counting processes and recurrent events beyond the cox
TRANSCRIPT
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Counting processes and recurrent eventsbeyond the cox model for Poisson process
Vincent Couallier
Institut Mathematique de Bordeaux UMR CNRS 5251, Universite de Bordeaux.
Dynamic predictions for repeated markers and repeated eventsWorkshop - GS0 2013
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Outline
1 Counting processes and recurrent events
2 The LEYP process as a dynamic intensity process
3 An application on recurrent failures of water networks
4 Conclusion
a joint work withKarim Claudio (PhD LYRE, Suez), Genia Babykina (Post Doc,CRAN), Yves Legat(IRSTEA)
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Counting processes and recurrent events
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Type of data
a1
a2 = 0
a3
a4 = 0
b1
b2
b3
b4
t11 t12
t21
t31 t32 t33
i = 1
i = 2
i = 3
i = 4
m1 = 2
m2 = 1
m3 = 3
m4 = 0
S E
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Main Aim of analysis of recurrent events
statistical analysis (and modeling) of non-independant occurence timesof an event.
• more than one event per individual.• interest in understanding the dependancy between times
heterogenity and risk factors (covariates observed on individuals)
• identify the covariates that influence the probability of event.• individual prediction of probability of occurrence knowing the
characteristics of the individual
Heterogeneity and frailties
• does the intensity of events differ from individual to individualbecause of covariates or past history ?
• dynamic intensity modeling versus frailty intensity modeling.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Main Aim of analysis of recurrent events
statistical analysis (and modeling) of non-independant occurence timesof an event.
• more than one event per individual.• interest in understanding the dependancy between times
heterogenity and risk factors (covariates observed on individuals)
• identify the covariates that influence the probability of event.• individual prediction of probability of occurrence knowing the
characteristics of the individual
Heterogeneity and frailties
• does the intensity of events differ from individual to individualbecause of covariates or past history ?
• dynamic intensity modeling versus frailty intensity modeling.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Main Aim of analysis of recurrent events
statistical analysis (and modeling) of non-independant occurence timesof an event.
• more than one event per individual.• interest in understanding the dependancy between times
heterogenity and risk factors (covariates observed on individuals)
• identify the covariates that influence the probability of event.• individual prediction of probability of occurrence knowing the
characteristics of the individual
Heterogeneity and frailties
• does the intensity of events differ from individual to individualbecause of covariates or past history ?
• dynamic intensity modeling versus frailty intensity modeling.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The counting process for recurrent events
t
N(t)
1
2
3
4
T1 T2 T3 T5T4
Counting process {N(t)}t≥0
Definition of the (stochastic) intensitywith respect to a historyH(t)t>0(filtration generated by the known history) :
λ(t) = limdt→0
1dtP [N(t + dt)− N(t) = 1 |H(t−)]
λ(t)dt = E [dN(t) |H(t−)] ,
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The marginal intensity : the rate function
H(t) contains jump times and ”external covariates” Z(t),H(t) = σ
(N(t),T1, ...,TN(t),Z(t)
)Note : λ(t) is a predictable process, it captures the nature of the recurrenceof events.
The rate function (ROCOF in reliability analysis)
r(t) = limdt→0
1dtP [N(t + dt)− N(t) = 1 |Z(t−)]
r(t)dt = E [dN(t) | Z(t−)] ,
Ex : (Chiang, 1968)
λ(t) = β0(t) + β1(t)Z + β2(t)N(t−)
r(t) = β0(t) + φ(β1, β2)Z + ψ(β0, β2)β2(t),
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Recall the Cox model for survival analysis
One event per subject→ Survival analysis : λ(t) = h(t)IN(t)=0
h(t) = limdt→0
1dt
P (T ∈ [t, t + dt[|T > t)
regression model for covariate→Multiplicative intensity model
λ(t) = λ0(t)eβ0+β1Z1+···+βpZp 1N(t−)=0
Remark :
A dead individual is no more ”at risk”censoring mechanism may be considered too→ ”At risk” process Y(t).
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
from Cox regression for lifetimes to Andersen and Gillintensity of counting processes
Remove The ”at risk” indicator Y(t) to remains ”at risk” after the event.
