Bayesian Reliability Modeling Using Monte Carlo Integration
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1 Joural o Modr Appld asal Mods Volu 4 Issu Arl Baysa laly Modl Us Mo Carlo Irao V A.. Caara Uvrsy o ou Florda vaara@. Crs P. Tsokos Uvrsy o ou Florda prop@as.us.du Follow s ad addoal works a: p://daloos.way.du/as Par o Appld ass Coos oal ad Bavoral s Coos ad asal Tory Coos odd Cao Caara V A.. ad Tsokos Crs P. 5 "Baysa laly Modl Us Mo Carlo Irao" Joural o Modr Appld asal Mods: Vol. 4 : Iss. Arl 8. DOI:.37/as/49668 Avalal a: p://daloos.way.du/as/vol4/ss/8 Ts ular Arl s rou o you or r ad op ass y Op Ass Jourals a DalCoos@Waya. I as apd or luso Joural o Modr Appld asal Mods y a auorzd dor o DalCoos@Waya.
2 Joural o Modr Appld asal Mods Copyr 5 JMAM I. May 5 Vol. 4 No /5/$95. Baysa laly Modl Us Mo Carlo Irao V A.. Caara Crs P. Tsokos Dpar o Maas Uvrsy o ou Florda T a o s arl s o rodu op o Mo Carlo Irao Baysa sao ad Baysa rlaly aalyss. Us su op approa sas o parars ad rlaly uos ar oad or r-parar Wull ad aa alur odls. Four dr loss uos ar usd: squar rror Hs-Tsokos Harrs ad a loar loss uo proposd s arl. lav y s usd o opar rsuls oad udr aov od loss uos. Ky words: sao loss uos Mo Carlo Irao Mo Carlo ulao rlaly uos rlav y. Iroduo I s arl op o Mo Carlo Irao Brr 985 s usd o oa approa sas o Bays rul a s ulaly usd o drv sas o rlaly uo. Mo Carlo Irao s usd o rs oa approa Baysa sas o parar r alur odl ad us s sa drly oa approa Baysa sas o rlaly uo. odly su op s usd o drly oa Baysa sas o rlaly uo. I prs odl or rparar Wull ad aa alur odls ar osdrd a ar rspvly dd as ollows: V A.. Caara ard a P.D. Maas/ass. Hs rsar rss lud ory ad applaos o Baysa ad pral Bays aalyss w pass o opuaoal asp o odl. Crs P. Tsokos s a Dsusd Prossor o Maas ad ass. Hs rsar rss ar sasal aalyss ad odl opraos rsar rlaly aalyss-ordary ad Baysa srs aalyss. a a > a a wr a ad ar rspvly loao sal ad sap parars ad α α α Γ α wr α ad ar rspvly sap ad sal parars. For s wo alur odls osdr sal parars ad o av as rado varals a ollow looral dsruo w s v y π 3. π For a o aov udrly alur odls approa Baysa sas wll oad or su parar ad rlaly uo w squard rror Hs-Tsokos Harrs ad a proposd loar loss uos. T loss uos 7
3 CAMAA & TOKO 73 alo w a sa o r ky ararss ar v low. quar rror loss uo T popular squar rror loss uo plas a sall w o sas ar ru valu ad proporoaly or w o r dvao ro ru valu o parar. Is populary s du o s aalyal raaly Baysa rlaly odl. T squard rror loss s dd as ollows: 4 Hs-Tsokos loss uo T Hs-Tsokos loss uo plas a avy paly o r ovr-or udrsao. Ta s plas a poal w o r rrors. T Hs-Tsokos loss uo s dd as ollows: HT. + + Harrs loss uo T Harrs loss uo s dd as ollows: H k. k 6 To our kowld proprs o Harrs loss uo av o ully vsad. Howvr s asd o prss a sys s.99 rlal o avra sould al o wras rlaly s.999 sould al o. Tus s s as ood. oar loss uo T loar loss uo ararzs sr o loss loarally ad ors usul aalyal raaly. Ts loss uo s dd as: l. l 7 I plas a sall w o sas wos raos o ru valu ar los o o ad proporoaly or w o sas wos raos o ru valu ar saly dr ro o. ad rprs rspvly ru rlaly uo ad s sa. Modoloy Cosdr a a rlaly o a sys a a v s proaly a sys als a a rar or qual o rlaly uo orrspod o r-parar Wull alur odl s v y a 8 ad or aa alur odl γ α > α > Γ α. 9 wr γ l l dos opl aa uo. W α s a r quao 9 os α! ad parular w α >. Cosdr suao wr r ar dpd rado varals X X... X w sa proaly dsy uo df ad a o av ralzaos a s X :... X :.... X :...
