Enhanced least squares Monte Carlo method for real-time decision optimizations for evolving natural hazards

Transcription

1 Dowloaded from orbdudk o: Ja 4 29 Ehaced leas squares Moe Carlo mehod for real-me decso opmzaos for evolvg aural hazards Aders Ae; Nshjma Kazuyosh Publcao dae: 22 Lk back o DTU Orb Cao (APA): Aders A & Nshjma K (22) Ehaced leas squares Moe Carlo mehod for real-me decso opmzaos for evolvg aural hazards Paper preseed a 6h IFIP WG 75 Workg Coferece Yereva Armea Geeral rghs Copyrgh ad moral rghs for he publcaos made accessble he publc poral are reaed by he auhors ad/or oher copyrgh owers ad s a codo of accessg publcaos ha users recogse ad abde by he legal requremes assocaed wh hese rghs Users may dowload ad pr oe copy of ay publcao from he publc poral for he purpose of prvae sudy or research You may o furher dsrbue he maeral or use for ay prof-makg acvy or commercal ga You may freely dsrbue he URL defyg he publcao he publc poral If you beleve ha hs docume breaches copyrgh please coac us provdg deals ad we wll remove access o he work mmedaely ad vesgae your clam

2 Ehaced leas squares Moe Carlo mehod for real-me decso opmzaos for evolvg aural hazards A Aders & K Nshjma Deparme of Cvl Egeerg Techcal Uversy of Demark Demark ABSTRACT: The prese paper ams a ehacg a soluo approach proposed by Aders & Nshjma (2) o real-me decso problems cvl egeerg The approach akes bass he Leas Squares Moe Carlo mehod (LSM) orgally proposed by Logsaff & Schwarz (2) for compug Amerca opo prces I Aders & Nshjma (2) he LSM s adaped for a real-me operaoal decso problem; however s foud ha furher mproveme s requred regard o he compuaoal effcecy order o faclae for pracce Ths s he focus he prese paper The dea behd he mproveme of he compuaoal effcecy s o bes ulze he leas squares mehod; e leas squares mehod s appled for esmag he expeced uly for ermal decsos codoal o realzaos of uderlyg radom pheomea a respecve mes a paramerc way The mplemeao ad effcecy of he ehaceme s show wh a example o evacuao a avalache rsk suao INTRODUCTION Real-me decso opmzao has become a eresg ad challegg opc wh he progress of real-me formao processg echology Releva applcaos cvl egeerg clude suaos where operaoal decsos have o be made respose o real-me formao o evolvg aural hazard eves I hese suaos all real-me formao avalable ca ad should be bes ulzed o fd he opmal decsos a respecve mes; akg o accou o oly possble fuure oucomes bu also opporues o make decsos fuure mes Ths ype of decso problem s geerally descrbed wh he framework of he preposeror/sequeal decso aalyss see Nshjma e al (29); however he developme of effce soluo schemes o he formulaed decso problems has remaed a echcal challege A effce soluo scheme s proposed by Aders & Nshjma (2) akg bass he Leas Squares Moe Carlo mehod (hereafer abbrevaed as LSM) whch s developed orgally by Logsaff & Schwarz (2) for Amerca opo prcg I Aders & Nshjma (2) he orgal LSM s exeded ad appled o a example for a real-me operaoal decso problem for shu-dow of he operao of a echcal facly he face of a approachg yphoo However due o mulple evaluaos of he expeced cosequeces for dffere possble fuure saes of he yphoo by meas of Moe Carlo smulao (MCS) he soluo scheme becomes less effce f he compuaoal me requred for MCS becomes doma The prese paper proposes a ehaced soluo scheme whch overcomes hs drawback The prese paper s orgazed as follows Seco 2 formulaes he real-me decso problems cosderao wh he framework preseed Nshjma & Aders (22) Seco 3 provdes a bref roduco o he exesos of he LSM Thereafer he proposed ehaceme o he exeded LSM s roduced Seco 4 preses a applcao example whch llusraes he performace of he ehaced LSM (elsm) Seco 5 cocludes he preseed work