Andersen & Gill, 82
λ(t)dt = E [dN(t) |Z1, . . .Zp] = λ0(t)eβ0+β1Z1+···+βpZp dt
The points of N(t)t≥0 form a Poisson process conditionally on thevalues Z1, . . .Zp.the process intensity does no depend on the past→ not dynamic.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Models for the intensity process in reliability analysis
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Some well known models for the intensity process
dynamic intensity : impact of occurrence of an event on the intensityN(t-) or event times themselves ?
observed ”inter-unit” heterogenity : impact of covariatesmultiplicative intensity with ”Cox-like” contribution.
unobserved ”inter-unit” heterogeneity : frailty : ? ?it may be hard to handle both frailty and dynamic parts.
Rk : in reliability (in this talk) :
an event = failure + instantaneous maintenance/repair
the dynamic part explains the maintenance actions.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The Poisson process with covariates
Cox regression model for recurrent events :
λ(t) = λ0(t)eβ′Z , The Z’s may be time-dependent
λ0 usually increasing, (⇒ ageing).eZ′β : impact of environment and/or individual heterogeneity
0 5 10 15 20 25 30
0.00
00.
005
0.01
00.
015
t
NH
PP
: λλ((
t))
h((t))eZ0ββ
(risque neutre)
(Andersen et Gill (Ann. Statist., 1982))
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The Poisson process with covariates
Cox regression model for recurrent events :
λ(t) = λ0(t)eβ′Z , The Z’s may be time-dependent
λ0 usually increasing, (⇒ ageing).eZ′β : impact of environment and/or individual heterogeneity
0 5 10 15 20 25 30
0.00
00.
005
0.01
00.
015
t
NH
PP
: λλ((
t))
h((t))eZ0ββ
(risque neutre)
(Andersen et Gill (Ann. Statist., 1982))
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The Poisson process with covariates
Cox regression model for recurrent events :
λ(t) = λ0(t)eβ′Z , The Z’s may be time-dependent
λ0 usually increasing, (⇒ ageing).eZ′β : impact of environment and/or individual heterogeneity
0 5 10 15 20 25 30
0.00
00.
005
0.01
00.
015
t
NH
PP
: λλ((
t))
h((t))eZ0ββ
(risque neutre)
h((t))eZ1ββ
(risque faible)
(Andersen et Gill (Ann. Statist., 1982))
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The Poisson process with covariates
Cox regression model for recurrent events :
λ(t) = λ0(t)eβ′Z , The Z’s may be time-dependent
λ0 usually increasing, (⇒ ageing).eZ′β : impact of environment and/or individual heterogeneity
0 5 10 15 20 25 30
0.00
00.
005
0.01
00.
015
t
NH
PP
: λλ((
t))
h((t))eZ0ββ
(risque neutre)
h((t))eZ1ββ
(risque faible)
h((t))eZ2ββ
(risque élevé)
(Andersen et Gill (Ann. Statist., 1982))
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
A renewal process for recurrent events
RP process (perfect repairs, AGAN models) :
λ(t) = λ0(t − TN(t))eβ′Z ,
λ0 usually increasing, λ0(0) = 0 (⇒ ageing).the intensity uses the duration since the last event t − TN(t).inter-arrival durations are i.i.d. random variablesdynamic intensity.
Þ
Doyen et Gaudoin (Rel. Eng. Syst. Saf. 2004) Kijima (J.Appl.Prob. 1989)
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
A generalized renewal process for recurrent events
GRP process (imperfect repair, virtual age models) :
λ(t) =(λ0(t)− ρλ0
(TN(t−)
))eβ
′Z ARI1
λ(t) =(λ0(t − ρTN(t−)
))eβ
′Z ARA1
λ0 usually increasing, λ0(0) = 0 (⇒ ageing).ρ captures the dynamic of the reduction of intensity
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The linear extended Yule process for recurrent events
LEYP process (imperfect repair) :
λ(t) = (1 + αN(t−))λ0(t)eZ(t)′β
a dynamical component (α > 0) uses the number of previous events.a baseline intensity λ0 (usually parameterized, λ0(., θ)).a Cox-like regression part eZ(t)′β .
αα == 0
t
NH
PP
: λλ((
t))
0 T1 T2 T3
0.00
00.
002
0.00
40.
006
0.00
80.
010
αα >> 0
t
LEY
P :
((1++
ααj))λλ
0((t))
0 T1 T2 T3
0.00
00.
005
0.01
00.