4 74 BAYIAN IABIITY MODING UING MONT CAO INTGATION T u vara uasd sa MVU o parar s oad ro ralzaos... wr... pa s dpd produr k s a squ o MVU s oad or s a s.... Us s ad r oo proaly dsy uo approa Baysa rlaly sas ar oad. π ad rprs rspvly lklood uo a uo o a pror dsruo o ad a proaly dsy uo o alld pora uo. Us sro law o lar urs [7] wr π π d Θ π l ^. No a rprss pao w rsp o proaly dsy uo ad s ay uo o w assurs ovr o ral also s posror dsy uo. For spal as wr quao ylds Θ π π d l quaos ad ply a posror pd valu o s v y π d Θ π d Θ π l π Ts approa s usd o oa approa Baysa sas o or dr loss uos udr sudy. Approa Baysa sas o parar ad rlaly ar oad y rpla y ad rspvly drvd prssos orrspod o approa Baysa sas o. T Baysa sas usd o oa approa Baysa sas o uo ar ollow w squard rror Hs-Tsokos Harrs ad proposd loar loss uos ar usd: π d Θ B π d Θ BHT + π d Θ π d Θ. Θ π d B H π d Θ
5 CAMAA & TOKO 75 Θ Θ π π d d B 3. Us quao ad aov Baysa dso ruls approa Baysa sas o orrspod rspvly o squard rror Hs-Tsokos Harrs ad proposd loar loss uos ar rspvly v y ollow prssos w rplas ar osdrd. π π 4 + HT π π 5. H π π 6 ad π π. 7 Frs us aov ral uoal ors o Baysa sas o o oa approa Baysa sas o rado parar r udrly alur odl. Furror s sas ar usd o oa approa Baysa rlaly sas. od us aov uoal ors o drly oa approa Baysa sas o rlaly uo. Tr-parar Wull udrly alur odl I s as parar dsussd aov wll orrspod o sal parar. T loao ad sap parars a ad ar osdrd d. T lklood uo orrspod o dpd rado varals ollow r-parar Wull alur odl s v y a a + 8 wr a. Furror a sow a s a su sas or parar ad a u vara uasd saor o s v y a.
6 BAYIAN IABIITY MODING UING MONT CAO INTGATION 76 T proaly dsy uo o Y X a wr X ollows Wull proaly dsy uo s y y y y p > > y y. 9 T o ra uo o Y s v y dy y y Us quao ad a a X s ar dpd o ra uo o u vara uasd saor o parar s a. quao orrspods o o ra uo o aa dsruo G. Tus odoal proaly dsy uo o MVU o s v y > > Γ a. Approa Baysa sas or sal parar ad rlaly uo ar oad w us o quaos 8 ad y rpla rspvly y ad quaos ad 7. T s a ar u vara uasd sas o sal parar wll play rol o s. Cosdr looral pror quaos ad 7 yld ollow approa Baysa sas o sal parar orrspod rspvly o squard rror Hs-Tsokos Harrs ad our proposd looral loss uos ar rpla y prsso o : a a HT a a H a a + + 5
7 CAMAA & TOKO 77 ad + a + a 6 T approa Baysa sas o rlaly orrspod o rs od ar ror v y a a > a 7 wr sads rspvly or aov approa Baysa sas o sal parar. Approa Baysa rlaly sas orrspod o sod od ar also drvd y rpla y quaos ad 7. T oad sas orrspod rspvly o squard rror Hs-Tsokos Harrs ad proposd loar loss uos ar rspvly v y ollow prssos ar rpla y prsso o : a a + a + 8 HT + H a + a a + a 9 a a + a ad a. a a a + a 3 Gaa udrly alur odl T lklood uo orrspod o dpd rado varals ollow wo-parar aa udrly alur odl a wr udr ollow or: α α α Γ α 3 wr.
