Introduction au Filtrage Particulaire Exemples en ... · compl´ement scientiﬁque Ecole Doctorale...

complement scientifique Ecole Doctorale MATISSE

IRISA et INRIA, salle Markov

jeudi 19 janvier 2012

Introduction au Filtrage Particulaire

Exemples en Navigation, Localisation et Poursuite

Francois Le Gland

INRIA Rennes et IRMAR

http://www.irisa.fr/aspi/legland/ed-matisse/


directeur de recherche INRIA, responsable de l’equipe

ASPI : Applications Statistiques des Systemes de Particules en Interaction

http://www.irisa.fr/aspi/

themes

• filtrage particulaire, et applications en

– localisation, navigation et poursuite

– assimilation de donnees sequentielle

• inference statistique des modeles de Markov caches

• simulation d’evenements rares, et extensions en

– simulation moleculaire

– optimisation globale

• analyse mathematique des methodes particulaires

http://www.irisa.fr/aspi/

contrats industriels

• avec France Telecom R&D, sur la localisation de terminaux mobiles

• avec Thales, sur la navigation par correlation de terrain

• avec DGA /Techniques Navales, sur l’optimisation du positionnement et de

l’activation de capteurs

collaboration reguliere avec l’ONERA (office nationale de recherche et d’etudes

aerospatiales) : encadrement de doctorants

projets ANR

• FIL, sur la fusion de donnees pour la localisation

• PREVASSEMBLE, sur les methodes d’ensemble pour l’assimilation de

donnees et la prevision

projets europeens

• HYBRIDGE et iFLY, sur les methodes de Monte Carlo conditionnelles pour

l’evaluation de risque en trafic aerien

Introduction to filtering

• Bayesian approach

• usefulness of a prior model

• some models

• Bayesian estimation (static case)

• exemple : Gaussian model (static case)

1 (Bayesian approach : 1)

Bayesian approach

objective is to estimate, in a recursive manner as much as possible, the unknown

state Xn of a system (typically, position and velocity of a mobile) in view of noisy

observations Y0:n = (Y0 · · ·Yn) related to the hidden state by a relation such as

Yk = hk(Xk) + Vk

many applications in localization, navigation and tracking of a mobile

in such cases, objective is to estimate position and velocity of a mobile, using :

(i) measurements provided by sensors

(ii) possibly a reference database, available for instance as a digital map

often needed to take into account

(iii) a prior model (not necessarily accurate) for the mobile displacement or

evolution, usually a Markov model


Bayesian approach, aka information fusion

prior information (state model)

+ likelihood (sensor model, consistency between state and measurement)

⇒ posteriori information

general principle : using Bayes rule

pX,Y (x, y) = pX|Y=y(x) pY (y) = pY |X=x(y) pX(x)

conditional distribution of X0:n given Y0:n (from which conditional distribution of

Xn given Y0:n is easily obtained, by marginalization) can be expressed in terms of

• conditional distribution of Y0:n given X0:n, often easy to evaluate for instance

in the additive model

Yk = hk(Xk) + Vk

• distribution of X0:n

recursive implementation possible thanks to Markov property


however, conditional distribution of Xn given Y0:n does not have an explicit

expression, except in a few special situations

• Markov chain with finite state space (forward Baum equation)

• Gaussian linear systems (Kalman filter)

in more general situations, growing interest (since beginning of 90’s) for

simulation–based methods of Monte Carlo type

particle filtering provides an implementation of Bayesian approach that is

• intuitive, easy to understand and implement

• flexible, many algorithmic variants available

• numerically efficient

• easy to analyze mathematically

questions

• why a prior model should be needed ? how to take it into account ?

• why (and how) should conditional distributions be computed ?

4 (some notations : 1)

some notations

if X is a random variable taking values in E, then mapping

φ 7−→ E[φ(X)] or equivalently A 7−→ P[X ∈ A]

defines a probability distribution µ on E, denoted as

µ(dx) = P[X ∈ dx]

and such that

E[φ(X)] =

∫

E

φ(x) µ(dx) = 〈µ, φ〉 or P[X ∈ A] = µ(A)

caracterizes uncertainty about X

5 (usefulness of a prior model : 1)

usefulness of a prior model

without additional information, observing

Yk = hk(Xk) + Vk

poses a new estimation problem at each new time instant (observations are not

accumulated)

in general, in each of these separated estimation problems, hidden state dimension

is greater than observation dimension

estimating hidden state Xk in view of observation Yk alone, and even if noise Vkwould not be present, is an ill–posed problem

• relation has more unknown variables than equations

• even in the favorable case where dimensions are equal, relation can have

several distinct solutions

• reconstructed sequence X0, X1, · · · , Xn can be poor as a trajectory, even

though each of its component separately is fine (temporal inconsistency)


common variational approach, to overcome underdetermination and / or solve

temporal inconsistency : introduction of additional information about the hidden

state sequence, e.g. in the form of cost functions involving initial state or

transitions between two successive states

for instance, minimize w.r.t. sequence x0:n = (x0, x1, · · · , xn), a criterion

J(x0:n) = c0(x0) +

n∑

k=1

ck(xk−1, xk) +

n∑

k=0

dk(xk)