015
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Generalization to Pena & Hollander model
The Pena-Hollander model
λ(t) = Uρ(N(t−), α)λ0(ε(t))ψ(
eZ(t)′β)
a frailty component : U (non observable source of heterogeneity).a dynamical component ρ(N(t−), α)
a baseline intensity λ0 (usually parameterized, λ0(., θ)).a Cox-like regression part eZ(t)′β .
Remark : LEYP ⊂ Pena & Hollander
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Some useful properties for dynamic predictionand statistical estimation for the LEYP model.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Marginal and conditional distributions of N(t)
λ(t) = (1 + αN(t−))λ0(t; Z(t); δ, β)
Λ0(t) =∫ t
0 λ0(s; Z(s); δ, β)ds
0 a b tN(a−) N(b)−N(a)
Truncated data Observed k failures
c
Prediction
N(c)−N(b)
N(t) ∼ NB(α−1, e−αΛ0(t))
E [N(t)] =eαΛ0(t) − 1
α
Var [N(t)] =eαΛ0(t)(eαΛ0(t) − 1)
α
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Marginal and conditional distributions of N(t)
λ(t) = (1 + αN(t−))λ0(t; Z(t); δ, β)
Λ0(t) =∫ t
0 λ0(s; Z(s); δ, β)ds
0 a b tN(a−) N(b)−N(a)
Truncated data Observed k failures
c
Prediction
N(c)−N(b)
N(t) ∼ NB(α−1, e−αΛ0(t))
E [N(t)] =eαΛ0(t) − 1
α
Var [N(t)] =eαΛ0(t)(eαΛ0(t) − 1)
α
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Marginal and conditional distributions of N(t)
0 a b tN(a−) N(b)−N(a)
Truncated data Observed k failures
c
Prediction
N(c)−N(b)
[N(b)− N(a) |N(a−) = k, Z(s), a < s < b] ∼ NB(α−1 + k, e−α[Λ0(b)−Λ0(a)]
)
[N(c)− N(b) |N(b−)− N(a) = k] ∼ NB(α−1 + k,
eαΛ0(b) − eαΛ0(a) + 1eαΛ0(c) − eαΛ0(a) + 1
)
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The likelihood
One individual, observed on [0,T], m events at times tj (j = {1, ...m}) :
L(θ) =
m∏j=1
λ(tj)
× exp
− m∑j=0
∫ t(j+1)
tjλ(u) du
Remark : In the following, parametric assumption on the baseline +truncated observation
λ(t) = (1 + αN(t−))δtδ−1eZ(t)′β = (1 + αN(t−))λ0(t; δ, β)
N individuals, observed on [ai, bi], i = 1 . . .NΛ0(t) =
∫ t0 λ0(s)ds
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Data for maximum likelihood estimation
a1
a2 = 0
a3
a4 = 0
b1
b2
b3
b4
t11 t12
t21
t31 t32 t33
i = 1
i = 2
i = 3
i = 4
m1 = 2
m2 = 1
m3 = 3
m4 = 0
S E
And : Z1, . . . ,Zp, fixed or external time-dependent covariates (no frailty)
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
The likelihood
Log-likelihood for truncated data, N individuals
ln L(θ) =
N∑i=1
( mi lnα+ ln Γ(α−1 + mi)− ln Γ(α−1)
− (α−1 + mi) ln(eαΛ0(bi;δ,β) − eαΛ0(ai;δ,β) + 1)
+
mi∑j=1
(lnλ0(tj; δ, β) + αΛ0(tj; δ, β)) )
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Appl. on recurrent failures of water networks
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
![Page 35: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/35.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
![Page 36: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/36.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
![Page 37: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/37.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
SEDIF : A public drinking water service in area of Paris.
stratification : only grey cast iron pipes are considered.
21450 pipes to provide drinking water (899km linear).
failures recorded on 1996-2006 (11 years).
mean age at inclusion : 35.8 years.
mean duration of observation : 10 years.
% of units with ≥ 1 failure : 12%.
% of units with > 1 failures : 3%.
![Page 38: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/38.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
20
40
60
80
100
Défaillances observées
Dates
Nom
bre
de d
éfa
illa
nces
1996 1998 2000 2002 2004 2006
FIGURE : observed monthly number of failures
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Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipesthe length L of the pipe influences the intensity
the climate influences the intensity (time dependent)→ X2 : air temperature average over 10-days periods.