8 78 BAYIAN IABIITY MODING UING MONT CAO INTGATION No a s a su sas or sal parar. Furror α s a u vara uasd saor o ad s o ra uo s v y α α α 33 w s o ra uo o aa dsruo G α. Tus α odoal dsy uo o MVU o s v y α α α α α > α Γ α 34 Approa Baysa sas or sal parar ad rlaly uo ar oad w us o quaos 3 ad 34 y rpla rspvly y ad quaos ad 7. T s a ar u vara uasd sas o sal parar wll play rol o s. Cosdr looral pror quao ad 7 yld ollow approa Baysa sas o sal parar orrspod rspvly o squard rror Hs-Tsokos Harrs ad proposd looral loss uos ar rpla y : prsso o + α α HT α + α + 35 α + α α + α H ad 36 + α α α + α 37 α α + α α + 38
9 CAMAA & TOKO 79 Approa Baysa sas o rlaly orrspod o rs od ar ror v y γ α α Γ α 39 HT + γ α α + α α Γ γ α α + α α Γ wr s approa Baysa sa o sal parar. T approa Baysa rlaly sas orrspod o sod od ar oad y rpla y quaos4 5 6 ad 7. T oad sas orrspod rspvly o squard rror Hs-Tsokos Harrs ad proposd loar loss uos ar v y ollow prssos ar rpla y : γ α Γ α α + α prsso o α + α 4 4 H α + α Γ α γ α γ α α + α Γ α γ α 4 ad γ α α + α Γα α + α 43
10 8 BAYIAN IABIITY MODING UING MONT CAO INTGATION lav y w sp o quard rror oss To opar our rsuls rro o rad a squar rror IM o approa Baysa rlaly sa s usd. Ta s IM d 44 D rlav y as rao o IM o approa Baysa rlaly sas us a all loss uo o a o popular squard rror loss. T rlav s o Hs-Tsokos Harrs ad proposd loar loss ar rspvly dd as ollows: ad HT H IM IM HT HT d d H IM H IM d d d. d I rlav y s sallr a o Baysa sa orrspod o squard rror loss s lss. T squard rror wll or rlav y s rar a o. I rlav y s approaly qual o o Baysa rlaly sas ar qually. Nural ulaos I ural sulaos Baysa ad approa Baysa sas o sal parar or aa alur odl ad looral pror wll opard w squard rror loss s usd ad sap parar α s osdrd d. od w approa wll pld ad approa Baysa rlaly sas wll oad or r-parar Wull ad aa alur odl udr squard rror Hs-Tsokos w Harrs ad loar loss uos rspvly. Coparso w Baysa sas ad approa Baysa sas o sal parar Us squar rror loss uo aa udrly alur odl ad looral pror Tal vs sas o sal parar w sap parar α s d ad qual o o. IM IM
11 CAMAA & TOKO 8 Tal. ooral pror Tru valu o Baysa sa o Approa Baysa sas o Nur o rplas
12 8 BAYIAN IABIITY MODING UING MONT CAO INTGATION T aov rsuls sow a oad approa Baysa sas o parar ar as ood o r a orrspod Baysa sas aus y ar ral losr o ru sa o aur. Approa Baysa laly sas o Tr-parar Wull ad Gaa Falur Modls or dr oss Fuos Us Mo Carlo sulao orao as rspvly rad ro r-parar Wull Wa ad wo-parar aa G α. For a o aov udrly alur odls r dr sapls ar rad o ry alur s ad r u vara uasd sas o sal parar ar oad. Tr-parar Wull Wa A ypal sapl o ry alur s a ar radoly rad ro Wa s v low: T oad u vara uasd sas o sal parar ar v low Ts u vara uasd sas wll usd alo w lklood uo ad