A New Bayesian Network Method for Computing Bottom Event's Structural Importance Degree using Jointree

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1 , pp A New Bayesa Network Method for Computg Bottom Evet's Structural Importace Degree usg Jotree Wag Yao ad Su Q School of Aeroautcs, Northwester Polytechcal Uversty, X a, PR Cha) wagyaorose@126.com, suqg@wpu.edu.c Abstract T Bayesa etwork methodology takg place of fault tree aalyss used for relablty assessmet has gotte lots of atteto recet years. O bass of the curret Bayesa etwork method used for calculatg structural mportace degree, a ew Bayesa etwork method s rased. Ths ew method ca avod repeat modfcato of the parameters the codtoal probablty tables, ad hece the termedate results computed durg ferece process ca be shared to decrease calculatg complexty. The ew method s proved to be correct a mathematc way ad a correspodg algorthm amed SID_Jotree for realzg ths ew method s desged, whch guaratees the ew method ca be realzed computer. Fally, the correctess ad effcecy of the ew method s valdated by usg two fault tree cases. Keywords: Bayesa etwork; fault tree; structural mportace degree; relablty assessmet; ferece 1. Itroducto As a model used for relablty assessmet, Bayesa etwork mapped from a fault tree ca acheve more useful results tha fault tree aalyss (FTA). Studes show that, all the results calculated through FTA ca be computed through Bayesa etwork ad the coverse s ot true. For example, f the top evet occurs, oe ca calculate the posteror probablty of the bottom evet usg Bayesa etwork, whch ca t be calculated usg FTA. The methods for evaluatg system usg a fault tree s equvalet Bayesa etwork have gotte lots of atteto home ad abroad. I specal, may researchers focus o the methods for solvg relablty dexes, such as mmal cut sets, mmal path cuts, structural mportace degree, ad probablty mportace degree. Amog these researchers, Lug Portale [1, 2] ad Nma Khakzad [3] studed how to solve the mmal cut sets ad how to calculate relablty degree a mathematcal way; Wedl [5] studed how to calculate the relablty degree ad some other tradtoal relablty dexes usg the exstet Bayesa etwork toolbox. Zhou Zhogbao [6] rased a ew method for computg the structural mportace degree, whch represets the ew state-of-the-art solvg such a tradtoal relablty dex usg Bayesa etwork. However, the ew method s complcated: for each bottom evet, oe eeds to ferece the equvalet Bayesa etwork oce. That s to say, for a system wth bottom evets, oe has to ferece tmes the Bayesa etwork. As creases, the effcecy of the method decreases. O the bass of researches stated above, ths paper presets a ew method for calculatg all the bottom evets structural mportace degree usg a Jotree extracted from the fault tree s equvalet Bayesa etwork, as well as the correspodg algorthm SID_Jotree. Jotree s a kd of structural that the termedate results ca be stored the edges of tself durg ferece, so through Jotree ISSN: IJUNESST Copyrght c 2015 SERSC

