A survey of rare event simulation methods for static input–output models

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A survey of rare event simulation methods for static input–output models

Jérôme Morio, Mathieu Balesdent, Damien Jacquemart, Christelle Vergé

To cite this version:

Jérôme Morio, Mathieu Balesdent, Damien Jacquemart, Christelle Vergé. A survey of rare event simulation methods for static input–output models. Simulation Modelling Practice and Theory, Elsevier, 2014, 49, pp.287-304. �10.1016/j.simpat.2014.10.007�. �hal-01081888�

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input-output models

JérmeMorio

1,∗

MathieuBalesdent

2

Damien Jaquemart

2,3

Christelle Vergé

2,4,5

Abstrat

Crude Monte-Carlo or quasi Monte-Carlo methods are well suited to haraterize events of

whihassoiatedprobabilities arenottoolowwithrespettothesimulationbudget.Forvery

seldomobservedevents,suhastheollisionprobabilitybetweentwoairraftinairspae,these

approahes donot lead to aurateresults. Indeed, the numberof available samples is often

insuient to estimate suh low probabilities (at least 10⁶ samples are needed to estimate a probability of order 10⁻⁴ with 10% relative error with Monte-Carlo simulations). In this artile,one reviewed dierent appropriatetehniquesto estimate rare event probabilities that

require afewernumberof samples.These methods an bedivided into four main ategories:

parameterizationtehniquesofprobabilitydensityfuntiontails,simulationtehniquessuhas

importanesamplingorimportanesplitting,geometri methodstoapproximateinput failure

spaeandnally,surrogatemodelling.Eahtehniqueisdetailed,itsadvantagesanddrawbaks

aredesribedandasynthesisthataimsatgivingsomeluestothefollowing questionisgiven:

"whihtehniquetousefor whihproblem?".

Keywords: Monte-Carlomethods,Rareevent,Input-outputmodel,Simulation

∗ orrespondingauthor

Emailaddresses: jerome.morioonera.fr (JérmeMorio),mathieu.balesdentonera .fr

(MathieuBalesdent),damien.jaquemartonera .fr(DamienJaquemart),

hristelle.vergeonera.f r(ChristelleVergé).

1

Onera-TheFrenhAerospaeLab,BP74025,31055ToulouseCedex,FraneTel.:+33562252663

2

Onera-TheFrenhAerospaeLab,BP80100,91123PalaiseauCedex,Frane

3

INRIARennes,ASPIAppliationsofinteratingpartile systemstostatistis, ampusdeBeaulieu,

35042Rennes,Frane

4

INRIABordeaux,351oursdelaLibération,33405TaleneCedex,Frane

5

CNES,18avenueEdouardBelin,31401ToulouseCedex9,Frane

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Rareeventestimationhasbeomealargeareaofresearhinthereliabilityengineering

andsystemsafetydomains.Asigniantnumberofmethodshasbeenproposedtoredue

theomputationburdenfortheestimationofrareeventsfromsamplingtoextremevalue

theory.Howeveritisoftendiulttodeterminewhihalgorithmisthemostadaptedto

agivenproblem.Moreover,theexisting surveyartilesonrareeventsare oftenfoused

onspeialgorithms[13℄.Thenoveltiesofthisartilearethustoprovideabroadview

of the urrent available tehniquesto estimate rare event probabilities desribed with

aunied notationand to provide someluesto answerthis question:whih rareevent

tehniqueisthemostadaptedtoagivensituation?

Thegeneralproblemonsideredin thisartileisanalysedin arstsetionandthenall

thedierentmethodsaredesribedseparately.Theiradvantagesanddrawbaksarealso

given.Finally,asynthesishelpsthereadertodeterminethemostappropriatemethodto

agivenrareeventestimationproblem.

Let us onsider a d-dimensional random vetor X with a probability density funtion (PDF) h0^, φ â ôntinuous ^positive ^salar ^funtion φ : R^d → R ând S â ^threshold.

The dierent omponents of X will be denoted X= (X¹, X², ..., X^d) ⁱⁿ ^the ^following.