3 2 REAL-TIME DECISION FRAMEWORK 2 Problem seg The decso suao cosdered he prese work s characerzed by he followg characerscs see Nshjma e al (29): (a) The hazard process evolves relavely slowly ad allows for reacve decso makg; (b) formao releva o predc he severy of he evolvg hazard eve ca be obaed pror o s mpac; (c) he decso makg s subjec o uceraes par of whch mgh be reduced a a cos; (d) decso makers have opos for rsk reducg acves whch may be commeced a ay me suppored by he formao avalable up o he me Here wag o commece he rsk reducg measures mples he reduco of uceray bu mgh also reduce avalable me o complee he rsk reducg acves; (e) ad o op of all he decsos mus be made fas ear-real me The decso makers are he requred o make decsos wheher hey commece oe of he rsk reducg acves whch a he same me ermaes he decso process (hece hereafer hese are called ermal decsos) or hey pospoe makg a ermal decso 22 Formulao of decso problem The decso problem characerzed above ca be formulaed accordace wh Nshjma & Aders (22) Deoe by A he decso se cossg of possble decso aleraves a me Here me s dscrezed I s assumed ha he decsos mus be ermaed before or a me ; hece {2 } The decso se A geerally depeds o he decsos made before me If a decso maker decdes o ermae he decso process o decso alerave s avalable a laer decso mes I s hus covee o dvde he decso se o ( c) ( s) ( c) ( s) wo muually exclusve subses; e A A A A A where A ( c) cosss of oe decso alerave a c wag (e ( c ) { ( c ) A a }) ad A s s he se cossg of rsk reducg decsos avalable Le E be a se of varables represeg possble formao avalable a me o he saes of he evolvg aural hazard eve cosderao * Gve ha o ermal decso s made up o me he opmal decso a a me s defed as he oe ha maxmzes he expeced uly a me codoal o he colleco of he formao up o me : max E[ U( Z a) e] for * aa EU [ ( Z a ) e] () max EU [ ( ) ] for ( s ) Z a e aa where for ad a ( c) ( c) * ( c) E[ U( Z a ) e] E[ U ( Z a ) a e ] f( e e) de (2) Here U( z a) s he uly whch s a fuco of he decso alerave a ad he realzao z of he hazard dex Z releva for he decso problem The hazard dex Z s defed hrough he uderlyg radom sequece { Y } represeg he evoluo of he aural hazard eve e ( e e e ) s he colleco of he formao avalable up o me Here s assumed ha y e ( ) ; amely he sae of he eve releva o he decso problem s kow o he decso maker whou uceray Thus he symbols y ad e are ulzed erchageably he followg f( e ) s he codoal probably desy/mass fuco of formao E gve E e From Equao 2 s see ha for he decso a c a me he opmzao requres o kow all opmal decsos a fuure mes 2 ; hece backward duco s requred Equao ca be rewre as: Here max h( e) c( e) for q( e) h( e) for q e E U Z a e (4) * [ ( ) ] (3)