that combines cost functions representing prior information about solution, and

cost functions representing for instance consistency between state and

measurement, of the form

dk(x) =12 |Yk − hk(x)|

2

interpretation : sought state xk should also satisfy observation equation

Yk = hk(xk) at least in some approximate sense

more generally, these cost functions could just represent a constraint (or a

property) that a sequence solution should fulfill


prior information is typically incorporated in terms of cost functions of the form

c0(x) =12 |x− µ|2 and ck(x, x

′) = 12 |x

′ − fk(x)|2

interpretation

• sought initial state x0 should be close to µ

• sought transition (xk−1, xk) should satisfy equation xk = fk(xk−1) at least

in some approximate sense

this regularization approach (from optimisation point of view) can be interpreted

as incorporating prior information (from statistical estimation point of view)


prior information can also be incorporated in terms of cost functions of the form

c0(x) = − log p0(x) and ck(x, x′) = − log pk(x

′ | x)

in this case, minimizing criterion

J(x0:n) = − log p0(x0)−

n∑

k=1

log pk(xk | xk−1) +

n∑

k=0

dk(xk)

is equivalent to maximizing (MAP, maximum a posteriori)

exp{−J(x0:n)} = p0(x0)

n∏

k=1

pk(xk | xk−1)

︸︷︷︸p0:n(x0:n)

exp{−

n∑

k=0

dk(xk)}

where p0:n(x0:n) denotes joint probability density of successive states (trajectory)

X0:n = (X0, X1, · · · , Xn) of a Markov chain characterized by

• initial probability density p0(x0)

• and transition probability densities pk(x′ | x)


pragmatical statistical approach : instead of looking for sequence minimizing (or

maximizing) a criterion, produce an estimator (MMSE, minimum mean square

error) and a quantitative measure of estimation error, i.e. evaluate path averages

(or integrals) of the form∫

E

· · ·

∫

E

f(x0:n) exp{−J(x0:n)} dx0:n

=

∫

E

· · ·

∫

E

f(x0:n) exp{−

n∑

k=0

dk(xk)} p0:n(x0:n) dx0:n

= E[f(X0:n) exp{−n∑

k=0

dk(Xk)} ]

such evaluation problem will be addressed later in an approximate manner, by

generating random sample paths approximately distributed according to

abovementionned Gibbs–Boltzmann distribution


Summary usefulness of a prior model

• to complement information coming from observations

• to connect observations received at different time instants (needed, just

because a mobile . . . usually moves between these time instants 6= fixed

parameter)

this can be a rough not necessarily accurate model, and it is usually a noisy model

(to acknowledge that a model is necessarily wrong, and in an attempt to quantify

modelling error, in a statistical manner)

11 (some models : 1)

some models

modeles (simplest to most general)

state space model (continuous state space, usually Rm)

• linear, Gaussian noise

• non–linear, Gaussian noise

• general state space model : non–linear, non Gaussian noise

hidden Markov model (HMM), and extensions

• hidden Markov model (hidden state sequence Markov, observations mutually

independant / Markov conditionally w.r.t. hidden states)

• partially observed Markov chains (hidden state and observation sequence

jointly Markov)

12 (some models : 2)

general state space

• discrete : finite, countable

• continuous : Euclidian space Rm, differentiable manifold

• hybrid continuous / discrete

• constrained

• graphical (collection de connected edges)

13 (Bayesian estimation (static case) : 1)

Bayesian estimation (static case)

random vector Z = (X,Y ) taking values in E × F (for example E = Rm,

F = Rd) with known joint probability distribution

for instance given by a probability density p(x, y)

objective : exploit observation of Y to improve knowledge about hidden

component X

given an estimator ψ(Y ) (based on observation Y ) of a statistics φ = φ(X)

taking values in Rp, its mean square error is defined as

E|ψ(Y )− φ(X)|2

trace of covariance matrix of estimation error

E[ (ψ(Y )− φ(X)) (ψ(Y )− φ(X))∗ ]

symmetric p× p semi–definite positive matrix

14 (Bayesian estimation (static case) : 2)

MMSE (minimum mean square error) estimator : an estimator φ such that

E[ (φ(Y )− φ(X)) (φ(Y )− φ(X))∗ ] ≤ E[ (ψ(Y )− φ(X)) (ψ(Y )− φ(X))∗ ]

in symmetric matrix sense, a fortiori such that

E|φ(Y )− φ(X)|2 ≤ E|ψ(Y )− φ(X)|2

for any other estimator ψ

implicit (non–constructive) definition −→ any more explicit formulation ?

Proposition MMSE estimator of statistics φ = φ(X) in view of observation Y

coincides with conditionnal mean of φ(X) given Y

φ(Y ) = E[φ(X) | Y ] =

∫

E

φ(x) P[X ∈ dx | Y ]

for instance

φ(y) =

∫

E

φ(x) p(x | y) dx with p(x | y) =p(x, y)

p(y)

15 (exemple : Gaussian model (static case) : 1)

exemple : Gaussian model (static case)

explicit MMSE estimator in Gaussian static case 6= dynamical case

Gaussian random vector Z = (X,Y ) taking values in Rm × R

d with mean vector

Z = (X, Y ), and covariance matrix QZ =

QX QXY

QY X QY

Proposition if covariance matrix QY is inversible, then conditional distribution of

X given Y is Gaussian, with mean vector

X(Y ) = X +QXY Q−1Y (Y − Y )

and covariance matrix (does not depend on observation Y )

