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Particle approximation of multiple object filtering problems
P. Del Moral
INRIA Bordeaux & IMB & CMAP Polytechnique Univ. New South Wales, Sydney (Jan. 2014)
IMACS Monte Carlo, July 15-19, 2013Some hyper-refs
Mean field simulation for Monte Carlo integration. Chapman & Hall, Series : Maths and Stat. (2013).
Particle approximations of a class of branching distribution flows arising in multi-target tracking.SIAM Control. & Opt. (2011). (joint work with Caron, Doucet, Pace)
On the Conditional Distributions of Spatial Point Processes.
Advances in Applied Probability (2011). (joint work with Caron, Doucet, Pace).
On the Stability & the Approximation of Branching Distribution Flows, with Applications to NonlinearMultiple Target Filtering.Stochastic Analysis and Applications (2011). (joint work with Caron, Pace, Vo).
Comparison of implementations of Gaussian mixture PHD filters. FUSION (2010).(joint work with Caron, Pace, Vo).
More references on the website http://www.math.u-bordeaux1.fr/delmoral/index.html [+ Links]
http://hal.archives-ouvertes.fr/docs/00/46/41/30/PDF/RR-7233.pdfhttp://hal.archives-ouvertes.fr/docs/00/46/41/27/PDF/RR-7232.pdfhttp://hal.archives-ouvertes.fr/docs/00/46/41/27/PDF/RR-7232.pdfhttp://hal.inria.fr/docs/00/51/65/07/PDF/RR-7376.pdfhttp://hal.inria.fr/docs/00/51/65/07/PDF/RR-7376.pdfhttp://ieeexplore.ieee.org/xpl/freeabs_all.jsp?reload=true&arnumber=5711953&tag=1http://www.math.u-bordeaux1.fr/~delmoral/index.htmlhttp://www.math.u-bordeaux1.fr/~delmoral/index.htmlhttp://www.math.u-bordeaux1.fr/~delmoral/index.htmlhttp://www.math.u-bordeaux1.fr/~delmoral/index.htmlhttp://ieeexplore.ieee.org/xpl/freeabs_all.jsp?reload=true&arnumber=5711953&tag=1http://hal.inria.fr/docs/00/51/65/07/PDF/RR-7376.pdfhttp://hal.archives-ouvertes.fr/docs/00/46/41/27/PDF/RR-7232.pdfhttp://hal.archives-ouvertes.fr/docs/00/46/41/30/PDF/RR-7233.pdfhttp://find/7/27/2019 Annecy Del Moral MTT
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Contents
Introduction/notation
Multiple objects branching signals
Multiple targets filtering models
General measure valued equations
Particle association measures
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Introduction/notationDefense Industrial Research projectSome basic notationSpatial Branching modelsFirst moments recursion
Multiple objects branching signals
Multiple targets filtering models
General measure valued equations
Particle association measures
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2 Industrial research project
1. Defense industrial Contract : ALEA INRIA team & DCNS (2009)
2. National Research project : ANR PROPAGATION[2,3Me] (2009-2012):
Passive radar tracking and optronics liabilities for the protection of
coastal infrastructures
ALEA INRIA team DCNS SIS, THALES, ECOMER, EXAVISION
Project members : + D. Arrivault, Fr. Caron, M. Pace. Visiting researchers :