λ(t) = (1 + αN(t−))δtδ−1eβ0+β1 ln(L)+β2X2(t)
0 100 200 300 400
−5
05
10
15
20
Time−dependent covariate (by decade)
Time (decades)
Z(t
)
TemperaturenJG
FIGURE : Temperature as a time-dependent covariate
![Page 40: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/40.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipesthe length L of the pipe influences the intensity
the climate influences the intensity (time dependent)→ X2 : air temperature average over 10-days periods.
λ(t) = (1 + αN(t−))δtδ−1eβ0+β1 ln(L)+β2X2(t)
0 100 200 300 400
−5
05
10
15
20
Time−dependent covariate (by decade)
Time (decades)
Z(t
)
TemperaturenJG
FIGURE : Temperature as a time-dependent covariate
![Page 41: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/41.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipesthe length L of the pipe influences the intensity
the climate influences the intensity (time dependent)→ X2 : air temperature average over 10-days periods.
λ(t) = (1 + αN(t−))δtδ−1eβ0+β1 ln(L)+β2X2(t)
0 100 200 300 400
−5
05
10
15
20
Time−dependent covariate (by decade)
Time (decades)
Z(t
)
TemperaturenJG
FIGURE : Temperature as a time-dependent covariate
![Page 42: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/42.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipesthe length L of the pipe influences the intensity
the climate influences the intensity (time dependent)→ X2 : air temperature average over 10-days periods.
λ(t) = (1 + αN(t−))δtδ−1eβ0+β1 ln(L)+β2X2(t)
0 100 200 300 400
−5
05
10
15
20
Time−dependent covariate (by decade)
Time (decades)
Z(t
)
TemperaturenJG
FIGURE : Temperature as a time-dependent covariate
![Page 43: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/43.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
The results (maximum likelihood estimation)
1
deviations (std) and 95% confidence intervals.
Parameter Model Estimate Standard
Dev.
95% CI
Estimate ± 1.96 std
a with 2X 0.96 0.078
[ ]0 81 1 12. , .
without 2X 0.98 0.079 [ ]0 82 1 13. , .
d with 2X 1.14 0.094 [ ]0 96 1 32. , .
without 2X 1.11 0.094 [ ]0 93 1 30. , .
0b with 2X -7.03 0.45 [ ]7 92 6 14- . ,- .
without 2X -7.52 0.45 [ ]8 41 6 63- . ,- .
