06_competing risk model

8/10/2019 06_Competing Risk Model

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Competing Risks

Competing risks occur when a piece of equipmentcan be taken out of service for several reasons. In

principle, one could imagine there being a time atwhich the equipment would be taken out of service

for each reason. However, one can observe only thesmallest such time, and usually also the associatedreason; hence the dierent out of service! reasonscompete to be the one that is actually observed.Competing risk theory seeks to provide models of the

"statistical# relationships between these times so thatone can determine quantities, such as the failuretime distribution for the equipment, which may notbe directly observable.


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Competing Risks Model asa Renewal Process

$odel data as renewal process% a sequence of iid variables&', &(). Competing risk models essentially describe asituation in which the minimum of two or more lifetimevariables is observed together with the indicator of whichvariable is the smallest. *he unobserved variable is said to be

censoredby the observed variable.

In the reliability area the variables involved are typicallyfrom the following classes%

*ime to critical failure from a particular failure mode*ime to preventive maintenance*ime to equipment being taken o line for some e+traneous

reason and being replacedrepaired


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Classical Competing Risksframework

-pplications of the classical competing risk modelassume renewals of the equipment; that is, that theequipment is restored to as good as new eachtime.

*he bestknown technique within competing risks isthat of the KaplanMeier (KM) estimator, whichprovides a nonparametric estimator assuming

independent censoring. *he total time on test"***# statistic used in reliability is the estimatorused under the assumption of e+ponentiallydistributed lifetimes and independent censoring


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Competing risksframeworkIf the variable / is the time of failure due to a failure

mode and 0 and is the time of 1$ actions. *hevariable / may be masked by 0. 2e only observethe smallest of / and 0, i.e the observed variable is

In other words we observe the least of /and 0 and observe which one it is.

-pplications of the competing risk modelassume renewal of equipment.

( )[ ])(,, ZXIZXMinY


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Competing riskframework2e are interested in the survival function of +

and 3to asses the failure rate corresponding todierent competing risks

*he competing risk theory deal with the subsurvival function and its related conditionalprobabilities

2here the su4+ 5 represent the subfunctions

)()(),()( tZPtStXPtSZX

>=>=

),()(),,()( *

XZtZPtSZXtXPtSZ

*

X ==


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Competing Risk Problem

*he competing risk problem is that of identifying theunderlying "marginal# distributions from the operationaldata. In general, one is unable to identify the marginalsfrom the data without making untestable assumptions.*he motivation for studying problems is to know how

things would change if one was able to abolish a failurecause, stop doing maintenance, or perform some otheraction that changes the underlying relationship of thevariables. 6rom an operational research point of view, thecompeting risk problem is one e+ample of the more

general modelling problem in which there is a need toconstruct models that support decision making, but thesemodels cannot always be easily identi7ed from e+istingdata.


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Coal Mill Failure modes

Internal of Mill !ternal of Mill

Mill rollers failure8oot cause% wear and tear,overload, 7re

"rinding table failure8oot cause% segmentdisplacement, fatigue, trampmetal and 7re

#ir port ring failure8oot cause% rotating vanesstrikes on static ring

Mill motor bearing failure8oot cause% stress from millstart and stop, mis

alignment, temperature andlife e+pired

"earbo! failure8oot cause% vibration, misalignment, foreign ob9ects

and load


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Competing risks inreliability

Competing risk models do not have to involvepreventive maintenance as a censoringmechanism, it is clear that preventivemaintenance provides an important category ofsuch models in the reliability conte+t. Hence, ine+tending the available classes of competing riskmodels, a better understanding of the impact ofpreventive maintenance is important.

*he construction of a competing risk modelsinvolving maintenance clearly relate to theconstruction of maintenance optimisation models.


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$ptimisation modelsIt is acknowledged that one of the reasons that many

models have found limited application is that theassumptions made are typically strong.

:ome maintenance optimisation models aredeveloped by ekker, "'


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$pportunisticmaintenance

$pportunistic maintenance can be de7nes as a strategyby which preventive maintenance actions are carried out ona component during the unavailability of other components"for e+ample arising through failure#, rather than accordingto a preset schedule.

*he concept of opportunistic maintenance modeling is ascarce commodity within industrial power plant.ngineering %udgments and risk analysis for good

reasons, held wave in the area of maintenance modeling.

pportunistic maintenance is one of many routines for upkeeping or, if necessary, improving the level of reliability of

components and systems.


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Competing riskincorporatingopportunisticmaintenance*he competing risk is to identify the

marginal distribution from competing risk

data, here we consider a simple

opportunistic maintenance policies.

*he focus of this work is to provideidentifythe needs and criteria for taking an

opportunity to do maintenance, the risk ofusing an opportunity, and the approacheswhich can best satisfy the need of

opportunistic maintenance.