2 oe ca reduce the ferece complexty by sharg the stored results [7-8]. I fact, through a ward ad outward ferece the Jotree (both ward ad outward ferece s equvalet to ferece oce usg varable elmato algorthm a Bayesa etwork, separately [8]), oe ca store all the termedate results eeded for calculatg each ode s posteror probablty, whch s the key kd of probablty for calculatg each ode s structural mportace degree. However, accordg to Zhou s method, oe has to chage other odes CPTs before calculatg oe bottom evet s structural mportace degree a Bayesa etwork. Oce ay CPT chaged a Bayesa etwork, the termedate results stored the edges of Bayesa etwork s correspodg Jotree become vald ad caot be shared for calculatg aother bottom evet s structural mportace degree. That meas oe has to calculate each bottom evet s structural mportace degree oe by oe. O the cotrary, the method rased ths paper avods chagg ay ode s CPT durg ferece. Thus, the termedate results stored the edges of the Jotree are stll vald for calculatg ay other evet s structural mportace degree. I addto, the ew method s proved correct a mathematcal way. Fally, such a method s exemplfed wth two fault cases. 2. Bayesa Network 2.1. Defto of Bayesa Network Bayesa etwork s a drected acyclc graph, cosstg of odes deoted by varables ad drected edges that pot from father odes to chldre odes, each edge deoted by oe-le arrows [7]. Every ode Bayesa etwork has a codtoal probablty table (CTP). To be specfc, the CPT of ode Y ca be represeted wth Pr( Y ( Y)), whch ( Y ) stads for the set of odey s fathers. Varable Y ad ts fathers ( Y ) make up a famly. Itutvely, every CPT stores ts famly s possble states, as well as the states correspodg probabltes. A smple Bayesa etwork s show Fg.1 (b). For a Bayesa etwork, whch all the varables make up set N { Y1, Y2,, Y }, jot probablty dstrbuto of the etwork ca be computed by multplyg all the odes CPTs: Pr( Y, Y,, Y ) Pr( Y ( Y )) [8] T X 1 A X 2 X 3 (a) 278 Copyrght c 2015 SERSC

3 X 1 A T Pr (T X 1,A) X 0 X 1 Pr(X ) 1 q 1 q T A X 2 X 3 X 2 X 3 A Pr (A X 2 X 3 ) Pr (T X 1 A) (b) Pr (A X 2 X 3 ) T,X 1,A T,A A X 3,A X 3,X 2 A X 1 T X 3 X 2 Pr (X 1 ) (c) Pr (X 3 ) Pr (X 2 ) Fgure 1. A Smple System Descrbed Dfferet Topologes (a) Fault Tree; (b) The equvalet Bayesa etwork; (c) A Jotree extracted from the Bayesa etwork wth each CPT assged to ther famly clusters, takg the cluster {T,X 1,A} as root, sold arrows dcate the drecto of ward calculato ad dotted arrows dcate the drecto of outward calculato. After a ward ad outward calculato, termedate results ca be stored the edges or clusters. Ad If termedate results ca be shared, Pr( X E e ) ca be read from cluster{ X } drectly Iferece Bayesa Networks All the ferece problems Bayesa etwork ca be boled dow to solvg the probablty dstrbuto of Pr( Q, E e ), whch Q stads for the set of query varables that people are terested ad E stads for the set of evdece varables that people have already kow ts values e. Mastream, varable elmato algorthm (VE) ad Jotree algorthm are two ma exact ferece algorthms Bayesa etwork. Jotree algorthm starts wth a jotree extracted from Bayesa etwork (see Fg. 1(c)).For a Bayesa etwork, t has may dfferet Jotrees accordg to dfferet costructo algorthms. I ths paper, oe kd of Jotree amed bary Jotree s adopted by usg the costructo algorthm stated the lterature [9,10]. Nodes of jotree are called clusters ad each cluster cotas oe or several varables. As stated before, f a cluster Copyrght c 2015 SERSC 279