4 h ( e ) max l ( a e ) (5) ( s ) a A l ( a e ) E[ U ( Z a ) e ] a A (6) ( s) c( e) E[ q (( e E )) e ] (7) The fuco q( e ) s he maxmzed expeced uly hereafer abbrevaed as MEU The fucos h( e ) ad c( e ) are amed soppg value fuco (SVF) ad coug value fuco (CVF) respecvely Noe ha whereas he evaluao of he SVF s sraghforward he sese ha does o requre backward duco he evaluao of CVF requres backward duco However o maer how complex he srucure of he decso opmzao problem may seem c( e ) s oly a fuco of e Furhermore f he uderlyg radom sequece { Y } follows s h -order Markov sequece c( e ) s a fuco effecvely of he las s formao es es2 e 3 ENHANCEMENT OF THE EXTENDED LSM 3 Exeded LSM The ma echcal challege of he opmzao problem formulaed Seco 22 s he evaluao of he CVF The CVF ca prcple be evaluaed by calculag he expeced uly for each combao of all possble dscrezed fuure saes ad possble decso opporues However pracce hs s o compuaoally feasble sce he oal umber of he possble combaos creases expoeally as a fuco of he umber The LSM crcumves hs by employg he leas squares mehod The dea behd he LSM s ha ay regular fuco ca be represeed by a lear combao of a approprae se of bass fucos; herefore he CVF s approxmaed as such for deals see Logsaff & Schwarz (2) I he coex of Amerca opo prcg hs meas ha f he prce of a sock follows a frs order Markov sequece he prce of s Amerca opo s a fuco oly of he curre sock prce Cosequely he CVF s approxmaed as a superposo of bass fucos whose argume s oly he curre sock prce The way o how hs dea s mplemeed he opmzao s explaed alog wh he exeded verso of he LSM (called exeded LSM) he followg I Aders & Nshjma (2) s demosraed ha he dea behd he LSM ca be appled for he case where he uderlyg radom sequece follows a homogeeous hgher-order Markov sequece There wo exesos are made: () he assumpos o he uderlyg radom sequece s relaxed from saoary frs-order Markov sequece o o-saoary hgher-order Markov sequece ad (2) he SVF s evaluaed by MCS Noe ha may egeerg applcaos he SVF cao be evaluaed aalycally ulke he case whe execug Amerca opos Moreover he MCS he secod exeso s compuaoally expesve ad he compuaoal effor creases proporoal o I he followg he seps of he exeded LSM are preseed: Sep : A se of b depede realzaos (pahs) of he radom sequece Y s geeraed by MCS accordg o he Markov raso desy f( y y ) wh he al codo Y y where y ( y y y ) These pahs are deoed by y ( y y y ) 2 b where y y for all pahs see Fgure (a) Sep 2: The SVF for all realzaos { y } 2 b are esmaed by addoal MCS Sep 3: Sarg a he me horzo as llusraed Fgure (a) for each pah he value of he MEU q(( y Y )) s defed by equag q ( y ) h ( y ) accordg o Equao 3 Sep 4: Movg o me he CVF s approxmaed Ths begs by relag each MEU q( y ) o y o oba he daase ( y q( y )) 2 b see he dos Fgure (b) Ths daase s ulzed o approxmae he CVF c ( y -) wh he leas squares mehod The

5 approxmaed CVF s llusraed by he curve Fgure (b) See Nshjma & Aders (22) for deals The approxmaed CVF s deoed by cˆ ( y ) Sep 5: Havg obaed cˆ ( y ) for me he realzaos of q (( y 2 Y )) e ( y ) 2 b are deermed as follows: q q h ( y ) f h ( y ) cˆ ( y ) ( y ) q( y) oherwse The procedure s repeaed backwards me ul hece q( y ) s obaed for all pahs Sep 6: A he esmae cˆ cˆ ( y ) s defed as he average of he realzaos q( y ) 2 b Fally q( y ) s obaed as he maxmum of cˆ ( y ) ad h ( y ) The opmal decso s he oe ha correspods o he maxmum (8) Fgure Illusrao of (a) hree pahs of a uderlyg radom sequece wh correspodg values q( y ) ( 23) a me ad (b) he esmao of he CVF usg he ses ( y q( y )) 32 Ehaceme of he exeded LSM As see Seco 3 addoal MCS are requred Sep 2 o esmae he SVF he exeded LSM The ehaced LSM (elsm) crcumves hs by applyg he leas squares mehod for he esmao of he SVF The geeral dea s explaed he followg Aalogous o Equao 5 he SVF h elsm y of he elsm s defed as maxmum of he codoal expeced