D. Clark, A. Doucet, J. Houssineau, S.S. Sing, B.N. Vo.
http://www.polemerpaca.com/Securite-et-surete-maritime/Protection-maritime-etendue-et-rapprochee/PROPAGATIONhttp://www.polemerpaca.com/Securite-et-surete-maritime/Protection-maritime-etendue-et-rapprochee/PROPAGATIONhttp://www.polemerpaca.com/Securite-et-surete-maritime/Protection-maritime-etendue-et-rapprochee/PROPAGATIONhttp://www.polemerpaca.com/Securite-et-surete-maritime/Protection-maritime-etendue-et-rapprochee/PROPAGATIONhttp://www.polemerpaca.com/Securite-et-surete-maritime/Protection-maritime-etendue-et-rapprochee/PROPAGATIONhttp://www.polemerpaca.com/Securite-et-surete-maritime/Protection-maritime-etendue-et-rapprochee/PROPAGATIONhttp://find/7/27/2019 Annecy Del Moral MTT
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Lebesgue integral on a measurable state space E
(, f) =(measure, function) (M(E) Bb(E))
(f) =
(dx) f(x)
Delta-Dirac Measure at a E
= a
a
(f) = f(x) a(dx) = f(a)Normalization of a positive measure M+(E) (when (1) = 0)
(dx) := (dx)/(1) = probability P(E)
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Lebesgue integral on a measurable state space E
(, f) =(measure, function) (M(E) Bb(E))
(f) =
(dx) f(x)
Delta-Dirac Measure at a E
= a
a
(f) = f(x) a(dx) = f(a)Normalization of a positive measure M+(E) (when (1) = 0)
(dx) := (dx)/(1) = probability P(E)
Example = 10 Law(X)
(1) = 10 & = Law(X) & (f) = E(f(X))
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Q(x, dy) integral operator from E into E
Two operator actions :
f Bb(E) Q(f) Bb(E) and M(E) Q M(E)
with
Q(f)(x) = Q(x, dx)f(x)
[Q](dx) =
(dx)Q(x, dx) ( [Q](f) := [Q(f)] )
and the composition
(Q1Q2)(x, dx) = Q1(x, dx)Q2(x, dx)Q Markov operator Q(1) = 1 =unit function notation : M or K (Markov transitions-kernel/Stochastic matrices)
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Boltzmann-Gibbs transformation : G 0 s.t. (G) > 0
G()(dx) =1
(G)
G(x) (dx)
Bayes rule (with fixed observation y)
(dx) = p(x)dx and G(x) = p(y|x)
G()(dx) =
1
p(y)p(y|x) p(x)dx = p(x|y) dx
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Boltzmann-Gibbs transformation : G 0 s.t. (G) > 0
G()(dx) =1
(G)
G(x) (dx)
Bayes rule (with fixed observation y)
(dx) = p(x)dx and G(x) = p(y|x)
G()(dx) =
1
p(y)p(y|x) p(x)dx = p(x|y) dx
Restriction
(dx) = P(X dx) = p(x)dx and G(x) = 1A(x)
G()(dx) =1
P(X A) 1A(x) p(x)dx = P(X dx | X A)
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Boltzmann-Gibbs transformation : G 0 s.t. (G) > 0
G()(dx) =1
(G)G(x) (dx)
(non unique) Markov transport equation
G()(dy) =
(dx)S(x, dy) G() = S
Example 1 : (G 1) accept/reject/recycling/interacting jumps
S(x, dy) = G(x)x(dy) + (1 G(x)) G()(dy)
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Boltzmann-Gibbs transformation : G 0 s.t. (G) > 0
G()(dx) =1
(G)G(x) (dx)
(non unique) Markov transport equation
G()(dy) =
(dx)S(x, dy) G() = S
Example 1 : (G 1) accept/reject/recycling/interacting jumps
S(x, dy) = G(x)x(dy) + (1 G(x)) G()(dy)
Note :
S1N
1jN Xj
(Xi, dy)
= G(Xi)Xi(dy) + (1 G(Xi))
1jNG(Xj)
1kN G(Xk)
Xj(dy)
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Other examples of transport equations G() = SExample 2 : s.t. G 1 a.e.
S(x, dy) = G(x) x(dy) + (1 G(x)) G()(dy)
Example 3 : a s.t. G > a
S
(x, dy) =a
(G)
x(dy) + 1
a
(G) Ga()(dy)Example 4 : G
S(x, dy) = (x) x(dy) + (1 (x)) [GG(x)]+ ()(dy)
with the acceptance rate
(x) = [G G(x)]/(G)
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Spatial Branching models (time index n N, state spaces En) 3 ingredients : Gn(x)
1, n(dx)
0, and Mn(xn1, dxn) Markov.