4 cossts of a varable wth all ts fathers, the cluster s called famly cluster. After a jotree s extracted, assg every CPT to ts famly cluster. The a jotree ca be vewed as a specal structure that oe ca store termedate results the edges or clusters durg ferece process ad these termedate results ca be shared whe solvg aother dfferet Bayesa etwork ferece problem. Such a data-shared mechasm ca avod more calculato solvg aother ferece problem. Because of these shared termedate data, oe ca compute the probablty dstrbuto Pr( Y, E e) of each odey by just performg Jotree algorthm oce. Here Y ca represet ay ode the etwork. But whe ay CPT s chaged, the termedate results stored the jotree become vald. Oe has to perform the Jotree algorthm aga oce ay CPT has bee chaged. VE algorthm does t store ay termedate results ferece process. Therefore, there s o shared data whe solvg aother ferece problem. Oe ca oly compute the probablty dstrbuto Pr( Y, E e) of someoe ode Y by VE algorthm. Here, Y stads oly some ode the etwork, ot ay ode. I fact, Jotree algorthm cotas two computato parts: ward calculato ad outward calculato. Each kd of calculato s equvalet to performg VE algorthm oce except that termedate results are store durg performg ward ad outward calculato whle VE algorthm stores othg. Takg Fg.1(c) for a stace, set evdece T=1. Usg Jotree algorthm, oe ca extract the results Pr( X, T 1), 1, 2,3 from the cluster { X } after a ward ad outward calculato. Whle oe eeds to perform VE algorthm three tmes order to calculate Pr( X, T 1), 1,2, Mappg Fault Trees to Bayesa Networks Deote a fault tree usg FT ad deote ts equvalet Bayesa etwork usg BN. O the bass of refereces [1-3], the steps for mappg FT to BN are gve as below. 1. For every bottom evet FT, buld oly oe ode BN to correspod to. The draw arrows that pot from odes whch are correspodet wth bottom evets to odes whch are correspodet wth termedate/top evets. Specfcally, BN, varable T deotes the ode that correspods to top evet, ad T=0/1 holds; X deotes the ode that correspods to bottom evet, ad X 0 /1 holds; Aj deotes the ode that correspods to termedate evet, ad Aj 0 /1 holds. I the followg, 1 represets the fault state ad 0 represets the ormal/workable state. 2. Determe every root ode CPT, deoted by Pr( X ). Root odes are mapped from bottom evets, therefore Pr( X ) s decded by each bottom evet s falure probablty q. To be specfc, Pr( X 1) q ad Pr( X 0) 1 q holds. 3. Determe o-root odes CPTs, deoted by Pr( T ( T)) or Pr( A ( A )). These o-root odes are mapped from termedate/top evets. These evets are caused by bottom evets some kd of determstc logc relatoshp, such as AND ad VOTE. The CPT of o-root ode must be gve correspodg to such a determstc relatoshp. The detaled method for geeratg o-root ode s CPT s dscussed [1, 6]. Accordg to the procedures above, the equvalet Bayesa etwork of Fgure 1 (a) s gve Fgure 1 (b). j j 280 Copyrght c 2015 SERSC

5 3. Methods for Solvg Structural Importace Degree 3.1. FTA Method The state of top evet s a structural fucto (or called state fucto), whch vares wth every bottom evet s state, deoted by ( X ) ( X1, X 2,, X ). Both ( X ) ad X take the value 1 or 0. Usg FTA, the structural mportace degree of bottom evet X s computed by Eq. (1) [11] based o FT. 1 1 I [ ( X 1, X k, X k, X k, X k,, X k ) (1) ( ) k, k k, k, k ( X 0, X1 k1, X 2 k2, X 1 k 1, X 1 k 1,, X k )] I Eq. (1), the summato symbol meas that all the varables sde the bracket