The funtion φ îs ^stati, î.e., ^does^not^depend ôn ^time, ând ^represents^for înstane ân

input-outputmodel. Thiskindofmodel isnotablyusedin numerousengineering appli-

ations[49℄.Weassumethat theoutputY =φ(X)îsâ^salar^random^variable.În^this

artile, we propose to review dierent algorithms that an be eient to estimate the

probabilityP =P(φ(X)> S)^when^this^quantity^is^rare^relatively^to^the^available^sim-

ulationbudgetN^,^thatîs^whenP < _N¹^.^For^the^sakeôfôniseness,^theîssueôfêxtreme

quantileestimation is notaddressed even ifthe vast majority of the methods that are

presentedinthepaperanbeadaptedtothisspeiase.Theaseofdynamisystems

modeledwith Markovhains isalsonotonsidered inthispaper.Speialgorithmex-

tensionsfor large omplexsystems modelled by anetwork ora oherent fault tree are

ompletely detailed in [10℄ and willnot be muhdevelopedhere. Itorrespondsto the

asewhere theinputs Xⁱ^, i = 1, ..., d ^follow ^a^Bernoullidistribution andthe output is equivalenttoanindiatorfuntion.

2. Monte-Carlomethods

AsimplewaytoestimateaprobabilityistoonsiderrudeMonte-Carlo(CMC) [11

16℄. Forthat purpose, onegenerates N independent and identially distributed (i.i.d.) samplesX₁, ...,X_N from thePDF h0 ând ômputes^theirôutputs ^with^the ^funtion φ^: φ(X₁), ..., φ(X_N)^. ^TheprobabilityP(φ(X) > S)^, âlso âlled ^failure probability, is then estimatedwith

Pˆ^CMC = 1 N

XN i=1

1_φ(_X

i)>S, ⁽¹⁾

where1_φ(_X

i)>Sîsêqual^to1 îfφ(X_i)> Sând0ôtherwise.^Thisêstimationônverges^to

therealprobabilityasshowsthelawoflargenumbers[13℄.Thepositiveandnegativeas-

petsofCMCaredesribedinTable1.Apossibleindiatoroftheestimationeienyis

notablyitsrelativedeviation.TherelativedeviationorrelativeerrorREôfânêstimator

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Simpleimplementation Slowonvergene

Informationonφ^not^needed^Signiant^simulation^budget^for^rare^events

Nobias

Table1

AdvantagesanddrawbaksofCMCmethods.

Pˆ ^ofP ^is^given^by^the^following^ratio:

RE( ˆP) = σPˆ

E( ˆP), ⁽²⁾

withσPˆ ^the^standard^deviation ôfPˆ ândE^themathematialexpetation.Therelative errorissaidboundedwhenRE( ˆP)^remains^bounded^whenP −→0^[17,18℄.În^thatâse,

thenumberofsamplesneededto getaspeiedrelativeerroris bounded whateverthe

rarityof φ(X)> S^. ^The^logarithmi êieny LE ânâlso ^be ^dened ^forân ûnbiased

estimatorPˆ ^with^[17,18℄,

LE( ˆP) = lim

P→0

log(E( ˆP²))

log(P) = 2. ⁽³⁾

Logarithmi eieny is a neessary but not suient ondition for bounded relative

error. Charaterizing the rare event probability estimate with these onepts is very

importanteveniftheyareoftendiultto verifyin pratie.

SinePˆ^CMC îsûnbiased,^the^relativeêrrorôf ^theêstimatorPˆ^CMC îs ^given^by^the^ratio

σP CM Cˆ

P ^with σPˆ^{CM C}^, ^the^standard^deviation ^ofPˆ^CMC^.^Knowing^the^trueprobabilityP

oftheevent(φ(X)> S)^,^one^has^[11,19℄

σPˆ^{CM C}

P = 1

√N

√P−P²

P . ⁽⁴⁾

Considering rare event probability estimation, that is when P ^takes ^low ^values, ^one

obtains

P→0lim σPˆ^{CM C}

P = lim

P→0

√1

N P = +∞. ⁽⁵⁾

Therelativedeviationisonsequentlyunbounded.Forinstane,toestimateaprobability

P ôf ôrder 10⁻⁴ ^with â10%^relative ^deviation, ât ^least 10⁶ ^samplesâre ^required. ^The

simulationbudgetisthusanissuewhentheomputationtimerequiredtoobtainasample

φ(X_i)^is ^not negligible.CMC is thus notadapted to rareevent estimation and awide olletionofstatistiandsimulationmethodshasbeendeveloped.Thefollowingsetions

desribethedierentavailablealternativestoCMC toimproveprobabilityestimations,

i.e., to redue the number of required samples, inrease the estimation auray, and

thusdereaseRE( ˆP)^.