0 ≤ R = QX −QXY Q−1Y QY X ≤ QX

Schur complement of matrix QY in block–matrix QZ

repeatedly used in correction step of Kalman filter (MMSE estimator in linear

Gaussian systems)

16 (linear Gaussian systems : Kalman filter : 1)

linear Gaussian systems : Kalman filter

hidden state sequence {Xk} taking values in Rm, such that

Xk = FkXk−1 + fk +Wk

and observation sequence {Yk} taking values in Rd, such that

Yk = HkXk + hk + Vk

assumptions :

• initial state X0 Gaussian vector with mean X0 and covariance matrix QX0

• state noise {Wk} Gaussian white noise with covariance matrix QWk

• observation noise {Vk} Gaussian white noise with covariance matrix QVk

• noise sequences {Wk} and {Vk} and initial state X0 mutually independant


interpretation of prior model

Xk = FkXk−1 + fk +Wk

in terms of uncertainty propagation

• even if state Xk−1 = x would be known exactly at time instant (k − 1), state

Xk at time instant k is uncertain, distributed as a Gaussian vector with mean

Fk x+ fk and covariance matrix QWk

• if state Xk−1 is uncertain at time instant (k − 1), distributed as a Gaussian

vector with mean Xk−1 and covariance matrix QXk−1, then this uncertainty is

propagated to time instant k : even if model noise would not be present,

state Xk at time instant k is uncertain, distributed as a Gaussian vector with

mean Fk Xk−1 + fk and covariance matrix FkQXk−1 F

∗k


Theorem [Kalman filter] if covariance matrix QVk is invertible

then conditional distribution of hidden state Xk given observation sequence Y0:kis Gaussian, with mean Xk and covariance matrix Pk, that satisfy following

recurrent equations

X−k = Fk Xk−1 + fk

P−k = Fk Pk−1 F

∗k +QW

k

and

Xk = X−k +Kk [Yk − (Hk X

−k + hk)]

Pk = [I −KkHk] P−k

where matrix

Kk = P−k H∗

k [Hk P−k H∗

k +QVk ]

−1

is called Kalman gain, and with initialization

X−0 = X0 = E[X0] and P−

0 = QX0 = cov(X0)

Non–linear and non Gaussian systems, with examples

• non–linear and non Gaussian systems

• hybridized inertial navigation (TAN, terrain aided navigation)

• visual tracking with color histogramme

• tracking a dim target (TBD, track–before–detect)

• indoor navigation

19 (non–linear and non Gaussian systems : 1)

non–linear and non Gaussian systems

◮ evolution of hidden state

Xk = fk(Xk−1,Wk) with Wk ∼ pk(dw)

interpretation as a model for propagating estimators and attached uncertainties

only requirement : easy to simulate model uncertainties

X0 ∼ µ0(dx) and Wk ∼ pk(dw)

more generally, for any x ∈ E

P[Xk ∈ dx′ | Xk−1 = x] = Qk(x, dx′)

only requirement : easy to simulate uncertain transitions

Xk ∼ Qk(x, dx′)


◮ relation between observation and hidden state

Yk = hk(Xk) + Vk with Vk ∼ qk(v) dv

only requirement : easy to evaluate likelihood function

gk(x) = qk(Yk − hk(x))

consistency between a possible state and true observation, e.g.

gk(x) ∝ exp{− 12 |Yk − hk(x)|

2}

more generally, for any x ∈ E

P[Yk ∈ dy | Xk = x] = gk(x, y)λk(dy)

only requirement : easy to evaluate likelihood function (with abuse of notation)

gk(x) = gk(x, Yk)


objective : recursively estimate hidden state Xk

in view of observations Y0:k = (Y0 · · ·Yk)

for instance, numerically approximate Bayesian filter

µk(dx) = P[Xk ∈ dx | Y0:k]


non–linear / non Gaussian systems : particle filtering

numerical approximation of Bayesian filter by weighted empirical probability

distribution

µNk =

N∑

i=1

wik δξik

with

N∑

i=1

wik = 1

associated with a system of N particles, caracterised by

• their positions (ξ1k · · · ξNk )

• and positive (nonnegative) normalized weights (w1k · · ·w

Nk )

system of interacting particles, that

• explore independently the state space, by following prior model

• and interact under a selection mechanism, related with how each particle is

consistent with current observation, by evaluating a likelihood function


SIS (sequential importance sampling) algorithm

• prediction : independently for any i = 1 · · ·N

ξik = fk(ξik−1,W

ik) with W i

k ∼ pk(dw)

• correction : for any i = 1 · · ·N

wik =

wik−1 qk(Yk − hk(ξ

ik))

N∑

j=1

wjk−1 qk(Yk − hk(ξ

jk))


alternatively (and preferably), weights can be used for re–sampling

SIR (sampling with importance resampling) algorithm : basic (bootstrap) version

• resampling (selection) : independently for any i = 1 · · ·N

within current population (ξ1k−1, · · · , ξNk−1), pick an individual ξ i

k−1

in a manner that is related with weights (w1k−1, · · · , w

Nk−1)

• prediction (mutation) : independently for any i = 1 · · ·N

ξik = fk(ξik−1,W

ik) with W i

k ∼ pk(dw)

• correction (weighting) : for any i = 1 · · ·N

wik =

qk(Yk − hk(ξik))