looral pror o oa approa Baysa rlaly sas. HT HT H H ad rprs rspvly approa Baysa rlaly sas oad w approa Baysa rlaly sas o sal parar ad os oad y dr opuao w squard rror Hs-Tsokos Harrs ad proposd loar loss uos ar usd. Ts sas ar v low Tal. Tal 3 vs approa Baysa rlaly sas oad drly us quaos ad 3. Gaa alur odl G α A ypal sapl o ry alur s a ar radoly rad ro G α s v low T oad u vara uasd sas o sal parar ar v low
13 CAMAA & TOKO 83 Approao Tal. HT H IM lav y w rsp o T aov approa Baysa sas yld ood sas o ru rlaly uo. Tal 3. T HT H
14 84 BAYIAN IABIITY MODING UING MONT CAO INTGATION Ts u vara uasd sas wll usd alo w lklood uo ad looral pror o oa approa Baysa rlaly sas. HT HT H H ad rprs rspvly approa Baysa rlaly sas oad w approa Baysa sa o ad os oad y dr opuao w squard rror Hs-Tsokos Harrs ad proposd loar loss uos ar usd. Ts sas ar v Tal 5 ad Tal 6. For opuaoal ov rsuls prsd Tal 3 ar usd o oa approa sas o aalyal ors o varous approa Baysa rlaly prssos udr sudy. T rsuls ar v Tal 4. Tal 6 vs approa Baysa rlaly sas oad drly y us quaos ad 43. For opuaoal ov rsuls prsd Tal 6 ar usd o oa approa sas o aalyal ors o varous approa Baysa rlaly prssos udr sudy. T rsuls ar v Tal 7. Tal 4. HT H Approao IM lav y w rsp o 3 Tal 5. HT H Approao 3 4 IM lav y w rsp o
15 CAMAA & TOKO 85 Tal 6. T HT H Tal 7. HT H Approao IM lav y w rsp o 3 T aov approa Baysa sas yld ood sas o ru rlaly uo.
16 86 BAYIAN IABIITY MODING UING MONT CAO INTGATION Coluso Us op o Mo Carlo Irao approa Baysa sas o sal parar wr aalyally oad or r-parar Wull alur odl udr dr loss uos. Us s sas approa Baysa sas o rlaly uo ay oad. Furror op o Mo Carlo Irao ay usd o drly approa sas o Baysa rlaly uo. od slar rsuls wr oad or aa alur odl. Fally ural sulaos o aalyal orulaos da: Approa Baysa rlaly sas ar ral ood sas o ru rlaly uo. W ur o rplas rass approa Baysa rlaly sas oad drly ovr or a loss uo o r orrspod Baysa rlaly sas. rs B G. K. 97. oo pral ays sao w applao o Wull dsruo. NAA Tal Moradu X Ho. V. & Cra A. T Iroduo o aaal sass. Nw York: T Malla Co. o G. H. & Kruko. G A pral Bays soo qu. Borka Marz J oo pral ays sao or o-paraar dsr dsruos. Borka Ta. F Uasd sao: uos o loao ad sal parars. Aals o Ma ad ass os H T pral ays approa o sasal dso prols. Aals o Ma ad ass Brr J. O. 985 asal dso ory ad Baysa aalyss d d.. Nw York: prr. 3 Approa Baysa rlaly sas orrspod o squard loss uo do o always yld s approaos o ru rlaly uo. I a Hs- Tsokos Harrs ad proposd loar loss uos ar sos qually o r.
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The Periodic Table of Elements 8 Uuo Uus Uuh (9) Uup (88) Uuq (89) Uut (8) Uub (8) Rg () 0 Ds (9) 09 Mt (8) 08 Hs (9) 0 h () 0 Sg () 0 Db () 0 Rf () 0 Lr () 88 Ra () 8 Fr () 8 Rn () 8 At (0) 8 Po (09)