should traverse the value 0 ad 1 except the bottom evet ode X Bayesa Network Method Exstet Bayesa Network Method: Eq. (1) provdes a exact way to solve bottom evet X s structural mportace degree. I essece, t s a exhaust method that oe should lst a system s every possble state. For example, f a system has bottom evets, the system wll have exp() states ad oe has to lst all these states to computg the dex. As crease, lstg all the possble states s a hardly work. I cotrast, the Bayesa etwork method proposed by Zhou [6] avods such a exhaust lst, whch makes exact computato feasble a larger system. Accordg to Zhou s method, ode X s structural mportace degree ca be computed usg Eq. (2) BN : I =Pr( T 1 X 1) Pr( T 1 X 0) () (2) X s the ode whose structural mportace degree to be calculated. However, for Eq. (2) there are the followg calculato codtos. Before usg Eq.(2), the o-root odes CPTs should rema uchaged just as ther orgal CPTs geerated the frst begg usg the mappg method Secto 2.3, as well as ode X s CPT; Meawhle, all the root odes CPTs except X s should be chaged to a uform dstrbuto as follows: Pr( X j 0) 1/ 2, Pr( X j 1) 1/ 2, j. Accordg to Eq. (2) ad ts calculato codtos, whe solvg X s structural mportace degree, the rest root odes CPTs have to be chaged. Therefore, termedate results caot be shared eve though Jotree algorthm s adopted (Obvously, the termedate results stored Jotree become vald whe solvg aother bottom evet because some CPTs have to be chaged aga). Oe has to solve each bottom evet s structural mportace degree oe by oe. Above all, usg the exstet Bayesa etwork method to solve a system wth bottom evets, Eq. (2) has to be computed tmes A New Bayesa Network Method: Here a ew method s rased ad wll be proved a mathematcal way. The ew Bayesa etwork method s expressed wth Eq. (3) as below: I () =2 Pr( T 1, X 1) Pr( T 1, X 0) (3) Before calculatg Eq. (3), set every root ode s CPT to a uform dstrbuto as follows: Pr( X j 0) 1/ 2, Pr( X j 1) 1/ 2. Copyrght c 2015 SERSC 281

6 Comparg the ew method wth Eq. (2), a major dfferece les that all the CPTs are set oce ad oe of them wll be chaged aga usg the ew method whle usg Eq. (2), some root odes CPTs have to be chaged aga whe solvg a dfferet bottom evet s structural mportace degree. No chagg CPTs ay more meas that termedate results wll be stll useful ad ca be shared whe solvg aother dfferet bottom evet s structural mportace degree usg Jotree algorthm (As stated Secto 2.2, Jotree algorthm ca store termedate results, ad f the CPTs are ever bee chaged, these termedate data wll vald forever ). Therefore, oe ca compute every bottom evet s structural mportace degree just performg Jotree algorthm oce. I what follows, Eq. (3) s proved a mathematcal way. Proof: As structural mportace degree s rrespectve of the falure probablty of each bottom evet of FT, set each bottom evet s falure probablty equals to 0.5. Obvously, for each root ode X BN, ts CPT s as below: Pr( X 0) 1/ 2, Pr( X 1) 1/ 2. j j As X1, X2,, X 1 ad X are depedet of each other, Eq. (4) ad Eq. (5) hold (4) Pr( X, X,, X, X,, X ) 1 2 Pr( X1, X2,, X ) 1 2 (5) Combg Bayer Theorem, Eq. (6) ad Eq. (7) ca be derved from Eq. (4) ad Eq. (5). Pr( T 1 X 1, X1 k1, X 2 k2,, X 1 k 1, X 1 k 1,, X k) (6) Pr( T 1, X 1, X k, X k,, X k, X k,, X k ) Pr( T 1 X 0, X1 k1, X 2 k2,, X 