N∑

j=1

qk(Yk − hk(ξjk))

25 (demos : 1)

some examples

twofold purpose

• modeling with non–linear and non (necessarily) Gaussian systems

• behaviour (more than actual performance) of basic bootstrap SIR algorithm

four examples

• non–linear and non Gaussian systems

• hybridized inertial navigation (TAN, terrain aided navigation)

• visual tracking with color histogramme

• tracking a dim target (TBD, track–before–detect)

• indoor navigation

26 (demos : 2)

1st example : hybridized inertial navigation (TAN)

funding : DGA, programme d’etude amont NCT (nouveaux concepts pour la

navigation par correlation de terrain), coordination Thales Communications

• inertial measurements

– linear acceleration and angular velocity of an aircraft (INS, inertial

navigation system)

→ double integration yields inertial estimation of aircraft position and velocity

• prior model for evolution of inertial estimation error

• noisy altimeter measurements of

– height of aircraft above sea level (baro–altimeter)

– height of aircraft above terrain (radio–altimeter)

→ noisy measurement of height of terrain below aircraft above sea level

• correlation with a digital map (DTED, digital terrain elevation data)

providing height of terrain for any position in horizontal plane

27 (demos : 3)

0 10 20 30 40 50 60 70 80 90 1005600

5700

5800

5900

6000

6100

6200

6300

6400

6500

6600

Figure 1: Actual trajectory and estimated trajectory by integration of noisy inertial

measurements alone — Terrain profile and noisy altimeter measurements

28 (demos : 4)

inertial measurements : noisy aircraft acceleration in horizontal plane

aINSk = ak + wINS

k

estimated trajectory by integration of inertial measurements only

rINSk = rINS

k−1 +∆k vINSk−1

vINSk = vINS

k−1 +∆k aINSk

state variable Xk = (δrk, δvk) : correction to inertial estimation of position and

velocity in horizontal plane, i.e. δrk = rk − rINSk and δvk = vk − vINS

k

state model : by difference (or by linearization, in more realistic modeling)

δrk = rk−1 +∆k vk−1 − (rINSk−1 +∆k v

INSk−1) = δrk−1 +∆k δvk−1

δvk = vk−1 +∆k ak − (vINSk−1 +∆k a

INSk ) = δvk−1 −∆k w

INSk

i.e. δrk

δvk

=

δrk−1 +∆k δvk−1

δvk−1

−∆k

0

wINSk

29 (demos : 5)

observation Yk : terrain height obtained as difference between

• height of aircraft above see level

zBARk = zk + wBAR

k

• height of aircraft about terrain

zALTk = (zk − hk(Xk)) + wALT

k

i.e.

Yk = zBARk − zALT

k = hk(Xk) + Vk with Vk = wBARk − wALT

k

relation h(x) with hidden state x = (δr, δv) : height of terrain (read of digital

map) below position rINSk + δr in horizontal plane

weighting : likelihood function

gk(x) ∝ qk(Yk − hk(x))

−→ demo

31 (demos : 7)

2nd example : visual tracking by color histogramme

source : Patrick Perez, Carine Hue, Jako Vermaak and Marc Gangnet, Color-based

probabilistic tracking , European Conference on Computer Vision (ECCV’02)

user selects a zone within first image of a sequence

objective : automatically track this same zone along whole sequence

. . .

initialization image 2 image 3 image 10

Figure 2: Tracking a face within a sequence of 10 images

assumption : color histogramme of the zone to be tracked remains constant along

whole sequence

32 (demos : 8)

initial zone is caracterized by a color histogramme (reference histogramme), build

with Nb most frequent colors in this zone

normalized number q∗(n) of pixels in initial zone, whose color is

n = 1 · · ·Nb

Figure 3: Initial zone and color histogramme associated with Nb=64

33 (demos : 9)

to evaluate consistency of a test zone in k–th image, characterized by position

(expressed in pixels) of its center x, compute its color histogramme

normalized number qk(x, n) of pixels in test zone centered in x in k–th

image of sequence, whose closest color is color n = 1 · · ·Nb

and define a mesure of distance (Hellinger distance) between the two normalized

color histogrammes

D(q∗, qk(x)) = 1−Nb∑

n=1

√q∗(n) qk(x, n) =

12

Nb∑

n=1

(√q∗(n)−

√qk(x, n))

2

34 (demos : 10)

state variable Xk = rk : position of center of tracked zone in k–th image

state model : simple random walk

rk = rk−1 +Wk

observation : k–th image within sequence

weighting of test zone centered in x in k–th image within sequence

gk(x) ∝ exp{−λD(q∗, qk(x))}

−→ demo

39 (demos : 15)

3rd example : tracking a dim target (TBD)

source : David J. Salmond and H. Birch, A particle filter for track-before-detect,