3. Statistial tehniques

Statistialtehniquesenabletoderiveaprobabilityestimateandassoiatedondene

intervalswithaxedset ofsamplesφ(X₁), ..., φ(X_N)^. ^The^main^statistial ^approahes,

extremevaluetheoryandlargedeviationtheory,modelthebehaviourofthePDFtails.

Letusreviewtheirtheoretialfounding.

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Extreme value theory(EVT) [20,21℄haraterizes thedistribution tailsof arandom

variable, basedon areasonablenumberof observations. Thanksto itsgeneralapplia-

tive onditions,thistheory hasbeenwidelyused fordesribing extrememeteorologial

phenomena with appliations suh as hydrology[22℄, snowfall [23℄, but also in nane

andinsurane[20,24℄,andengineering[25℄.

3.1.1. Law ofsample maxima

EVT isnotablyveryuseful whenonehasto work withonly axedset of data.One

onsequentlyassumesinthefollowingthat anitesetofi.i.d.samplesφ(X₁), ..., φ(X_N)

oftheoutput isavailable,butalsothat oneannotgeneratenewsamplesofφ(X)^.^The

assoiated orderedsampleset is denedwith φ(X₍₁₎)≤φ(X₍₂₎)≤...≤φ(X_(N₎)^. ^EVT

enablesto estimateforsomethresholdS^theprobabilityP(φ(X)> S)^.

Thefounder theorem of EVT [20,26,27℄isthat, under someonditions,themaxima of

ani.i.d.sequeneonvergetoageneralizedextremevalue(GEV) distributionGξ^,^whih

admitsthefollowingumulativedistributionfuntion (CDF)

Gξ(x) =







exp(−exp(−x)), ^for ξ= 0, exp

−(1 +ξx)⁻¹^ξ

, ^for ξ6= 0.

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ThesetofGEVdistributionsisomposedofthreedistinttypes,haraterizedbyξ= 0^,ξ > 0 ând ξ <0 ^that ôrrespond ^to ^the^Gumbel, ^Fréhet ând ^Weibulldistributions respetively.Letusdene G^,^the^CDFôf^theî.i.d.^samplesφ(X₁), ..., φ(X_N)^.

Theorem3.1 SupposethereexistaN ^andbN^,^withaN >0^suh^that,^for ^all y∈R P

φ(X_(N₎)−bN

aN ≤y

=G^N(aNy+bN)^N→∞−→ G(y),

where Gîs â^non ^degenerate ^CDF, ^then Gîs â^GEV distributionGξ^.În ^this âse, ône

denotes G∈M DA(ξ)(MDA=maximumdomain ofattration).

ThesequenesaN ând bN âreômputedⁱⁿ ^[20℄^for ^most^well-known^PDF. Ân âpproxi-

mationofP(φ(X)> S)^[20℄ ^for^large^valuesôfSând N ânâlso^beôbtained:

Pˆ^{EV T}(φ(X)> S)≈ 1 N

1 +ξ

S−bN

aN

−¹_ξ

. ⁽⁷⁾

TheGEVapproahisnotablyusedwhenonlysamplesofmaximaareavailable.Inthat

ase,thedierentparametersoftheGEVdistributionareobtainedbydeterminingmax-

imumlikelihood orprobabilityweightedmomentestimators.Whensamples ofmaxima

arenotavailable,itisrequiredtogroupthesamplesφ(X₁), ..., φ(X_N)^into^bloks^and^t

theGEVusingthemaximumofeahblok(blokmaximamethod).Themaindiulty

istodetermineaneientsamplesizeforthedierentbloks.

3.1.2. Peak overthreshold approah

Insteadofgroupingthesamplesintoblokmaxima,POTonsidersthelargestsamples

φ(X_i)^to^estimate^theprobabilityP(φ(X)> S)^.

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to haraterizethe distribution of samples above athreshold u^, ^whih ^is ^given ^by ^the

generalizedParetoCDF.AnalternativeistouseaPoissonpointproesswhihountsthe

numberofthresholdexeedanes.Thisapproahisnotdevelopedinthisartile,butone

anreferto[27℄ formoredetails.TherstpaperlinkingtheEVTwith thedistribution

ofathresholdexeedaneis[28℄.Later,DeHaanobtainsaresultofthesametype,with

aslightlysimpliedonlusion,usingslowvaryingfuntions[29℄.Thefollowingtheorem

[20℄anbethenobtained:

Theorem3.2 LetusassumethatthedistributionfuntionGôfî.i.d.^samplesφ(X₁)^,..., φ(X_N)îsôntinuous.^Set y^∗= sup{y, G(y)<1}= inf{y, G(y) = 1}^.^Then,^the ^two^fol-

lowing assertionsareequivalent

(i) G∈M DA(ξ)^,

(ii) thereexistsapositive andmeasurablefuntion u7→β(u) ^suh^that

u7→ylim^∗ sup

0<y<y^∗−u|G^u(y)−Hξ,β(u)(y)|= 0,

where Gû(y) = P(φ(X)−u ≤ y|φ(X) > u)^, ând H_ξ,β(u) îs ^the ^CDF ôf â generalized Pareto distribution(GPD) withshape parameterξând^sale^parameter β(u)^.