1 k 1, X 1 k 1,, X k) (7) Pr( T 1, X 0, X1 k1, X 2 k2,, X 1 k 1, X 1 k 1,, X k) 2 I FT, setece ( X1 k1, X 2 k2,, X 1 k 1, X k, X 1 k 1, X k) 0 meas that top evet certaly wo t happe whe all the bottom evets take the state k1, k2,, k 1, k, k 1,, k lsted the Paretheses. I BN, such a setece meas that top evet T takes state 1wth a probablty of. Aalogously, FT setece ( X k, X k,, X k, X k, X k, X k ) 1 meas that the top evet curtas to occur whe all the bottom evets take the state k1, k2,, k 1, k, k 1,, k lsted the Paretheses. I BN, such a setece meas that top evet T takes state 1wth a probablty of. Accordg to the llustrato above, Eq. (8) holds. ( X1 k1, X 2 k2,, X 1 k 1,, X 1 k 1,, X k) (8) Pr( T 1 X1 k1, X 2 k2,, X 1 k 1,, X 1 k 1,, X k) Take Eq. (6), Eq. (7) ad Eq. (8) to Eq. (1), Eq. (9) ca be acheved. 1 I 2 Pr( T 1, X 1, X k,, X k, X k,, X k ) ( ) k, k, k, k T X X1 k1 X 1 k 1 X 1 k 1 X k (9) k, k, k, k 0 2 Pr( 1, 0,,,,,, ) [Pr( T 1, X 1) Pr( T 1, X 0)] Proof eds. 282 Copyrght c 2015 SERSC

7 Above all, for a system wth bottom evets, oe ca compute all bottom evets structural mportace degree wth two tmes ferece usg the ew method: a ward computato ad a outward computato The Correspodg New Algorthm SID_Jotree: Eq. (3) s a ew mathematcal method for computg each bottom evet s structural mportace degree. As a complemet to the ew method, the correspodg ew algorthm SID_Jotree s desged. I fact, two tmes ferece (a ward ad outward) s performed after callg SID_Jotree, whch results every bottom evet s structural mportace degree. Algorthm SID _Jotree (N, N, CPTs, X, X ) Iputs: N: set whch cotas all the varables Bayesa etwork; N : the umber of varables Bayesa etwork; CPTs: every ode s CPT; X: set whch cotas all root odes (bottom evets) Bayesa etwork; X : the umber of root odes; Outputs: I () 1,2,, X : each bottom evet s structural mportace degree 1: for(=1,, X ) 2: Set Pr(X =1)=1/2, Pr(X =0)=1/2; 3: Covert a Bayesa etwork to a jotree; 4: for(=1,, N ) 5: { I ode s CPT, delete the parameters that are compatble wth T=1; 6: Assg ode s CPT to ts famly cluster the jotree buld le 3; } 7: Choose oe cluster the jotree as root; 8: Perform ward calculato; 9: Perform outward calculato; 10: for(=1,, X ) 11: {Extract results Pr(T=1, X =0) ad Pr(T=1, X =1) from cluster { X }; 12: I () 2[Pr( T 1, X 1) Pr( T 1, X 0)] ;} 4. Case Studes Two cases are dscussed usg the old ad ew Bayesa etwork method, separately. Ad the results are compared. The effcecy ad feasblty of the ew Bayesa etwork method s the ma cocer ths paper. I addto, the ew method s a geeral methodology whch ca be appled to ay system, ot lmted to ay cocrete specal systems. Therefore, the followg fault trees omt each evet s physcal meag A Smple Fault Tree Takg Fgure 1 for the frst case, Fgure 1 (a) s a fault tree, the equvalet Bayesa etwork mapped from the tree show Fgure 1 (b), ad the correspodg Jotree extracted from the Bayesa etwork show Fgure 1 (c). All the bottom evets CPTs should be set to Pr( X 0) 1/ 2, Pr( X 1) 1/ 2 ad the top evet T should be set to T=1 the Jotree accordg to the ew method. After the ward ad outward calculato, the results Pr(X 1 T=1),Pr(X 2 T=1) ad Pr(X 3 T=1) ca be read from the cluster {X 1 },{X 2 }ad Copyrght c 2015 SERSC 283