1b with 2X 0.67 0.020 [ ]0 63 0 71. , .
without 2X 0.65 0.020 [ ]0 61 0 69. , .
2b with 2X -0.10 0.0.003 [ ]0 11 0 09- . ,- .
The inclusion of the time-dependent covariate to the model reduces the
![Page 44: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/44.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipesGlobal prediction of failures on the water network
✶✷✻ ✻✳ ❆♣♣❧✐❝❛(✐♦♥ + ❞❡. ❞♦♥♥/❡. 0/❡❧❧❡.
▲❡% ❝♦♥❝❧✉%✐♦♥% %✉, ❧❡% ❞♦♥♥.❡% ❞✬❖%❧♦ %♦♥1 ❛%%❡③ ❞✐,❡❝1❡%✳ ▲✬❛❜%❡♥❝❡ ❞❡ ❧❛ ❝♦✈❛,✐❛❜❧❡ 1❡♠♣♦,❡❧❧❡
❡♥1,❛9♥❡ ❧❛ %✉,❡%1✐♠❛1✐♦♥ ❞✉ ♥♦♠❜,❡ ❞❡ ❞.❢❛✐❧❧❛♥❝❡% ❞❛♥% ❧❡ ❢✉1✉, ✿ ✈♦✐, ✜❣✉,❡% ✻✳✼✭❞✮✱ ✭❡✮ ❡1 ✭❢✮✳ ▲❡%
❡%1✐♠❛1✐♦♥% ❢❛✐1❡% B ❧✬❛✐❞❡ ❞✉ ▲❊❨E ❛✈❡❝ ❧❛ ❝♦✈❛,✐❛❜❧❡ 1❡♠♣♦,❡❧❧❡✱ ♣❛, ❡①❡♠♣❧❡ %✉, ❧❡ ,.%❡❛✉ ❡♥1✐❡,✱
❝♦♥❞✉✐%❡♥1 B ❧✬❡,,❡✉, ❞❡ ♣,.❞✐❝1✐♦♥ ♠♦②❡♥♥❡ ❞❡ ✹✳✸✷ ❞.❢❛✐❧❧❛♥❝❡% ♣❛, ♠♦✐% ✈%✳ ✹✳✹ ❞.❢❛✐❧❧❛♥❝❡% ♣❛,
♠♦✐% ❧♦,%J✉❡ ❧❛ ❝♦✈❛,✐❛❜❧❡ 1❡♠♣♦,❡❧❧❡ ❡%1 ♦♠✐%❡✳ ▲❛ ♠K♠❡ 1❡♥❞❛♥❝❡ ❡%1 ♦❜%❡,✈.❡ ❞❛♥% ❞✬❛✉1,❡% ❝❛%
❞❡ ✜❣✉,❡% ✭✈♦✐, ✓ E,.❞✐❝1✐♦♥% ✔ ❞❛♥% ❧❡ 1❛❜❧❡❛✉ ❇✳✽✮✳ ◆♦✉% ❡♥ ❝♦♥❝❧✉♦♥% J✉❡ ❧❛ ❝♦✈❛,✐❛❜❧❡ 1❡♠♣♦,❡❧❧❡
❝♦,,✐❣❡ ❧✬❡%1✐♠❛1✐♦♥ ❞❡% ♣❛,❛♠Q1,❡%✳
❉✬❛✉1,❡ ♣❛,1✱ ❧❡ ▲❊❨E ❡%1 ❧.❣Q,❡♠❡♥1 ♠❡✐❧❧❡✉, J✉❡ ❧❡ ◆❍EE✳ E❛, ❡①❡♠♣❧❡✱ ❧✬❡,,❡✉, ❞❡ ♣,.❞✐❝1✐♦♥
♠♦②❡♥♥❡ ❡%1 ❞❡ ✹✳✸✷ ♣♦✉, ▲❊❨E ❝♦♥1,❡ ✹✳✸✼ ❢♦✉,♥✐❡ ♣❛, ❧❡ ◆❍EE ❞❛♥% ❧❛ ♠K♠❡ ❝♦♥✜❣✉,❛1✐♦♥✳ E♦✉,
❧❡ %♦✉%✲❡♥%❡♠❜❧❡ ❞❡ 1,♦♥U♦♥% ♣♦%.% ❛♣,Q% ✶✾✹✻✱ ❧❡ ◆❍EE %❡♠❜❧❡ K1,❡ ✐❞❡♥1✐J✉❡ ❛✉ ▲❊❨E ✭❧✬❡,,❡✉,
♠♦②❡♥♥❡ ❞❡ ✷✳✹✼ ❞❛♥% ❧❡ ❝❛% ❞✉ ◆❍EE ❡1 ❞❡ ✷✳✹✽ ❞❛♥% ❧❡ ❝❛% ❞✉ ▲❊❨E✮✳ ❈❡ ,.%✉❧1❛1 ❡%1 ❝♦❤.,❡♥1
❛✈❡❝ ❧❛ ❝♦♥❝❧✉%✐♦♥ ❢❛✐1❡ B ♣❛,1✐, ❞❡% ,.%✐❞✉% ❞❡ ♠❛,1✐♥❣❛❧❡ ✭%❡❝1✐♦♥ ✻✳✸✳✶✮ ✿ ❧❡ ◆❍EE %❡♠❜❧❡ K1,❡
♠✐❡✉① ❛❞❛♣1. ❛✉① ❝♦✉,1% ❤✐%1♦,✐J✉❡%✳
▲❛ ❝♦♥❝❧✉%✐♦♥ ❣.♥.,❛❧❡ ❝♦♥❝❡,♥❛♥1 ❧❡% ❞♦♥♥.❡% ❞✉ ❙❊❉■❋ ❡%1 J✉❡ ❧❛ ❝♦✈❛,✐❛❜❧❡ 1❡♠♣♦,❡❧❧❡ ♣❡,♠❡1
❞❡ ♠✐❡✉① ❡①♣❧✐J✉❡, ❧✬.✈♦❧✉1✐♦♥ ❞✉ ♣,♦❝❡%%✉% ❞❡ ❞.❢❛✐❧❧❛♥❝❡✳ ▲❛ ✜❣✉,❡ ✻✳✾ ✐❧❧✉%1,❡ ❧❡% ♣,.❞✐❝1✐♦♥%
❛♥♥✉❡❧❧❡% ❞✉ ♥♦♠❜,❡ ❞❡ ❞.❢❛✐❧❧❛♥❝❡%✱ ❢❛✐1❡% ❡♥ ♣,❡♥❛♥1 ❡♥ ❝♦♠♣1❡ ❧❛ 1❡♠♣.,❛1✉,❡ ♠♦②❡♥♥❡ ♣❛,
❞.❝❛❞❡✱ ❧❡ ♥♦♠❜,❡ ❞❡ ❥♦✉,% ❞❡ ❣❡❧ ♣❛, ❞.❝❛❞❡ ❡1 ❧❛ 1❡♠♣.,❛1✉,❡ ❜♦,♥.❡ B ✶✵ ✝ ✱ ❛✐♥%✐ J✉❡ ❧❡%
♣,.❞✐❝1✐♦♥% ❢❛✐1❡% %❛♥% ❧❛ ❝♦✈❛,✐❛❜❧❡ 1❡♠♣♦,❡❧❧❡✳
0100
200
300
400
500
Temps
Nom
bre
de d
éfa
illances
1996 1998 2000 2002 2004 2006
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Observé (année)Observé (décade)Temp.