ules ( j ) ( j) ( s) lelsm a y wh respec o he ermal decsos a A Here he fucos ( j l ) elsm a y are esmaed wh he leas squares mehod usg he realzaos { y } b smlar o he esmao of he CVF descrbed Seco 3; e by lear combao of bass fucos { ()} K ( j) L wh ukow coeffces r k k k ( j) K ( j) elsm k k k l ( a y ) L ( y ) r (9) There he leas squares mehod s ulzed o esmae he coeffces r j ( j j 2 j ) T r r r K by mmzg he sum of he squared dsaces bewee he observed realzaos of he depede varable ( j l ) elsm a y he daase ad her fed values; he marx form hs s expressed by ( j) ( j) 2 r arg m r u Lr 2 () where 2 deoes he Euclda orm L s a b K marx cossg of values of bass fucos { L ()} K k k whch are fucos of realzaos of y ad u ( j) he b vecor of ( j) observed fuure ules u( z a ) 2 b gve he realzao z of he hazard dex relaed o he pah y ad decso ( j) ( j) a s made a me Noe ha u( z a ) s a realzao of ( j l ) elsm a y Furhermore o avod a bas roduced by he leas squares esmao wh he deermao of he MEU Equao 8 s chaged o: q elsm * * ˆ u ( ) f ˆ z a helsm ( y) celsm ( y) ( y) q elsm ( y ) oherwse () where u ( z a ) s he observed fuure uly of pah for he opmal ermal decso * * * a

6 4 EXAMPLE The am of hs seco s o demosrae how he elsm ca be appled o a egeerg decso problem ad o compare s performace o ha of he exeded LSM For hs purpose a decso suao of he evacuao of people he face of a avalache eve s cosdered 4 Problem seg Cosder a vllage locaed earby a moua slope havg a crcal agle for sow avalaches Gve prevalg wer codos ad crcal sow heghs a decso has o be made wheher o evacuae people from he vllage Assume ha he occurrece of a severe avalache causg sgfca damages o he vllage depeds oly o he addoal sow hegh S ; e S s he hazard dex Furher f S exceeds he hreshold s ( 8[ mm]) a severe avalache occurs Weaher forecas by a meeorologcal agecy predcs ha sowfall ca occur wh he ex hours whch creases he lkelhood of he occurrece of he avalache However he durao ad he esy of he sowfall are ucera New formao becomes avalable every 8 hours from he meeorologcal agecy; e he me erval bewee he subseque decso phases s se o 8 hours ( d 8 ) A each decso phase a decso s made accordg o formao avalable Three decso aleraves are assumed; e o evacuae he people a o () (2) ( c) o evacuae a ad o wa a I s assumed ha he evacuao akes 6 hours o complee 42 Cosequece model The cosequeces are posulaed as follows see also Table : The cosequece s equal o C Ev wo cases: () whe he evacuao has bee aed bu he avalache does o occur ad (2) whe he evacuao s compleed before he avalache occurs A cosequece of C D s curred f he avalache occurs ad he people are o evacuaed or he evacuao was aed bu o compleed No cosequece s curred oly he case whe o evacuao s aed ad o avalache occurs Table Codos ad assocaed cosequeces posulaed he cosequece model Addoal sow hegh he me perod [ ] People S s 8[ mm] S s 8[ mm] No evacuaed C D Evacuaed C C Ev Ev 43 Probablsc sowfall model A hypohecal probablsc sowfall model s assumed whch s adaped from a rafall model developed by Hydma & Gruwald (2) Le X deoe he radom sequece represeg he amou of sowfall he me perod ( d ] Hereafer hs me perod s deoed by ( ] (e he me u s d 8 ) ad hus { X } for smplcy The dsrbuo of X s a mxure comprsg a dscree compoe coceraed a x ad a couous compoe for x The dscree compoe of X represes he o-occurrece of sowfall ad s characerzed by he Beroull sequece J whose codoal probably fuco s: ( y y2) PJ ( Y y Y y2) lµ ( ( y y 2)) (2) where Y ( J X ) ad l() deoes he log fuco whch s defed as l exp / ( exp) f ad l oherwse ad 2 µ ( y y 2) j 2j2 3log( x c) 4log( x2 c2 ) 5 (3) The couous compoe of X s srcly posve ad characerzes he esy of he sowfall If J X s descrbed by he couous codoal desy g( x y ) x g ( ) follows he Gamma dsrbuo wh shape parameer ad mea ( y ) where log( ( y )) j log( x c ) (4)

