Reduction of complexity in an heterogeneous population with two - - PowerPoint PPT Presentation
Motivations The population process Population with two timescales Reduction of complexity in an heterogeneous population with two timescales Colloque Jeunes Probabilistes et Statisticiens, avril 2016 Sarah Kaakai, joint work Nicole El Karoui
Motivations The population process Population with two timescales
LPMA, Universit´ e Paris 6 18 avril 2016
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Motivations The population process Population with two timescales
1 Motivations 2 The population process 3 Population with two timescales
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Motivations The population process Population with two timescales
Initial motivation: model and better understand human longevity.
◮ Longevity data are often aggregated data such as national data. ◮ Issue: strong heterogeneity in populations. Ex: Expected lifetime at
for individuals with no diploma (INSEE).
◮ Social structure of populations is not stable and can change at fast
pace.(Evolution of French diploma repartition for 30-45 years old)
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Motivations The population process Population with two timescales
More ”micro” description of the population needed to understand aggregated variables.
◮ Aggregation methods are important in several fields (economics,
biology, operations research..)
◮ Goal : model complex link between the micro and macro. ◮ Separation of timescales between different types of phenomenons:
permits to make some averaging approximations and thus to better understand the aggregation process. Our framework: Study of a stochastic model of population when ”social changes” occur at a fast pace in comparison to the demographic time scale of a life time.
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Motivations The population process Population with two timescales
1 Motivations 2 The population process Setup Representation of the population process as a sum of counting processes Markov framework 3 Population with two timescales
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Motivations The population process Population with two timescales
◮ We consider a population divided in p homogeneous subgroups:
individuals in the same subgroup are indistinguishable and have the same ”demographic” behavior.
◮ Population process: describes the evolution of the number of
individuals in each subgroup. c` adl` ag process Z = (Z 1, .., Z p) ∈ Np, adapted to a given filtration (Ft)t≥0 where Z i
t is the number of
individual in subgroup i at time t.
◮ Total size of the the population: ¯
Z =< Z, 1 >= Z i.
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Motivations The population process Population with two timescales
◮ Two kinds of events occur:
Demographic events: birth or death in one of the subgroups. ∆ ¯ Zt = ±1, ∆Z i
t = ±1.
Mixing events: An individual changes of characteristics and moves from a subgroup i to j. ∆ ¯ Zt = 0, ∆Z j
t = 1,∆Z i t = −1.
◮ Assumption: No events happen simultaneously. ◮ (Z i t) only increase/decrease of one when an event happens in the
subpopulation i. But Z is not a multivariate birth and death process since components have simultaneous jumps when mixing events happen.
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Motivations The population process Population with two timescales
◮ Canonical basis in Rp: (ei)1≤i≤p. e∞ = 0. ◮ Mixing events: natural notation (i, j) for mixing event from i to j.
Set of all mixing events: Imix = {(i, j) ∈ 1, p2; i = j}.
◮ Demographic events: we define the ”metaphysical” infinite
population {∞} of individuals not born yet and who died. Notation (∞, i) ((i, ∞)) for the event birth (death) in subgroup i. Set of demographic event: Idem = (1, p × {∞}) ∪ ({∞} × 1, p).
◮ Set of all events: I = Idem ∪ Imix. ◮ Jump associated to event of type (i, j): φ(i, j) = ej − ei.
{(i, j) happens at time t} = {∆Zt = φ(i, j)}.
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Motivations The population process Population with two timescales
Definition For all (i, j) ∈ I, we denote by N = (Nij)(i,j)∈I the a multivariate counting process with: Nij
t =
1{∆Zs=φ(i,j)} t ≥ 0, (1) Counting processes easily characterized by their intensity. Gives martingale property and makes computations very simple. Markov framework Assumption (Intensity of the counting processes) For all (i, j) ∈ I the counting process Nij has the {Ft}-intensity qij(Zs) : Mij
t = Nij t −
t
0 qij(Zs)ds is a {Ft}-local martingale.
We assume that there are no explosions of the processes. N is a strong Markov process.