Temp. bornéenJGSansX2
❋✐❣✉$❡ ✻✳✾ ✕ ◆♦♠❜$❡ ❞❡ ❞'❢❛✐❧❧❛♥❝❡. ♣$'❞✐0 ♣❛$ ❛♥♥'❡✳ ❉♦♥♥'❡. ❞✉ ❙❊❉■❋✳ ✿ 0❡♠♣'$❛0✉$❡ ❡0
▲✬❛♥❛❧②%❡ ❞.1❛✐❧❧.❡ ❞❡% ❡%1✐♠❛1✐♦♥% ❢❛✐1❡% %✉, ❧❡% ❞♦♥♥.❡% ❞✉ ❙❊❉■❋ ,.✈Q❧❡ ❧❡% 1❡♥❞❛♥❝❡% %✉✐✲
✈❛♥1❡% ✿
✶✳ ▲❡ ▲❊❨E ♥✬❡%1 ♣❛% ✉♥ ♠♦❞Q❧❡ ❝♦♥%✐❞.,❛❜❧❡♠❡♥1 ♠❡✐❧❧❡✉, J✉❡ ❧❡ ◆❍EE ✿ ❡♥ 1❡,♠❡% ❞✬❡,,❡✉,
❞❡ ♣,.❞✐❝1✐♦♥ ❧❡ ▲❊❨E ❛✈❡❝ ♠.1❤♦❞❡ ❡1 %❛♥% ❝♦✈❛,✐❛❜❧❡ 1❡♠♣♦,❡❧❧❡ ❞♦♥♥❡ J✉❛%✐♠❡♥1 ❧❡
♠K♠❡ ,.%✉❧1❛1 J✉❡ ❧❡ ◆❍EE ✭✈♦✐, ✓ E,.❞✐❝1✐♦♥% ✔ ❞❛♥% ❧❡% 1❛❜❧❡❛✉① ❇✳✶✶ ❡1 ❇✳✶✷✮✳
![Page 45: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/45.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
individual prediction of failure on [U,V], at time T (≤ U < V) ?
at time T, use the adjusted model, the known history of the unit (age,covariate, number of past failures) to compute the Negative Binomialdistribution for Ni(V)− Ni(U).
ranking of the sample by ordering P[Ni(V)− Ni(U)]i=1...n.
use this ranking to provide a preventive maintenance policy.
or do a graph of the Lift Curve to validate the prediction efficiency ofthe model.
![Page 46: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/46.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
individual prediction of failure on [U,V], at time T (≤ U < V) ?
at time T, use the adjusted model, the known history of the unit (age,covariate, number of past failures) to compute the Negative Binomialdistribution for Ni(V)− Ni(U).
ranking of the sample by ordering P[Ni(V)− Ni(U)]i=1...n.
use this ranking to provide a preventive maintenance policy.
or do a graph of the Lift Curve to validate the prediction efficiency ofthe model.
![Page 47: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/47.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
individual prediction of failure on [U,V], at time T (≤ U < V) ?
at time T, use the adjusted model, the known history of the unit (age,covariate, number of past failures) to compute the Negative Binomialdistribution for Ni(V)− Ni(U).
ranking of the sample by ordering P[Ni(V)− Ni(U)]i=1...n.
use this ranking to provide a preventive maintenance policy.
or do a graph of the Lift Curve to validate the prediction efficiency ofthe model.