TheexpressionoftheGPD distributionfuntion isthefollowing

Hξ,β(x) =









1−exp

−^xβ

, ^forξ= 0, 1−

1 + ^ξx_β−1/ξ

, ^forξ6= 0.

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This theorem is in fat useful to estimate a probability of exeedane. Indeed, the

probabilityP(φ(X)> S)^an^be^rewritten^as

P(φ(X)> S) =P(φ(X)> S|φ(X)> u)P(φ(X)> u). ⁽⁹⁾

forS > u^. Â^naturalêstimateôfP(φ(X)> u)îs^given^by Pˆ^CMC(φ(X)> u) = 1

N XN i=1

1_φ(_X

i)>u. ⁽¹⁰⁾

With theTheorem3.2 andforsigniantvalueofu^,^one^obtains

Pˆ(φ(X)> S|φ(X)> u) = 1−H_ξ,β(u)(S−u). ⁽¹¹⁾

TheestimateofP(φ(X)> S)^is^then^built^with Pˆ^{P OT}(φ(X)> S) = 1

N XN i=1

1_φ(_X

i)>u

!

× 1−H_ξ,β(u)(S−u)

. ⁽¹²⁾

ThemathematialjustiationofEq.11andEq.12isnotablydisussedin[21℄,[30℄,[31℄,

or[32℄ foragivenset of samplesto determine ifthis set issuitable forthe appliation

of POT. Three parameters haveto be determined in the POTprobability estimate of

Eq. 12: thethreshold uând ^the ôuple(ξ, β(u))^. ^The^hoie ôf u îs^veryînuent^sine

it determines the samples that are used in the estimation of (ξ, β(u))^. ^Indeed, ^a ^high

thresholdleadstoonsideronlyasmallnumberofsamplesintheestimationof(ξ, β(u))

andthustheirestimateanbethenspoiledbyalargevarianewhereasalowthreshold

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Noneedtoresample Complexestimationoftheadequateparameters

(u, ξ, β(u))^or^of^the^blok^maxima^size.

CanbeappliedwitharelativelylowvalueofN ^Less^eient^than^simulation

methodswhenresamplingispossible

Table2

AdvantagesanddrawbaksofEVT.

introduesabiasintheprobabilityestimate[33℄.Thereareseveralmethodstodetermine

avaluablethresholdu^knowing^the^samples.^The^most^well-known^ones^are^the^Hill^plot

andthemeanexessplot[20℄.Thesemethodsareneverthelessveryempirialsinethey

are based on graphial interpretation. It is often neessary in pratie to ompare the

estimatesofu^given^by^the^dierent^methods.Ône^the^valueôfuîs^set,^the^parameters (ξ, β(u)) âre ôften êstimated ^by ^maximum ^likelihood ^[34℄ ôr ^more ôasionally^by ^the

method of moments [35℄.Theestimate Pˆ^{P OT}(φ(X)> S)^givenⁱⁿ ^Eq. ¹² ^for S > u ^is

thenompletely dened.A reviewofthese dierentmethods anbefound in[36℄.It is

notpossible,toourknowledge,toontroltheprobabilityerrorestimateinEVT.Never-

theless,theuseofboostraponsamplesφ(X₁), ..., φ(X_N)^[37℄^an^give^someinformation ontheeieny ofEVT.

3.1.3. Blok maximaversus POT

ThePOTmethodtakesintoaountallrelevanthighsamplesφ(X₁), ..., φ(X_N)^whereas

theblokmaximamethod anmiss someofthese highsamplesand, onthesametime,

onsidersomelowersamplesinitsprobabilityestimation.Thus,POTseemstobemore

appropriateforthedesignofsample PDFtail.Nevertheless, theblokmaxima method

is preferablewhen the available samples are notexatlyi.i.d. orwhen only samples of

maxima are available. For instane, the samples of a monthly river maximum height

orrespond to this situation. Finally, the tuning of blok maxima size turns out to be

easierthanthetuning ofPOTthresholduⁱⁿ^many^situations^[38℄.^The^advantages^and

drawbaksofEVTarepresentedinTable2.

