportat varables ad wth domat low-order terms (type B) for whch d S 5, ad fuctos wth equally mportat varables ad wth domat teracto terms (type C) for whch d S ¼ d T ¼. For fuctos of type A ad B, QMC s eve the hgh-dmesoal case superor to MC whle for fuctos of type C, QMC loses ts advatage over MC because of the mportace of hgher order terms the correspodg AOVA decomposto. The results of umercal tests verfy the predcto of the suggested classfcato. Ackowledgemets We express our grattude to I.M. Sobol for helpful dscussos. We also ackowledge the facal support of the Uted Kgdom s Egeerg ad Physcal Sceces Research Coucl Grat EP/D506743/. W.M. would lke to ackowledge the facal support of the Erest-Solvay Stftug, Esse, Germay.

10 S. Kuchereko et al. / Relablty Egeerg ad System Safety 96 (20) Refereces [] Borgoovo E. Measurg ucertaty mportace: vestgato ad comparso of alteratve approaches output. Rsk Aal 2007;26(3): [2] Borgoovo E. A ew ucertaty mportace measure output. Relab Eg Syst Safety 2007;92: [3] Caflsch RE, Morokoff WJ, Owe AB. Valuato of mortgage backed securtes usg Browa brdges to reduce effectve dmeso. J Comput Face 997;(): [4] Owe A. The dmeso dstrbuto ad quadrature test fuctos. Stat Sca 2003;3: 7. [5] Bratley P, Fox B, ederreter H. Implemetato ad tests of low-dscrepacy sequeces. ACM Tras Mod Comput Sm 992;2(3): [6] Paskov S, Traub J. Faster evaluato of facal dervatves. J Portfol Maage 995;22():3 20. [7] Lemeux C, Owe A. Quas-regresso ad the relatve mportace of the AOVA compoet of a fucto. I: Fag K-T, Hckerell FJ, ederreter H, edtors. Mote Carlo ad quas-mote Carlo. Berl: Sprger-Verlag; [8] Sloa I, Wozakowsk H. Whe are quas-mote Carlo algorthms effcet for hgh dmesoal tegrals? J Complexty 998;4: 33. [9] ederreter H. Radom umber geerato ad quas-mote Carlo methods. Socety for Idustral ad Appled Mathematcs; 992. [0] Sobol I. O quas-mote Carlo tegratos. Math Comput Sm 998;47: [] Sobol I. O the dstrbuto of pots a cube ad the approxmate evaluato of tegrals. Comp Math Math Phys 967;7:86 2. [ Russa]. [2] Schler C. Error treds quas-mote Carlo tegrato. Comput Phys Commu 2004;93: [3] Morokoff W, Caflsch R. Quas-radom sequeces ad ther dscrepaces. SIAM J Sc Stat Comput 99;5: [4] Morokoff W, Caflsch R. Quas Mote-Carlo tegrato. J Comput Phys 995;22: [5] Papageorgou A, Traub J. Faster evaluato of multdmesoal tegrals. Comput Phys 997;(6): [6] Sobol I. O a estmate of the accuracy of a smple multdmesoal search. Sovet Math Dokl 982;26: [7] Sobol I. Sestvty estmates for olear mathematcal models. Matematcheskoe Modelrovae 990;2():2 8. [ Russa, Traslated I.M. Sobol, Sestvty estmates for olear mathematcal models, Mathematcal Modelg ad Computatoal Expermet 993;26: ]. [8] Saltell A, Cha K, Scott E. Sestvty aalyss. Wley; [9] Sobol I. Multdmesoal quadrature formulas ad Haar fuctos. Moscow: auka; 969. [ Russa]. [20] Hoeffdg W. A class of statstcs wth asymptotcally ormal dstrbutos. A Math Stat 948;9: [2] Homma T, Saltell A. Importace measures global sestvty aalyss of olear models. Relab Eg Syst Safety 996;52: 7. [22] Sobol I. Global sestvty dces for olear mathematcal models ad ther Mote Carlo estmates. Math Comput Sm 200;55: [23] Saltell A. Makg best use of model evaluatos to compute sestvty dces. Comput Phys Commu 2002;45: [24] Mautz W. Global sestvty aalyss of geeral olear systems. Master s thess, Imperal College Lodo, CPSE; [Supervsors: C. Pateldes ad S. Kuchereko.]. [25] Sobol I, Taratola S, Gatell D, Kuchereko S, Mautz W. Estmatg the approxmato error whe fxg uessetal factors global sestvty aalyss. Relab Eg Syst Safety 2007;92: [26] Asotsky D, Myshetskaya E, Sobol I. The average dmeso of a multdmesoal fucto for quas-mote Carlo estmates of a tegral. Comput Math Math Phys 2006;46: [27] Capstck S, Kester B. Multdmesoal quadrature algorthms at hgher degree ad/or dmeso. J Comput Phys 996;23: [28] Sobol, I. Computato of defte tegrals. The Mote Carlo method. Pergamo Press; 966. [29] Sobol I. umercal Mote Carlo methods. Moscow: auka; 973. [ Russa]. [30] Moskowtz B, Caflsch R. Smoothess ad dmeso reducto quas- Mote-Carlo methods. Math Comput Mod 996;23(8/9): [3] Papageorgou A. The Browa brdge does ot offer a cosstet advatage quas-mote Carlo tegrato. J Complexty 2002;8():7 86. [32] Sobol I, Kuchereko S. Global sestvty dces for olear mathematcal models. Rev Wlmott Mag 2005;2:56 6. [33] Bratley P, Fox B. Sobol s quasradom sequece geerator. ACM Tras Math Software 988;4: [34] Levta YL, Markovch, Roz S, Sobol I. Short commucatos o quasradom sequeces for umercal computatos. USSR Comput Math Math Phys 988;28(3): [Egl Trasl]. [35] Saltell A, Sobol I. About the use of rak trasformato sestvty aalyss of model output. Relab Eg Syst Safety 995;50: [36] Saltell A, Ao P, Azz I, Campologo F, Ratto M, Taratola S. Varace based sestvty aalyss of model output. Desg ad estmator for the total sestvty dex. Comput Phys Commu 200;8(2):