American Control Conference (ACC’01)

radar image : rectangular pixel array, where echo intensity received in a pixel is

coded in terms of grey level, ranging from darkest (echo with low intensity) to

brighter (echo with high intensity)

in principle, if a target is present in physical space, it will appear in image plane in

the form of a pixel brighter than other pixels

to detect (and localize) target, it is therefore sufficient to search for brightest

pixel, i.e. with highest intensity, or simply to use thresholding

repeating this procedure on each radar image successively, it is therefore possible

first to detect, then to track, target in image sequence

high SNR : threshold detection on each image separately, then tracking

vs. low SNR : tracking before (or even without) detection

40 (demos : 16)

observation, frame #7

−100 −50 0 50 100−100

−50

0

50

100avec la position réelle

−100 −50 0 50 100−100

−50

0

50

100

détection, seuil = 2 sigma

−100 −50 0 50 100−100

−50

0

50

100

−1 0 1 20

20

40

60

80histogramme des intensités

Figure 4: Observed image, actual position, histogramme, detection (visible target)

41 (demos : 17)

observation, frame #3

−100 −50 0 50 100−100

−50

0

50

100avec la position réelle

−100 −50 0 50 100−100

−50

0

50

100

détection, seuil = 2 sigma

−100 −50 0 50 100−100

−50

0

50

100

−2 −1 0 1 20

10

20

30

40histogramme des intensités

Figure 5: Observed image, actual position, histogramme, detection (dim target)

42 (demos : 18)

state variable Xk = (rk, vk) : target position and velocity

state model : constant velocity motion

allows to search for a consistent motion between successive images

rk = rk−1 +∆k vk−1 and vk = vk−1 +Wk

hence rk

vk

=

rk−1 +∆k vk−1

vk−1

+

0

I

Wk

observation Yk ≡ (Yk(s) , s ∈ S) : intensity of each pixel in k–th image

Yk(s) = I(Xk, s) +Bk(s)

caused either by

• noise only

• or noise and target, if target is present in region of physical space

corresponding to considered pixel in image plane

43 (demos : 19)

relation I(x, s) with hidden state x = (r, v) : intensity of pixel s ∈ S caused by a

target present in position r in physical space

I(x, s) = I0δ2

2π σ2PSF

exp{−|r(s)− r|2

2σ2PSF

} 1{s ∈ C(r)}

where r(s) denotes position in physical space corresponding to pixel center s,

where δ > 0 denotes pixel size in physical space, and where set C(r) denotes

9–points neighbourhood in image plane around pixel corresponding to position r

in physical space


gk(x) ∝ exp{−1

2σ2B

∑

s∈S

|Yk(s)− I(x, s)|2}

∝ exp{1

σ2B

∑

s∈C(r)

I(x, s) Yk(s)−1

2σ2B

∑

s∈C(r)

|I(x, s)|2}

−→ demo

47 (demos : 23)

4th example : indoor navigation

funding : ANR, project FIL (fusion d’information pour la localisation), programme

Telecommunications, coordination Thales Alenia Space

• navigation measurements (PNS, pedestrian navigation system)

– walked distance and direction change between two time instants

→ integration yields PNS estimation of user position and orientation

• prior model for user motion, i.e. position and orientation evolution, based on

incremental PNS measurements

• noisy measurements of

– distance between user and a ranging beacon with known position and with

limited range

• user detection (or non–detection) by a beacon

• fingerprinting : knowledge of motion constraints due to obstacles, available

from a map of building

48 (demos : 24)

0 25 50 75 100 125 150

0

25

50

75

100

Figure 6: Admissible trajectory and estimated trajectory from PNS measurements

alone

49 (demos : 25)

Figure 7: Detection by a ranging beacon

50 (demos : 26)

incremental PNS measurements : walked distance and direction change

dPNSk = dk + wwalk

k et αPNSk = αk + wturn

k

estimated trajectory by integration of PNS measurements alone

rPNSk = rPNS

k−1 + u(θPNSk−1 ) d

PNSk

θPNSk = θPNS

k−1 + αPNSk

state variable Xk = (rk, θk) : user position and orientation

state model : incremental update based on PNS measurements

rk = rk−1 + u(θk−1) (dPNSk − wwalk

k )︸︷︷︸dk

and θk = θk−1 + (αPNSk − wturn

k )︸︷︷︸αk

i.e. rk

θk

=

rk−1 + u(θk−1) d

PNSk

θk−1 + αPNSk

−

u(θk−1) 0

0 1

wwalk

k

wturnk

51 (demos : 27)

observation Yk : distance to an active beacon located at position a and with

range R

Yk = h(Xk) + Vk

relation h(x) with hidden state x = (r, θ) : distance |r − a| to beacon


gk(x) ∝ qk(Yk − h(x))

if user is / is not detected by beacon, then

gk(x) ∝ 1{h(x) ≤ R} or else gk(x) ∝ 1{h(x) ≥ R}

−→ demo

54 (conclusion : 1)

to be continued next time . . .

#1 more general models, from non–linear and non Gaussian systems to hidden

Markov models and partially observed Markov chains, so as to handle e.g.