3.2. Largedeviationtheory

Thelargedeviationtheory(LDT)haraterizestheasymptotibehaviourofPDFse-

quenetails[3941℄andmorepreisely,itanalyseshowaPDFsequenetaildeviatesfrom

itstypialbehaviourdesribedbythelawoflargenumbers.LDTanbeusedtoevaluate

theonvergeneofrareeventalgorithms[4246℄.LetusdeneHN =J(φ(X₁), ..., φ(X_N))

arandomvariableindexed byN ^withJ âôntinuous^salar^funtion,H îts^mathemat-

ial expetation and VN = HN −H^. ^One^says ^that VN ^satises ^the ^priniple ^of ^large

deviationswithaontinuousratefuntion I^if^the^following^limit^exists:

Nlim→∞

1

N ln[P(|VN |> γ)] =−I(γ). ⁽¹³⁾

TheexisteneofthislimitimpliesforalargevalueofN ^that

P(|VN |> γ)≈exp (−N I(γ)). ⁽¹⁴⁾

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The probability deays exponentially as N ^grows ^to înnity^, ât â ^rate ^depending ôn γ^. ^Thisapproximationis awell-known resultof LDT. Ifthe limitdoesnotexist, then

P(|VN |> γ)^has^a^too^singular^behaviour^or^dereases^faster ^thanexponentialdeay.If thelimitisequalto0^,^then^the^tailP(|VN |> γ)^dereases^withN ^slower^thanexp (−N a)

witha >0^.^Theômputation ôf^the^rate^funtion I îs^notôbvious^but ân^beôbtained

throughtheGärtner-Ellistheorem[47℄.Letusdenethefuntionλ(θ)^ofVN ^with

λ(θ) = lim

N→∞

1

N ln [E(exp (N θVN))], ⁽¹⁵⁾

withθ∈R^.

Theorem3.3 Gärtner-Ellis theorem If the funtion λ(θ) ^of ^the ^variable VN ^exists

andisdierentiable for allθ∈R^,^thenVN ^satises^the^priniple ^of ^large^deviations^and

I(γ)^is^given ^by

I(γ) = sup

θ∈R

[θγ−λ(θ)].

In the spei ase of a salar funtion J^, ^one ^an ^derive ^the ^Cramér ^theorem ^from

Gärtner-Ellistheorem[47℄.

Theorem3.4 Cramér theorem If VN = _N¹ PN

i=1J(φ(X_i)) ^where ^the ^random ^vari-

ables J(φ(X_i))âreî.i.d, ^the ^rate^funtion îs^given ^by I(γ) = sup

θ∈R

[θγ−λ(θ)],

with

λ(θ) = ln [E(exp (θJ(φ(X))))].

Thistheoremonlyholdsforlighttaildistributions.

Let us onsider theMonte-Carlo probabilityestimate given in Eq. 1.In that ase, one

has J(φ(.)) = 1_φ(.). The random variable J(φ(X_i)) ^follows ^a^Bernoulli distribution of meanP^. ^The^sequeneVN ^is^dened ^with

VN = 1 N

XN i=1

1_φ(_X

i)>S

!

−P. ⁽¹⁶⁾

The funtions λ(θ) ând I(γ) ân^be ^derived ^for ^some ^well-known ^PDF. În ^the âse ôf

BernoullidistributionsofmeanP^,^one^has

λ(θ) =Pexp(θ) + 1−P, ⁽¹⁷⁾

and

I(γ) =γlnγ P

+ (1−γ) ln 1−γ

1−P

. ⁽¹⁸⁾

Oneanthen obtainthe onvergene speed oftheMonte-Carloprobabilityestimatein

funtion ofthenumberofsampleswiththefollowingequation

Nlim→∞

1

N ln[P(|VN |> γ)] =−I(γ) =−γlnγ P

−(1−γ) ln 1−γ

1−P

. ⁽¹⁹⁾

The quantity I(γ)ôrresponds^to ^the ^relativeêntropy (Kullbak-Leiblerdivergene) of a oin toss with bias γ ^with ^respet ^to ^true ^value P^. În â ^lot ôf situations, the large deviationratefuntionistheKullbak-Leiblerdivergene[47℄.