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Motivations The population process Population with two timescales
Functionals of population process can be written as sum of their jumps: f (Zt) = f (Z0) +
(f (Zs− + ∆Zs) − f (Zs−))1{∆Zs=0}
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Motivations The population process Population with two timescales
Functionals of population process can be written as sum of their jumps: f (Zt) = f (Z0) +
(f (Zs− + ∆Zs) − f (Zs−))1{∆Zs=0} f (Zt) = f (Z0) +
t (f (Zs− + φ(i, j)) − f (Zs−))dNij
s 10/20
Motivations The population process Population with two timescales
Functionals of population process can be written as sum of their jumps: f (Zt) = f (Z0) +
(f (Zs− + ∆Zs) − f (Zs−))1{∆Zs=0} f (Zt) = f (Z0) +
t (f (Zs− + φ(i, j)) − f (Zs−))dNij
s
Martingale problem f (Zt) − f (Z0) − t
f (Zs)ds is a martingale, with Kmix (dem)f (z) =
qij(z) (f (z + φ(i, j)) − f (z)) Z is a continuous time Markov chain.
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Motivations The population process Population with two timescales
1 Motivations 2 The population process 3 Population with two timescales Mixing excursions Average aggregated framework
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Motivations The population process Population with two timescales
Separation of timescale
◮ Let’s define the nth demographic event T d n :
T d
n = inf{t > T d n−1; ∆ ¯
Zt = 0} (T0 = 0).
◮ Hypothesis: mixing intensities >> demographic intensities.
t T d
1
T d
2
T d
3
Idea:
◮ Decouple mixing and demographic events by isolating ”mixing
excursions” which evolve on finite spaces between two demographic events.
◮ Process reinterpreted as a Markov chain of killed trajectories, reborn
after a demographic event.
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Motivations The population process Population with two timescales
t T d
1
T d
2
T d
3 ◮ Time between two demographic events: Dn = T d n+1 − T d n ◮ nth mixing excursion of the population process; process after
nth demographic event and killed at the (n + 1)th: En
t =
n
if t < Dn ∂ if t ≥ Dn. (3) Population process is equivalent to the sequence of Mixing excursions (En)n∈N: Zt =
1{T d
n ≤t}En
t−T d
n
∀t ≥ 0. (4)
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Motivations The population process Population with two timescales
Population killed at the first demographic event:
◮ Only mixing events happen on {t < T d 1 }
f (Zt) = f (Z0) +
t (f (Zs− + φ(i, j)) − f (Zs−)) dNij
s . ◮ T d 1 is the first jump time of ¯
Ndem =
◮ Happens with intensity qdem(Zt)1{t<T d
1 } with
qdem(·) =
p
q∞i(·) + qi∞(·). Mixing excursions are ”pure mixing” processes, killed at a rate , of generator: K∂f (z) = Kmixf (z) − qdem(z)f (z) .
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Motivations The population process Population with two timescales
Pure mixing process
◮ X = (X x)x∈Np processes with semigroup Pmix generated by Kmix. ◮ Conservative process: for all initial population z ∈ Np, X z has value
in the finite space U¯
z of populations of size ¯
z =< z, 1 >: U¯
z = {x ∈ Np ¯
z = ¯ x}. Subordinated process
◮ The semigroup P∂ is subordinated to the pure mixing process,i.e:
P∂
s f ≤ Pmix s
f ∀f : Np → R+, s ≥ 0.
◮ P∂ can be realized by the expectation of a pure mixing process
discounted by a multiplicative functional of the process. Here: P∂
s f (z) = EX[f (X z s )e− s
0 qdem(X z u )du].
(5)
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Motivations The population process Population with two timescales
Hypothesis: mixing intensities >> demographic intensities.
◮ The population depends on a small parameter ǫ. The two timescale
population process Z ǫ has generator 1 ǫ Kmix + Kdem.
◮ Killed ”fast” mixing representation:
P∂,ǫ
s
f (z) = EX ǫ[f (X z,ǫ
s
) exp(− s
0 qdem(X z,ǫ u )du)]
with X ǫ some ”fast” pure mixing process of generator 1
ǫKmix. 16/20
Motivations The population process Population with two timescales
Hypothesis: mixing intensities >> demographic intensities.
◮ The population depends on a small parameter ǫ. The two timescale
population process Z ǫ has generator 1 ǫ Kmix + Kdem.