• regime / mode switching

• correlation between state noise and observation noise

#2 for each of these models (or just for most general model), representation of

P[X0:n ∈ dx0:n | Y0:n]

as a Gibbs–Boltzmann distribution, with recursive formulation, and idem for

P[Xn ∈ dxn | Y0:n]

#3 particle approximation (SIS and SIR algorithms) from either representations

#4 asymptotic behaviour as sample size goes to infinity

#5 numerous algorithmic variants

complement scientifique Ecole Doctorale MATISSE

IRISA et INRIA, salle Markov

jeudi 26 janvier 2012

Variantes Algorithmiques et Justifications Theoriques

Francois Le Gland

INRIA Rennes et IRMAR



Modeles generaux

au–dela des systemes lineaires gaussiens

• systemes non–lineaires / non–gaussiens

• modeles de Markov caches


some notations (continued)

if X is a random variable taking values in E, then mapping

φ 7−→ E[φ(X)] or equivalently A 7−→ P[X ∈ A]

defines a probability distribution µ on E, denoted as

µ(dx) = P[X ∈ dx]

and such that

E[φ(X)] =

∫

E

φ(x) µ(dx) = 〈µ, φ〉 or P[X ∈ A] = µ(A)

caracterizes uncertainty about X


transition probability kernel M(x, dx′) on E

collection of probability distributions on E indexed by x ∈ E

acts on functions according to

M φ(x) =

∫

E

M(x, dx′) φ(x′)

and acts on probability distributions according to

µM(dx′) =

∫

E

µ(dx)M(x, dx′)

seen as a mixture distribution caracterized by

〈µM,φ〉 =

∫

E

[

∫

E

µ(dx)M(x, dx′)] φ(x′)

=

∫

E

µ(dx) [

∫

E

M(x, dx′) φ(x′)]

= 〈µ,M φ〉


product of two transition probability kernels

4 (systemes non–lineaires / non–gaussiens : 1)

systemes non–lineaires / non–gaussiens

modele a priori pour l’etat cache a valeurs dans E

Xk = fk(Xk−1,Wk) avec Wk ∼ pWk (dw)

condition initiale X0 ∼ η0(dx)

observation a valeurs dans Rd dans un bruit additif possedant une densite

Yk = hk(Xk) + Vk avec Vk ∼ qVk (v) dv

les v.a. X0, W1, · · · ,Wk, · · · et V0, V1, · · · , Vk, · · · sont

independantes mais pas necessairement gaussiennes

pour la suite, il suffit de savoir

• simuler une v.a. selon η0(dx) ou selon pWk (dw)

• evaluer pour tout v ∈ Rd, la fonction qVk (v)


Proposition la suite {Xk} est une chaıne de Markov a valeurs dans E, c–a–d

que conditionnellement aux etats passes X0:k−1, l’etat present Xk ne depend que

de Xk−1 (statistique exhaustive)

P[Xk ∈ dx | X0:k−1] = P[Xk ∈ dx | Xk−1]

et la loi de la suite {Xk} est completement determinee par

• la loi initiale

P[X0 ∈ dx] = η0(dx)

• et le noyau de transition

P[Xk ∈ dx′ | Xk−1 = x] = Qk(x, dx′)


Preuve compte tenu que X0:k−1 et Wk sont independants

E[φ(Xk) | X0:k−1] = E[φ(fk(Xk−1,Wk)) | X0:k−1]

=

∫

Rp

φ(fk(Xk−1, w)) pWk (dw)

ne depend que de Xk−1, c–a–d que

E[φ(Xk) | X0:k−1] = E[φ(Xk) | Xk−1]

d’ou la propriete de Markov , et∫

E

Qk(x, dx′) φ(x′) = E[φ(Xk) | Xk−1 = x] =

∫

Rp

φ(fk(x,w)) pWk (dw)

ce qui definit implicitement le noyau de transition 2


Remarque si fk(x,w) = bk(x) +w et si la distribution de probabilite pWk (dw) de

la v.a. Wk admet une densite encore notee pWk (w), c’est–a–dire si

Xk = bk(Xk−1) +Wk avec Wk ∼ pWk (w) dw

alors

Qk(x, dx′) = pWk (x′ − bk(x)) dx

′

c’est–a–dire que le noyau Qk(x, dx′) admet une densite

en effet, le changement de variable x′ = bk(x) + w donne immediatement

Qk φ(x) =

∫

Rm

φ(bk(x) + w) pWk (w) dw =

∫

Rm

φ(x′) pWk (x′ − bk(x)) dx′


Remarque en general, le noyau Qk(x, dx′) n’admet pas de densite

en effet, conditionnellement a Xk−1 = x, le v.a. Xk appartient necessairement au

sous–ensemble

M(x) = {x′ ∈ Rm : il existe w ∈ R

p tel que x′ = fk(x,w)}

et dans le cas ou p < m ce sous ensemble M(x) est generalement, sous certaines

hypotheses de regularite, une sous–variete differentielle de dimension p dans

l’espace Rm

il ne peut donc pas y avoir de densite par rapport a la mesure de Lebesgue de Rm

pour la distribution de probabilite Qk(x, dx′) du v.a. Xk


Proposition la suite {Yk} verifie l’hypothese de canal sans memoire, c–a–d que

pour tout instant n

• conditionnellement aux etats caches X0:n les observations Y0:n sont

mutuellement independantes, ce qui se traduit par

P[Y0:n ∈ dy0:n | X0:n] =

n∏

k=0

P[Yk ∈ dyk | X0:n]

• pour tout k = 0 · · ·n, la loi conditionnelle de Yk sachant X0:n ne depend que

de Xk, ce qui se traduit par

P[Yk ∈ dyk | X0:n] = P[Yk ∈ dyk | Xk]

avec la densite d’emission

P[Yk ∈ dy | Xk = x] = qVk (y − hk(x)) dy


on definit la fonction de vraisemblance

gk(x) = qVk (Yk − hk(x))

qui mesure l’adequation de x ∈ Rm avec l’observation Yk

la loi conditionnelle jointe des observations Y0:n sachant les etats X0:n verifie