Branching rule (spawning) :
Random mapping x gn(x) N offsprings, with E(gn(x)) = Gn(x)
survival rates en(x) + cemetery states : Gn en(x)Gn(x)
Spontaneous births: Spatial Poisson with intensity n(dx) Free motion between branching times : Mn-evolutions
Random occupation measure (after the n-th evolution step)
Xn =Nn
i=1
Xin
En := {types, locations, labels, excursions, paths,. . . }
S i l B hi d l ( i i d N E )
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Spatial Branching models (time index n N, state spaces En) First moment recursion = branching intensity distribution
n+1(f) := E (X
n+1(f)) = n(Qn+1(f)) + n+1(f)
withQn+1(x, dy) = Gn(x)Mn+1(x, dy)
S ti l B hi d l (ti i d N t t E )
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Spatial Branching models (time index n N, state spaces En) First moment recursion = branching intensity distribution
n+1(f) := E (X
n+1(f)) = n(Qn+1(f)) + n+1(f)
withQn+1(x, dy) = Gn(x)Mn+1(x, dy)
Sketched proof (n = 0): Xn+1 =Nn+1i=1
Xin+1 =
Nni=1
gin(Xin)
j=1
Xi,jn+1
E (Xn+1(f) | Xn, gn(Xn)) =Nn
i=1
gi
n(Xi
n) Mn+1(f)(Xi
n)
E (
Xn+1(f)
| Xn) =
Nn
i=1Gn(X
in) Mn+1(f)(X
in) =
Xn(Qn+1(f))
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Continuous time models
Geometric clocks exponential rates time mesh(tn tn1) 0
Xn = XtnGn = survival
[spawning
mean offsprings + (1
spawning)
1]
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Continuous time models
Geometric clocks exponential rates time mesh(tn tn1) 0
Xn = XtnGn = survival
[spawning
mean offsprings + (1
spawning)
1]
(tn tn1) 0 G = 1 + V dt and M = Id + L dt and tn t
d
dtt(f) = t(L
V(f)) + t(f) with LV = L + V
Schrodinger operator
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n = 0 Classical Feynman-Kac models
Feynman-Kac representation ( Application domains)
n+1(f) = 0(1) E0
f(Xn+1)
0pnGp(Xp)
Particle approximations = Genetic type algo = Particle filters = ...
Qn+1(x, dy) = Gn(x) Selection potential
Mn+1(x, y)
Mutation transition
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Introduction/notation
Multiple objects branching signalsEvolution equationsStability propertiesThree typical scenariosAn extended Feynman-Kac modelMean field particle interpretationsSome convergence results
Multiple targets filtering models
General measure valued equations
Particle association measures
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More general spatial Branching models (hyp. 0 = 0)
n = n1Qn + n and n := n/n(1)
(n(1), n) := p,n (p(1), p)
Some problems Problem 1: Mass n(1) unstable n(1) or n(1) 0 as n Problem 2: Xn =
Nni=1 Xin generally NOT POISSON random field.
Problem 3:
non degenerate numerical sampling method?
Problem 4: non degenerate approximation ofn?