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$pportunity agereplacement model(&ekker and &i%kstra)/D time of failure. pportunities for 1$ arise following a

1oisson process, 1@"#. *he maintenance policy is to usethe 7rst opportunity occurring after the equipment hasbeen operating for time t0. 2e denote the time of 7rst

opportunity byZ

.

In the competing risk problem we try to show that thedistribution 6/"fXor SX# of / can be identi7ed in terms ofthe subsurvivor functions :5"t#.

n the interval E@, t@# / is not censored and on the intervalE@, F# the variable / is independently censored by ane+ponentially distributed variable. *he overall model isidenti7able. Crowder $.G., "(@@'#, indqvist et al., "(@@=#.


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$pportunity agereplacement modelsimulation e!ample

*he lifetime Xmay be censored byZ, the 7rst opportunity after a giventime t0. 6or illustration we have initially taken t0D@.= which is half the

mean value ofX. *he conditional subsurvivor functions generated by

'@@@ samples.

*he simulated empirical conditional subsurvivor function tells us the

probability of an event at time greater than t, given that the event iseventually observed.


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Repair'alert model

*he model assumes that with a 7+ed probability p, andindependently of the lifetime, a signal is given o bythe equipment, which is sub9ected to repair instantly.

*he time at which this signal is given o is dependent

on the underlying h"t# of the equipment, that is, thedensity function for the 1$ time given failure time is

the conditional survivor function have to be ordered inthis case, with ,0,)(

)(Xt

XH

th

).(~

)(~ **

tStSZX


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$pportunity'alert model

2e now assume that the model only gives the time at whicha signal would be given and that actual repair can onlyoccur opportunistically after the signal has been given.

*he dierence between this model and the repairalert

model is that repair can only occur at an opportunityfollowing the alert.

6or failures taking place at short lifetimes, there is a lowchance that an opportunity will take place prior to failure,and there is an increased probability that the failure willactually be observed "increased with respect to the repairalert model#.


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$pportunity'alertimulation e!ample

2e assume / to be 2eibull "shape D'.J and scaleD '#, and probabilityof alert to be @.?. - simulation with '@@@ sample gives the empiricalconditional subsurvivor function

Conditional subsurvivor functions "$ signi7es opportunisticmaintenance, that is opportunity taken after a signal, while 6 signi7esfailure#.

time

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

t

probability KM

X

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

OM

F


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ock'opportunity model

*he idea in this model is that discrete eventoccur that change the underlying failure rate"events such as internale+ternal shocks#

*he model requires us to de7ne the failure timedistribution through a process rather thanthrough the more conventional survival functionapproach

2e de7ne an increasing sequence of shock timesand for each period between shocks

, a failure rate

,...,,0 210 SSS =

],( 1 ii SS

],0(),( 1 iii SStt


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*he distribution for the failure time is de7nedconditionally% Kiven that the equipmentsurvives the i-1th shock, , the probability

that it survives the ith shock, , is equal to

H"t# has an e+ponential distribution with mean

'. 2e 7rst simulate an e+ponential variable V

with mean ', and inductively simulate shocktimes until the integrated conditional ha3ard

e+ceed V.

1

0 )(exp

ii SS

i

dtt

iSX>


1> iSX

++=101

00

1 )(...)()(ii SS

i

SS

i dttdttSH


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ock'opportunity modelIn order to model the censoring process%

how opportunities arise, andwhen opportunities are taken.

If denotes the constant ha3ard between shocks. *his

implies that the times between successive shocks follow ane+ponential distribution. *he e+pression for the conditionaldistribution function in terms of the shocks is given as

2here is the ha3ard between shocks and is thelargest event time less than t.

.1),......,,|( )()(

21

1

1 1 jji

j

i ii StSSt

n eSSStF + =

= +

i

jS

i


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2e assume that there are two shocks . *he7rst shock occurs with an e+ponential distributedtime after the equipment has been taken intoservice "as new#, and has mean @.(. *he secondshock occurs with an e+ponential distributed time

after the 7rst shock, and has mean @.?.

In addition we assume that the equipment hasfailure rates , for

the time intervals and equal to@.@', @.' and ' respectively. *his speci7es thedistribution of the failure time.

321 and, ),,[),,0[ 211 SSS ),,[ 3 S

21 and SS


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*hree competing risk models that could be used to analysedata arising in a power plant. ne is an e+isting opportunitymaintenance model that has been described from acompeting risk point of view.

*he 7rst model presented here has independent censoring,and hence from a dataanalysis point of view is identi7able onthe basis of competing risk data M the times of failure or 1$events together with the label of the event type.

*he second and third models illustrate dependent censoring.

*he Aaplan $eier estimator M often recommended in reliabilityte+ts M would be inappropriate in both cases, and indeed isoveroptimistic in its assessment of equipment lifetime.


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*ank +ou

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06_competing risk model

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