8 {X 3 }separately. The accordg to the Eq. (3), all these three bottom evets structural mportace degree ca be computed. Apparetly, ward ad outward two tmes ferece are eeded here.the fal results are show Table 1. Table 1. Results of Case 1 Bottom evets Old Bayesa etwork method New Bayesa etwork method X X X Number of ferece 3 tmes 2 tmes 4.2. A Fault Tree for a Power Dstrbuto System Fgure 2(a) s a fault tree for a power dstrbuto system [12-13]. Accordg to the steps for mappg fault trees to Bayesa etworks, the acheved equvalet BN s show Fgure 2. (b) ad the Jotree mapped from the Bayesa etwork s show Fgure 2 (c). It easly ca be see that all the bottom evets correspodg clusters ca be foud the Jotree, separately. Accordg to the ew method, set every bottom evet s CPT to Pr( X 0) 1/ 2, Pr( X 1) 1/ 2 ad T=1. The the results Pr( X 0 T 1), Pr( X 1 T 1) ca be read from the Jotree after a ward ad outward computato. The all the bottom evets structural mportace degree ca be computed accordg to Eq. (3), results show Table 2, whch the results computed usg the Eq.(2) (the old method) also provded. From the results Table 1 ad Table 2, t ca be see that the same results ca be computed by these two methods, whch proves the correctess of the ew method. I case 1, there are three bottom evets, three tmes ferece are performed usg old method whle two tmes ferece are performed usg the ew algorthm; I case 2, there are eght bottom evets, eght tmes ferece are ecessary usg the old method whle stll two tmes ferece are performed usg the ew algorthm. It ca be cocluded that as the umber of bottom evets creases, the umber of ferece creases whe usg the old method. However, the umber of ferece s rrespectve of whe usg the ew method (oly a ward ad outward ferece s ecessary). 284 Copyrght c 2015 SERSC

9 T T A 1 A2 A 1 A 2 X 7 A 3 X 8 A 4 X 7 A 3 A 4 X 8 X 6 A 5 X 4 X 2 X 5 X 6 A 5 A 6 X 4 X 5 X 4 A 6 X 5 X 1 X 2 X 3 X 1 X 2 X 3 (a) (b) X 1 Pr(X 1 ) X 5 Pr(X 5 ) X 1 X 2 X 3 X X 4 X 5 A 5 A 4 6 Pr(A 6 X 1 X 2 X 3 ) Pr(X 4 ) A Pr(X 6 ) 6 X 6 Pr(A 5 X 4 X 5 A 6 ) X 1 X 2 X 3 X 2 X 3 X 1 X 2 X 3 X 5 X 1 X 2 X 3 X 5 A 6 X 2 X 5 A 6 X 2 X 4 X 5 Pr(A 2 X 8 A 4 ) A 2 X 8 A 4 Pr(X 7 ) X 7 A 3 A 4 X 8 A 6 A 3 A 5 X 6 X 2 X 4 X 5 X 2 X 4 X 5 X 2 X 4 X 5 A 5 A 6 A 5 X 6 A 5 Pr(A 3 X 6 A 5 ) X 8 Pr(X 8 ) X 2 X 4 X 5 X 6 A 4 A 5 X 2 X 3 Pr(X 2 ) Pr(X 3 ) A 2 A 3 A 4 X 7 X 8 A 3 A 4 X 7 X 8 A 1 A 3 X 7 Pr(A 1 X 7 A 3 ) A 3 A 4 A 5 X 6 X 8 A 2 A 3 X 7 A 1 A 2 A 3 X 7 A 1 A 2 T A 1 A 2 A 4 A 5 X 6 X 8 Pr(T A 1,A 2 ) A 4 A 5 X 6 Pr(A 4 X 2 X 4 X 5 ) Fgure 2. A Power Dstrbuto System Descrbed Dfferet Topologes (a) A fault tree for a power dstrbuto system; (b) Equvalet Bayesa etwork; (c) Jotree extracted from Bayesa etwork wth each CPT assged to ther famly clusters, takg the cluster {T,A 1,A 2 } as root, sold arrows dcate the drecto of ward calculato ad dotted arrows dcate the drecto of outward calculato. (c) X 2 X 4 A 4 A 5 Bottom evets Table 2. Results of Case 2 Old Bayesa etwork method New Bayesa etwork method Copyrght c 2015 SERSC 285