7 The he raso probably desy fuco of X s defed as (see Fgure 2): f( x y y2) ( ( y y2)) ( x) ( y y2) g( x y ) (5) where s he Drac dela fuco The addoal sow hegh s obaed by mulplyg he sow esy by he facor F whch accous for he desy of he sow; e s S S ( y ) F x S Fx (6) j j s s s Hece S (he hazard dex) a me s characerzed by he dex S a me ad a sochasc process composed of a secod- ad a frs-order Markov process (he secod erm he rghmos equao) The values of he parameers of he model are summarzed Table 2 The me frame s se o hree days; e 9 Table 2 Parameers of he probablsc sowfall model Parameer Value Parameer Value j j S c ( c c c ) 2 3 (535) α ( ) 5 ( ) 5 β ( ) 2 3 ( ) F s Fgure 2 Illusrao of f ( x y y ) 2 44 Soluo wh he elsm Here he MEU Equao 3 s defed by he expeced cosequece; e he mmum operaor s used ad he equaly sg of Equao 8 s ured The seps Seco 3 are execued wh he exeded LSM ad he elsm o oba he opmal decso Sep : By MCS geerae b depede realzaos of { Y } ad S ( S S S ) 2 b where S S( y ) ad ( y j x) The realzaos y y2 y are smulaed accordg o he probably desy fucos Equaos 2 ad 5; he pahs are deoed by y ( y y y ) where y y y y ad 2 b Sep 2: For each y he value h h( y y ) of he SVF s esmaed A me 9 he cosequece relaed o each realzao ad decso s assumed o be kow; e eher s exceeds he hreshold s or o hus hmc helsm for all Furher for 2 m () wh he exeded LSM: Smulao of addoal M pahs m ( m y y y y y ) m ( j) m 2 M for whch he observed cosequeces u( s a ) j 2 are deermed m Here s s he realzao of he addoal sow hegh relaed o he pah realzao y m Defe lˆ MC( j ) M ( m j a y y u s )/ m a M he ˆ ˆ () ˆ (2) h m{ l ( a y y ) l ( a y y )} (7) MC MC MC (2) wh he elsm as explaed Seco 32: Defe ˆ ˆ () ˆ (2) helsm m{ lelsm ( a y y ) lelsm ( a y y )} (8) where ˆ j j lelsm ( a y y ) L r j 2 The vecor j ( j) r of he coeffces relaed o a s compued by Equao L deoes he h row of marx L ; L cosss of values of bass fucos wh argumes y y ad S ; eg for s order lear bass fucos

8 x x s x x s L b b b x x s For se ˆ ( j ) ˆ ( j ) ˆ ( j ) b ( j l ) lmc( a y y ) lelsm( a y y ) u ( s a )/ b j 2 Sep 3: Sarg a me for boh LSM approaches he values of qmc qmc( y y ) ad q elsm are se equal o h MC ad h elsm respecvely for all Sep 4: Movg o me he values of c ( y y 2) are smlarly esmaed for boh approaches usg he leas squares mehod as descrbed Seco 3 Sep 5: The for each pah deerme he values of q ( y y 2) : ˆ () for he exeded LSM wh he esmae h MC obaed by meas of MCS: ˆ ˆ h ˆ MC f h MC c MC q MC (2) oherwse q MC (2) for elsm wh he esmae ˆ h elsm obaed by meas of he leas squares mehod: * ˆ u ˆ f h elsm c elsm q elsm (2) oherwse q elsm * where u deoes he observed fuure cosequece pah for he opmal ermal decso a * As Seco 3 movg aoher me sep back he same procedure s repeaed Ths s coued ul me ad for each pah q MC ad q elsm are deermed Sep 6: Execue Sep 6 of Seco 3 45 Resuls To evaluae he performace of he elsm compared o he exeded LSM boh mehods are appled o solve he decso problem of he example The opmal decso a he al me s obaed