Th i M M G G [ ]
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Three scenarios Mn = M, Gn = G [g, g+], n = 1. G = 1 := M (independent of)
n(1) = 0(1) + n(1) and n tv = O(1/n)2. g+ < 1 := /(1) with given by
:=n0
Qn Poisson equation (Id Q) =
and|n(f) (f)| |n(f) (f)| c gn+ f
Th i M M G G [ ]
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Three scenarios Mn = M, Gn = G [g, g+], n = 1. G = 1 := M (independent of)
n(1) = 0(1) + n(1) and n tv = O(1/n)2. g+ < 1 := /(1) with given by
:=n0
Qn Poisson equation (Id Q) =
and|n(f) (f)| |n(f) (f)| c gn+ f
Continuous time models G = eVt
& M = L t + Id
t(f) =
t0
E
f(Xs) exp
s
0
V(Xr)dr
ds
t
Poisson equation L
V = , with LV = L + V
The 3 d sce a io (M M G G [ ] )
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The 3-rd scenario (Mn = M, Gn = G [g, g+], n = )
g > 1
(f) := Q(f)/Q(1) (independent of)
limn
1
nlog n(1) = log (G) and n tv c en
= [quasi-invariant meas., Yaglom meas., ground states, Feynman-Kacsg fixed points, infinite population stationary measure, . . . ]
Hyper-refs :
On the stability of interacting processes with applications to filtering and geneticalgorithms. (joint work with A. Guionnet) Annales IHP (2001).
Particle approximations of Lyapunov exponents connected to Schrodinger operators andFeynman-Kac semigroups. (joint work with L. Miclo) ESAIM: P&S (2003).
Particle Motions in Absorbing Medium with Hard and Soft Obstacles.
(joint work with A. Doucet) Stochastic Analysis and Applications (2004).
Nonlinear equations
http://www.math.u-bordeaux1.fr/~pdelmora/ihp.pshttp://www.math.u-bordeaux1.fr/~pdelmora/ihp.pshttp://www.math.u-bordeaux1.fr/~pdelmora/exponent.pshttp://www.math.u-bordeaux1.fr/~pdelmora/exponent.pshttp://www.math.u-bordeaux1.fr/~pdelmora/exponent.pshttp://www.math.u-bordeaux1.fr/~pdelmora/obstacle.pshttp://www.math.u-bordeaux1.fr/~pdelmora/obstacle.pshttp://www.math.u-bordeaux1.fr/~pdelmora/exponent.pshttp://www.math.u-bordeaux1.fr/~pdelmora/exponent.pshttp://www.math.u-bordeaux1.fr/~pdelmora/ihp.pshttp://www.math.u-bordeaux1.fr/~pdelmora/ihp.pshttp://find/7/27/2019 Annecy Del Moral MTT
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Nonlinear equations
n+1 n(1) nQn+1 + n+1(1) n+1
Nonlinear & interacting mass + proba measures equations
n+1(1) = n(1) n(Gn) + n+1(1)
n+1 = Gn (n)Mn+1,(n(1),n)
with the Markov transitions:
Mn+1,(m,)(x, dy) := n (m, ) Mn+1(x, dy) + (1 n (m, )) n+1(dy)
and the collection of [0, 1]-parameters
n (m, ) =m (Gn)
m (Gn) + n+1(1)
An extended Feynman Kac model
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An extended Feynman-Kac model
nupdating n := Gn (n) = nSn,n prediction n+1 := nMn+1,(n(1),n)
A couple of equations:
The total mass evolution
n+1(1) = n(1) n(Gn) + n+1(1)
The nonlinear filtering/Feynman-Kac type conservative equations
n+1 = nSn,n Mn+1,(n(1),n) := n Kn+1,(n(1),n) Markov transition
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Mean field interacting particle models
Nn =1
N 1iNin N n and Nn (1) N n(1)
the total mass evolution [deterministic]
Nn+1(1) := Nn (1)
Nn (Gn) + n+1(1)
Mean field particle model
in+1 = r.v. with distribution Kn+1,(Nn (1),Nn )(in, dxn+1)
(Local) Stochastic perturbation model:
Nn+1 := Nn Kn+1,(Nn (1),Nn ) +
1N
WNn+1
Theoretical convergence results
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g