◮ Killed ”fast” mixing representation:
P∂,ǫ
s
f (z) = EX ǫ[f (X z,ǫ
s
) exp(− s
0 qdem(X z,ǫ u )du)]
with X ǫ some ”fast” pure mixing process of generator 1
ǫKmix. ◮ Realization on the same probability space: X x,ǫ τ
= X x
τ ǫ
∀τ ≥ 0.
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Motivations The population process Population with two timescales
Hypothesis: mixing intensities >> demographic intensities.
◮ The population depends on a small parameter ǫ. The two timescale
population process Z ǫ has generator 1 ǫ Kmix + Kdem.
◮ Killed ”fast” mixing representation:
P∂,ǫ
s
f (z) = EX ǫ[f (X z,ǫ
s
) exp(− s
0 qdem(X z,ǫ u )du)]
with X ǫ some ”fast” pure mixing process of generator 1
ǫKmix. ◮ Realization on the same probability space: X x,ǫ τ
= X x
τ ǫ
∀τ ≥ 0.
◮ Killed ”fast” mixing process with changed time pure mixing:
P∂,ǫ
s
f (z) = EX[f (X z
s ǫ ) exp
s qdem(X z
u ǫ )du
(6)
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Motivations The population process Population with two timescales
Hypothesis: mixing intensities >> demographic intensities.
◮ The population depends on a small parameter ǫ. The two timescale
population process Z ǫ has generator 1 ǫ Kmix + Kdem.
◮ Killed ”fast” mixing representation:
P∂,ǫ
s
f (z) = EX ǫ[f (X z,ǫ
s
) exp(− s
0 qdem(X z,ǫ u )du)]
with X ǫ some ”fast” pure mixing process of generator 1
ǫKmix. ◮ Realization on the same probability space: X x,ǫ τ
= X x
τ ǫ
∀τ ≥ 0.
◮ Killed ”fast” mixing process with changed time pure mixing:
P∂,ǫ
s
f (z) = EX[f (X z
s ǫ ) exp
ǫ
ǫqdem(X z
u )du
(6)
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Motivations The population process Population with two timescales
Pure mixing defined on disjoint equivalence classes (Un)n∈N. Assumption ∀n ∈ N, the pure mixing process restricted to Un is strongly irreducible. Thus the mixing process restricted to Un admits a unique stationary measure νn. In particular,∀z ∈ Np, and bounded functions f : 1 t t f (X z
s )ds −
→
t→+∞ ν¯ z(f ) =
z
f (x)ν¯
z(dx)
a.s, EX[f (X z
t )] −
→
t→+∞ ν¯ z(f ).
P∂,ǫ
s
f (z) = EX[f (X z
s ǫ )e−ǫ
s
ǫ
qdem(X z
u )du] −
→
ǫ→0 ν¯ z(f )e−sν¯
z(qdem).
In particular, the distribution of the first demographic event tends to an exponential variable of intensity ν ¯
Z0(qdem). 17/20
Motivations The population process Population with two timescales
◮ Aggregated process ¯
Z ǫ =< Z ǫ, 1 > is not a Markov process (intensities depend on the whole structure of the population, and are not constant between two events due to mixing events)
◮ Birth and death intensities:
qb(·) =
p
q(∞,i)(·) and qd(·) = p
i=1 q(i,∞)(·) ◮ When timescale of mixing process becomes instantaneous:
aggregated process converges to an ”averaged” markovian process. Theorem (Convergence of the aggregated process) Let ¯ Z ¯
z,ǫ be the aggregated process starting with an initial size of ¯
z. ∀¯ z ∈ N, ¯ Z ¯
z,ǫ converge in distribution towards a Birth and Death process
¯ Z ¯
z with birth and death intensities defined respectively by:
Q(¯ z, ¯ z + 1) = ν¯
z(qb)
Q(¯ z, ¯ z − 1) = ν¯
z(qd)
∀¯ z ∈ N.
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Motivations The population process Population with two timescales
◮ Dynamic of the aggregated process can be approximated by a
simpler Markovian process.
◮ The aggregated birth and death intensities are ”averaged” with
respect to the stationary distribution of the mixing process.
◮ Perspective: Understand the link between this limit and an
”intermediate” limit given by quasi stationary distributions.
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Motivations The population process Population with two timescales
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