by esmag he expeced cosequeces for he hree decsos aleraves Varous ypes ad degrees of bass fucos are mplemeed; eg lear Legedre ad Chebyshev polyomals Applyg hese bass fucos s foud ha he resuls do o sgfcaly dffer Thus oly he resuls obaed wh lear bass fucos are preseed Fgure 3 llusraes he fdgs for dffere parameer segs of he LSM There Fgure (a) shows for creasg umber b of pahs b {333} he covergece of he cosequece esmaes for he hree decsos For each b he esmaes are calculaed by he average of compuaos of he dcaed mehod To be able o compare he resuls dffere ye fxed ses of radom umbers are used o geerae he pahs Sep Hece he esmaes for he ermal decsos are decal for all mehods; hey are preseed by sold les wh crcles The followg resuls are obaed for b : lˆ 5 () 92 ˆ(2) ( c) l 8969 ad eg cˆ elsm 855 wh he elsm The opmal decso s a whch s depede of he ype of LSM; see Fgure 3 (a) Furher he fgure shows ha he esmae ĉ obaed by he exeded LSM wh M s based Therefore s o cosdered Fgure 3 (b) whch llusraes he covergece rae erms of he coeffce of varao (COV) of he esmaes ĉ as a fuco of he compuaoal me [sec] The fgure shows a sgfca mproveme wh he elsm erms of compuaoal me; a reduco by he facor of A applcao of he proposed approach pracce s preseed Fgure 4 Fgure 4 (a) llusraes a hypohecal me seres of he addoal sow hegh { S 6 } where he hreshold s s exceeded wh he me erval (34] Applyg he elsm subsequely for each me sep s foud ha he opmal decso a me s a ( c) whereas a me s foud () o be a gve ha he sow hegh a me he fgure s realzed (9)

9 (a) a () (b) Evacuae 4 elsm Exeded LSM M= 3 M= Do o evacuae a 9 (2) 2 elsm M= Wa a (c) 8 M= M= Toal umber of pahs (b ) Calculao me [sec] Fgure 3 Comparso of he resuls of he exeded LSM (wh varous umbers M of addoal MCS) ad elsm (a) Covergece of he average expeced cosequeces wh creasg oal umber of pahs (b) Illusrao of he decreasg COV of ĉ relaed o he creasg calculao me for oe LSM compuao as he umber b of pahs creases Expeced cosequece (a) (b) Threshold s a 4 4 a (2) 2 2 a (c) Tme sep Tme sep Fgure 4 Illusrao of (a) a hypohecal me seres of S ad (b) he correspodg me seres of he 5 esmaed expeced cosequece of he hree decso aleraves calculaed wh he elsm ad b S [mm] COV Expeced cosequece 5 CONCLUSION The prese paper proposes a ehaceme of he exeded LSM he coex of real-me operaoal decso problems for evacuao he face of emergg aural hazards The proposed approach (elsm) s appled o a example ad s foud ha he elsm sgfcaly mproves he compuaoal effcecy; by he facor up o 6 ACKNOWLEDGEMENT Ths research was parly suppored by he Swss Naoal Scece Foudao (Projec umber: ) REFERENCES Aders A & Nshjma K 2 Adapo of opo prcg algorhm o real me decso opmzao he face of emergg aural hazards Proceedgs of h Ieraoal Coferece o Applcaos of Sascs ad Probably Cvl Egeerg M H Faber J Köhler ad K Nshjma Zurch Swzerlad Hydma R J & Gruwald G K 2 Geeralzed addve modellg of mxed dsrbuo Markov models wh applcao o Melboure's rafall Ausrala & New Zealad Joural of Sascs 42 (2): pp Logsaff F A & Schwarz E S 2 Valug Amerca Opos by Smulao: A Smple Leas- Squares Approach The Revew of Facal Sudes 4 (): pp 3-47 Nshjma K & Aders A 22 Opmzao of sequeal decsos by leas squares Moe Carlo mehod Proceedgs of 6h IFIP WG 75 Workg Coferece Armea Yereva Nshjma K Graf M & Faber M H 29 Opmal evacuao ad shu-dow decsos he face of emergg aural hazards Proceedgs of ICOSSAR29 H Furua D M Fragopol ad M Shozuka Osaka Japa