![Page 48: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/48.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
individual prediction of failure on [U,V], at time T (≤ U < V) ?
at time T, use the adjusted model, the known history of the unit (age,covariate, number of past failures) to compute the Negative Binomialdistribution for Ni(V)− Ni(U).
ranking of the sample by ordering P[Ni(V)− Ni(U)]i=1...n.
use this ranking to provide a preventive maintenance policy.
or do a graph of the Lift Curve to validate the prediction efficiency ofthe model.
![Page 49: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/49.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
recurrent events of failure-repair on water network pipes
individual prediction of failure on [U,V], at time T (≤ U < V) ?
at time T, use the adjusted model, the known history of the unit (age,covariate, number of past failures) to compute the Negative Binomialdistribution for Ni(V)− Ni(U).
ranking of the sample by ordering P[Ni(V)− Ni(U)]i=1...n.
use this ranking to provide a preventive maintenance policy.
or do a graph of the Lift Curve to validate the prediction efficiency ofthe model.
![Page 50: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/50.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Conclusion - Bilan
![Page 51: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/51.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Conclusion
many models to handle dynamic intensity of recurrent events.
the LEYP is one of them, with useful properties
semiparametric framework has not been investigated (yet)
recurrent events problems : epidemiology / biostatistics / reliabilityanalysis : bridges exist.
dynamic versus frailty models : a risk to badly identify dynamiccomponents (in fact due to frailty component) : The negative Binomialdistribution is also the marginal for Gamma mixed Poisson processes(Poisson with Gamma frailty)
![Page 52: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/52.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Conclusion
many models to handle dynamic intensity of recurrent events.
the LEYP is one of them, with useful properties
semiparametric framework has not been investigated (yet)
recurrent events problems : epidemiology / biostatistics / reliabilityanalysis : bridges exist.
dynamic versus frailty models : a risk to badly identify dynamiccomponents (in fact due to frailty component) : The negative Binomialdistribution is also the marginal for Gamma mixed Poisson processes(Poisson with Gamma frailty)
![Page 53: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/53.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Conclusion
many models to handle dynamic intensity of recurrent events.
the LEYP is one of them, with useful properties
semiparametric framework has not been investigated (yet)
recurrent events problems : epidemiology / biostatistics / reliabilityanalysis : bridges exist.
dynamic versus frailty models : a risk to badly identify dynamiccomponents (in fact due to frailty component) : The negative Binomialdistribution is also the marginal for Gamma mixed Poisson processes(Poisson with Gamma frailty)
![Page 54: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/54.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Conclusion
many models to handle dynamic intensity of recurrent events.
the LEYP is one of them, with useful properties
semiparametric framework has not been investigated (yet)
recurrent events problems : epidemiology / biostatistics / reliabilityanalysis : bridges exist.
dynamic versus frailty models : a risk to badly identify dynamiccomponents (in fact due to frailty component) : The negative Binomialdistribution is also the marginal for Gamma mixed Poisson processes(Poisson with Gamma frailty)
![Page 55: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/55.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Conclusion
many models to handle dynamic intensity of recurrent events.
the LEYP is one of them, with useful properties
semiparametric framework has not been investigated (yet)
recurrent events problems : epidemiology / biostatistics / reliabilityanalysis : bridges exist.
dynamic versus frailty models : a risk to badly identify dynamiccomponents (in fact due to frailty component) : The negative Binomialdistribution is also the marginal for Gamma mixed Poisson processes(Poisson with Gamma frailty)
![Page 56: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/56.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Conclusion
many models to handle dynamic intensity of recurrent events.
the LEYP is one of them, with useful properties
semiparametric framework has not been investigated (yet)
recurrent events problems : epidemiology / biostatistics / reliabilityanalysis : bridges exist.
dynamic versus frailty models : a risk to badly identify dynamiccomponents (in fact due to frailty component) : The negative Binomialdistribution is also the marginal for Gamma mixed Poisson processes(Poisson with Gamma frailty)
![Page 57: Counting processes and recurrent events beyond the cox](https://reader034.vdocuments.site/reader034/viewer/2022050516/6272608812bac3676806ef98/html5/thumbnails/57.jpg)
Counting processes and recurrent events The LEYP process Appl. on recurrent failures of water networks Conclusion
Thank you for your attention.