10 X X X X X X X X Number of ferece 8 tmes 2 tmes 5. Coclusos O the bass of the exstet exact methods for solvg bottom evet s structural mportace degree cludg the tradtoal FTA method ad the Bayesa etwork method rased by Zhou Zhogbao, combg wth Bayesa etwork s ferece characterstcs, a ew exact method for solvg such a dex together wth the correspodg algorthm SID_Jotree s rased ad proved to be correct a mathematcal way. Two cases are used to llustrate the ew Bayesa etwork method whch a Jotree ca be used effectvely. Both practce ad theory show that, as the bottom evets umber creases, oly two tmes computato s requred usg the ew method stead of tmes computato usg the exstet old method. As creases, the ew method s more effectve, whch s a feasble way to compute each bottom evet s structural mportace degree exactly ad ca be a useful assstat aalyzg a system s relablty. Refereces [1] L. Portale ad A. Bobbo, Bayesa etworks for depedablty aalyss: a applcato to dgtal cotrol relablty, I Proceedgs of the ffteeth coferece o ucertaty artfcal tellgece, Morga Kaufma Publshers Ic., (1999), July, pp [2] H. Lagseth ad L. Portale, Bayesa etworks relablty, Relablty Egeerg & System Safety, vol. 92, o. 1, (2007), pp [3] N. Khakzad, F. Kha ad P. Amyotte, Safety aalyss process facltes: Comparso of fault tree ad Bayesa etwork approaches, Relablty Egeerg & System Safety, vol. 96, o. 8, (2011), pp [4] Y. Xaowe, Q. Wexue ad X. Lyag, A Method for System Relablty Assessmet Based o Bayesa Networks [J], Acta Aeroautca Et Astroautca Sca, vol. 6, o. 013, ( Chese), (2008). [5] G. Wedl, A. L. Madse ad S. Israelso, Applcatos of object-oreted Bayesa etworks for codto motorg, root cause aalyss ad decso support o operato of complex cotuous processes, Computers & chemcal egeerg, vol. 29, o. 9, (2005), pp [6] Z. Zhogbao, D. Doudou ad Z. Jglu, Applcato of Bayesa Networks Relablty Aalyss [J], Systems Egeerg-Theory & Practce, vol. 6, (2006), pp ( Chese). [7] A. Darwche, Modelg ad reasog wth Bayesa etworks, New York: Cambrdge Uversty Press, (2012). [8] Z. Lawe ad G. Hapeg, A Itroducto to Bayesa Networks, Bejg: Scece Press ( Chese), (2006). [9] P. P. Sheoy, Bary jo trees for computg margals the Sheoy-Shafer archtecture, Iteratoal Joural of approxmate reasog, vol. 17, o. 2, (1997), pp [10] K. Lgasubramaa, S. M. Alam ad S. Bhaja, Maxmum error modelg for fault-tolerat computato usg maxmum a posteror (MAP) hypothess, Mcroelectrocs Relablty, vol. 51, o. 2, (2011), pp [11] D. Qog, Safety system egeerg, X a: Northwest Idustral Uversty press ( Chese), (2009). [12] S. Mahadeva, R. Zhag ad N. Smth, Bayesa etworks for system relablty reassessmet, Structural Safety, vol. 23, o. 3, (2001), pp Copyrght c 2015 SERSC

11 [13] E. Castllo, C. Solares ad P. Gómez, Estmatg extreme probabltes usg tal smulated data, Iteratoal Joural of Approxmate Reasog, vol. 17, o. 2, (1997), pp Authors Wag Yao, She receved her B.Eg. Arcraft Desg ad Egeerg (2011) ad s curretly dog a doctorate Arcraft Desg at School of Aeroautcs, Northwester Polytechcal Uversty as a full-tme studet (Sce 2011). Her curret research terests clude the Arcraft system relablty assessmet ad fault dagoss. Su Q, He receved hs PhD Arcraft Desg from Northwester Polytechcal Uversty. Now he s full professor of Arcraft Desg at School of Aeroautcs, Northwester Polytechcal Uversty. He s the charma of the structure desg ad stregth of Cha avato commttee. He has publshed a total of more tha 100 academc papers (10/SCI, 50/EI, 20/ISTP). Hs curret research terests clude the arcraft structural desg, arcraft system desg ad relablty evaluato. Copyrght c 2015 SERSC 287

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