Independent local sampling error fluctuations
(WNn )n0
N iid centered Gaussian fields (Wn)n0
Theoretical convergence results
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Independent local sampling error fluctuations
(WNn )n0
N iid centered Gaussian fields (Wn)n0
Functional CLT(s) (withNn :=
Nn (1) Nn
)
V,Nn :=
N (Nnn) & V
,Nn :=
N (Nn
n)
N V
n & V
n
Theoretical convergence results
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Independent local sampling error fluctuations
(WNn )n0
N iid centered Gaussian fields (Wn)n0
Functional CLT(s) (withNn :=
Nn (1) Nn
)
V,Nn :=
N (Nnn) & V
,Nn :=
N (Nn
n)
N V
n & V
n
Uniform cv results (under some mixing conditions on Mn)
supn0
ENn
n (f)
p
c(p)/Np/2 (
uniform concentration)
Theoretical convergence results
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Independent local sampling error fluctuations
(WNn )n0
N iid centered Gaussian fields (Wn)n0
Functional CLT(s) (withNn :=
Nn (1) Nn
)
V,Nn :=
N (Nnn) & V
,Nn :=
N (Nn
n)
N V
n & V
n
Uniform cv results (under some mixing conditions on Mn)
supn0
ENn
n (f)
p
c(p)/Np/2 (
uniform concentration)
Unbiased particle total mass with variance (N n)
E1 Nn (1)/n(1)
2
c n/N
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Introduction/notation
Multiple objects branching signals
Multiple targets filtering modelsConditioning principlesPHD filtering equationStability properties
General measure valued equations
Particle association measures
Conditioning principles for marked point processes
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Poisson point process X with intensity (dx1) Q(x1, dx2) onE = (E1 E2)
X:= mN(X1,X2) = 1iN
(Xi1,Xi2) and Xj := mN(Xj) = 1iN
Xij
Conditioning principles for marked point processes
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Poisson point process X with intensity (dx1) Q(x1, dx2) onE = (E1 E2)
X:= mN(X1,X2) = 1iN
(Xi1,Xi2) and Xj := mN(Xj) = 1iN
Xij
2 Bayes rules: Normalization p(x2|x1) Markov operator p(x1|x2)
Q(x1, dx2) =Q(x1, dx2)
Q(x1,E2)and (dx1) Q(x1, dx2) = (Q) (dx2) Q(x2, dx1)
Conditioning principles for marked point processes
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Poisson point process X with intensity (dx1) Q(x1, dx2) onE = (E1 E2)
X:= mN(X1,X2) = 1iN
(Xi1,Xi2) and Xj := mN(Xj) = 1iN
Xij
2 Bayes rules: Normalization p(x2|x1) Markov operator p(x1|x2)
Q(x1, dx2) =Q(x1, dx2)Q(x1,E2)
and (dx1) Q(x1, dx2) = (Q) (dx2) Q(x2, dx1)
2 conditional distributions formulae:
E (F1(X1) | X2 ) = F1 (mN(x1)) 1iN
Q(Xi2, dx
i1)
E (F2(X2) | X1 ) =
F2 (mN(x2))
1iNQ(Xi1, dx
i2)
Conditioning principles for marked point processes
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g p p p p
(X1,X2) = (X,Y), X Poisson Signal (dx) YPoisson Obs.
(Xi = x) (Yi = y) (x) g(x, y) (dy) + (1 (x)) c(dy)
Clutter Y Poisson with intensity (dy) = h(y) (dy)
Conditioning principles for marked point processes
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(X1,X2) = (X,Y), X Poisson Signal (dx) YPoisson Obs.
(Xi = x) (Yi = y) (x) g(x, y) (dy) + (1 (x)) c(dy)
Clutter Y Poisson with intensity (dy) = h(y) (dy)
Observables Y0 = Y 1=c ( = detection rate)
(f) := E (X(f) | Yo)= ((1 )f) +
Yo(dy) (1 (y)) g(y,.)()(f)
with the conditional clutter probability density
(y) = h(y)/[h(y) + (g(y, .))]