P[Y0:n ∈ dy0:n | X0:n = x0:n] =

n∏

k=0

qVk (yk − hk(xk)) dy0 · · · dyn


Preuve compte tenu que V0:n et X0:n sont mutuellement independants

E[φ0(Y0) · · ·φn(Yn) | X0:n]

= E[φ0(h0(X0) + V0) · · ·φn(hn(Xn) + Vn) | X0:n]

=

∫

Rd

· · ·

∫

Rd

φ0(h0(X0) + v0) · · ·φn(hn(Xn) + vn) P[V0:n ∈ dv0:n]

=

n∏

k=0

∫

Rd

φk(hk(Xk) + vk) P[Vk ∈ dvk]

=

n∏

k=0

∫

Rd

φk(hk(Xk) + vk) qVk (vk) dvk =

n∏

k=0

E[φk(Yk) | Xk]

=n∏

k=0

∫

Rd

φk(yk) qVk (yk − hk(Xk)) dyk 2

12 (modeles de Markov caches : 1)

modeles de Markov caches

plus generalement, on peut supposer que les etats caches {Xk} forment une

chaıne de Markov a valeurs dans un espace E qui peut etre tres general, par

exemple

• un espace hybride continu / discret

• un sous–ensemble defini par des contraintes

• une variete differentielle

• un graphe, etc.

y compris trajectoriel, de noyau de transition

P[Xk ∈ dx′ | Xk−1 = x] = Qk(x, dx′)

et de loi initiale

P[X0 ∈ dx] = η0(dx)


la loi jointe des etats X0:n verifie

P[X0:n ∈ dx0:n] = η0(dx0)

n∏

k=1

Qk(xk−1, dxk)


on suppose egalement que les observations {Yk} verifient l’hypothese de canal

sans memoire, c–a–d que pour tout instant n

• conditionnellement aux etats caches X0:n les observations Y0:n sont

mutuellement independantes, ce qui se traduit par

P[Y0:n ∈ dy0:n | X0:n] =

n∏

k=0

P[Yk ∈ dyk | X0:n]

• pour tout k = 0 · · ·n, la loi conditionnelle de Yk sachant X0:n ne depend que

de Xk, ce qui se traduit par

P[Yk ∈ dy | X0:n] = P[Yk ∈ dy | Xk]

avec la densite d’emission

P[Yk ∈ dy | Xk = x] = gk(x, y)λFk (dy)

ou la mesure positive λFk (dy) definie sur F ne depend pas de x ∈ E


par abus de notation on definit la fonction de vraisemblance

gk(x) = gk(x, Yk)

qui mesure l’adequation de x ∈ E avec l’observation Yk

la loi conditionnelle jointe des observations Y0:n sachant les etats X0:n verifie

P[Y0:n ∈ dy0:n | X0:n = x0:n] =n∏

k=0

gk(xk, yk) λF0 (dy0) · · ·λ

Fn (dyn)

la situation est completement decrite par

- - - -Xk−1 Xk Xk+1

? ? ?

Yk−1 Yk Yk+1

ou les fleches representent la dependance entre variables aleatoires


pour la suite il suffit de savoir

• simuler pour tout x ∈ E, une v.a. selon le noyau de transition Qk(x, dx′)

• evaluer pour tout x′ ∈ E, la fonction de vraisemblance gk(x′)

pour tout instant k = 1 · · ·n

Filtre bayesien pour les modeles de Markov caches

• representation probabiliste

• equation recurrente

17 (filtre bayesien : modeles de Markov caches : representation probabiliste : 1)

modeles de Markov caches : representation probabiliste

Theorem le filtre bayesien µn defini par

〈µn, φ〉 = E[φ(Xn) | Y0:n]

admet la representation probabiliste

〈µn, φ〉 =〈γn, φ〉

〈γn, 1〉avec 〈γn, φ〉 = E[φ(Xn)

n∏

k=0

gk(Xk) ]

Remarque l’esperance porte seulement sur les etats caches successifs X0:n : les

fonctions de vraisemblance gk(x) sont definies par abus de notation comme

gk(x) = gk(x, Yk)

pour tout k = 0, 1 · · ·n, et dependent implicitement des observations Y0:n, mais

celles–ci sont considerees comme fixees


Preuve d’apres la formule de Bayes, et d’apres la propriete de canal sans

memoire, la loi jointe des etats caches X0:n et des observations Y0:n verifie

P[X0:n ∈ dx0:n, Y0:n ∈ dy0:n] = P[Y0:n ∈ dy0:n | X0:n = x0:n] P[X0:n ∈ dx0:n]

= P[X0:n ∈ dx0:n]

n∏

k=0

gk(xk, yk)λF0 (dy0) · · ·λ

Fn (dyn)

en integrant par rapport aux variables x0:n, on obtient la loi jointe des

observations Y0:n, c–a–d

P[Y0:n ∈ dy0:n] =

∫

E

· · ·

∫

E

n∏

k=0

gk(xk, yk) P[X0:n ∈ dx0:n]λF0 (dy0) · · ·λ

Fn (dyn)

= E[

n∏

k=0

gk(Xk, yk) ] λF0 (dy0) · · ·λ

Fn (dyn)


d’apres la formule de Bayes, il vient

P[X0:n ∈ dx0:n, Y0:n ∈ dy0:n]