Ex.: full detect and no clutter = 1 & h = 0 Yo = Y
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Conditional mean number of targets and their distributions
(1) = Y(1)and
(f) := (f)
(1) = Y(dy) =Y/Y(1)g(y,.)()(f) Bayes rule
with := /(1)
Single target Yo
= Y Classical filtering updating equations = g(Y,.)()
PHD filt i ti [Si l b hi d l (Q )]
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PHD filtering equation [Signal branching model (Qn, n)]
Hyp.: Xn+1 Poisson n+1 = nQn + n with obs. Y0n+1 as before
= PHD filtering equations:
n+1 := nQn + nn(f) := n((1 n)f) + Yon (dy) (1 n (y)) ngn (y,.)(n)(f)
A class of measure valued equations PHD; Bernoulli filters, etc.
n+1 = nQn+1,n
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Stability properties of meas. valued equations
n = n/n(1) Nonlinear semigroup (n(1), n) = p,n(p(1), p)
Stability Theorem :
p,n(m
, )
p,n(m, )
c e(np)
Regularity prop. 3 natural conditions on the PHD filter/model
1. small clutter intensities
2. high detection probability
3. high spontaneous birth rates
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Introduction/notation
Multiple objects branching signals
Multiple targets filtering models
General measure valued equationsNonlinear evolution equationsMean field particle approximation
Particle association measures
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Mean field particle models
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Mean field particle models
Nn =1
N 1iNin N n and Nn (1) N n(1)
with
Nn+1(1) = Nn (1) Nn (Gn,Nn (1)Nn )
in+1 = random var. with law Kn+1,(Nn (1)Nn )(in, dx)
Same theorems as before with uniform convergence estimates
Mean field particle models
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Mean field particle models
Nn =1
N 1iNin N n and Nn (1) N n(1)
with
Nn+1(1) = Nn (1) Nn (Gn,Nn (1)Nn )
in+1 = random var. with law Kn+1,(Nn (1)Nn )(in, dx)
Same theorems as before with uniform convergence estimates
Abstract general models
numerical scheme with local errors Interacting Kalman type filters particle associations measures ( GM-PHD)
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Introduction/notation
Multiple objects branching signals
Multiple targets filtering models
General measure valued equations
Particle association measures
Association measures [ = 1 & h = 0 & Qn = Mn]
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Ex. : Computable (exact or approximate) filters
The mappings ynn+1() := gn(yn,.
)()Mn+1
{Kalman, EKF, Ensemble Kalman filters, particle filters,. . . }
Initial association measure
1 :=
Y0(dy0) y01 (0) N1 :=
YN0 (dy0) y01 (0)
for instance
YN0 =1
N
1iN
Yi0 i.i.d. samples from Y0 or (if possible) YN
0 = Y0
Particle association measures [ = 1 & h = 0 & Qn = Mn]
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2 Y1(dy1)
y12 (
N1 )
=
Y1(dy1) YN0 (dy0)
y01 (0)(g1(y1, .))YN0 (dy0) y01 (0)(g1(y1, .)) [y12 y01 ] (0)
YN
0,1(d(y0, y1)) [y1
2 y0
1
] (0)
for instance
YN0,1 =1
N
1iN
(Yi0,1,Yi1,1)
i.i.d. samples from the N Y1(1) supported measures
Y1(dy1) YN0 (dy0)y01 (0)(g1(y1, .))YN0 (dy0) y01 (0)(g1(y1, .)) (y0,y1)
and so on ...
Particle association measures - Track management
http://find/7/27/2019 Annecy Del Moral MTT
48/48
Association particle tree genealogies
Nn+1
:= YN0,n(d(y0, . . . , yn)) ynn+1 . . . y01 (0)with
YN0,n :=1
N
1iN
(Yi0,n,Yi1,n,...,Yin,n)
Stochastic models and cv analysis :
General case :
(miss-detect, survival, spontaneous birth) = as before virtual obs.
Abstract models of the form n+1 = nQn+1,n . Mean field particle models Association particle measures. Lp-bounds Concentration sub-Gaussian inequalities.
http://find/