= P[X0:n ∈ dx0:n]

n∏

k=0

gk(xk, yk) λF0 (dy0) · · ·λ

Fn (dyn)

= P[X0:n ∈ dx0:n | Y0:n = y0:n] P[Y0:n ∈ dy0:n]

= P[X0:n ∈ dx0:n | Y0:n = y0:n] E[

n∏

k=0

gk(Xk, yk) ] λF0 (dy0) · · ·λ

Fn (dyn)

et on obtient

P[X0:n ∈ dx0:n | Y0:n = y0:n] =

n∏

k=0

gk(xk, yk) P[X0:n ∈ dx0:n]

E[n∏

k=0

gk(Xk, yk) ]

pour toute suite y0:n d’observations


pour toute fonction φ definie sur E

E[φ(Xn) | Y0:n = y0:n] =

∫

E

· · ·

∫

E

φ(xn)

n∏

k=0

gk(xk, yk) P[X0:n ∈ dx0:n]

E[

n∏

k=0

gk(Xk, yk) ]

=

E[φ(Xn)

n∏

k=0

gk(Xk, yk) ]

E[

n∏

k=0

gk(Xk, yk) ]

comme cette identite est verifiee pour toute suite y0:n d’observations

〈µn, φ〉 = E[φ(Xn) | Y0:n] =

E[φ(Xn)

n∏

k=0

gk(Xk) ]

E[

n∏

k=0

gk(Xk) ]

=〈γn, φ〉

〈γn, 1〉2


de la meme maniere, le predicteur bayesien µ−n defini par

〈µ−n , φ〉 = E[φ(Xn) | Y0:n−1]

admet la representation probabiliste

〈µ−n , φ〉 =

〈γ−n , φ〉

〈γ−n , 1〉avec 〈γ−n , φ〉 = E[φ(Xn)

n−1∏

k=0

gk(Xk) ]

22 (filtre bayesien : modeles de Markov caches : decomposition d’importance : 1)

filtre bayesien : modeles de Markov caches : decomposition d’importance

decomposition (non unique)

γ0(dx) = g0(x) η0(dx) = gimp0 (x) ηimp

0 (dx)

et

Rk(x, dx′) = Qk(x, dx

′) gk(x′) = g

impk (x, x′) Qimp

k (x, dx′)

comme le produit de

• la fonction de ponderation positive gimp0 (x) ou gimp

k (x, x′)

• et la distribution de probabilite ηimp0 (dx) ou le noyau markovien Qimp

k (x, dx′)

respectivement, ou en pratique la decomposition doit etre telle qu’il est facile

• de simuler pour tout x ∈ E, une v.a. selon le noyau markovien Qimpk (x, dx′)

• d’evaluer pour tout x, x′ ∈ E, la fonction de ponderation gimpk (x, x′)

avec la convention gimp0 (x, x′) = g

imp0 (x′)

attention : la fonction de ponderation gimpk (x, x′) depend de la transition (x, x′)

et pas seulement de l’etat d’arrivee x′

23 (filtre bayesien : modeles de Markov caches : decomposition d’importance : 2)

pour une decomposition d’importance donnee

〈γn, φ〉 = E[φ(Xn)

n∏

k=0

gk(Xk) ]

=

∫

E

· · ·

∫

E

φ(xn) η0(dx0)n∏

k=1

Qk(xk−1, dxk)n∏

k=0

gk(xk)

=

∫

E

· · ·

∫

E

φ(xn) ηimp0 (dx0)

n∏

k=1

Qimpk (xk−1, dxk)

n∏

k=0

gimpk (xk−1, xk)

= E[φ(X impn )

n∏

k=0

gimpk (X imp

k−1, Ximpk ) ]

ou la suite {X impk } est une chaıne de Markov, caracterisee par

• la loi initiale ηimp0 (dx)

• et le noyau de transition Qimpk (x, dx′)

et avec une fonction de ponderation / selection gimpk (x, x′) qui depend de la

transition (x, x′) et pas seulement de l’etat d’arrivee x′

24 (filtre bayesien : modeles de Markov caches : equation recurrente : 1)

modeles de Markov caches : equation recurrente

Theorem la suite {µk} verifie l’equation recurrente suivante

µk−1prediction

−−−−−−−−−−−→ µ−k = µk−1Qk

correction−−−−−−−−−−−→ µk = gk · µ−

k

avec la condition initiale µ−0 = η0

Remarque dans l’enonce du theoreme, la notation

µk−1Qk(dx′) =

∫

E

µk−1(dx)Qk(x, dx′)

designe la distribution de melange obtenue par l’action du noyau markovien

Qk(x, dx′) sur la distribution de probabilite µk−1(dx), et la notation

gk · µ−k =

gk µ−k

〈µ−k , gk〉

designe le produit (projectif) de la distribution de probabilite a priori µ−k (dx

′) et

de la fonction de vraisemblance gk(x′)


Preuve du Theoreme on procede en deux etapes, correspondant respectivement

aux etapes de prediction et de correction, et en raisonnant d’abord sur les versions

non normalisees, puis en normalisant

◮ etape de prediction expression de µ−n en fonction de µn−1

on remarque immediatement que

〈γ−n , 1〉 = E[

n−1∏

k=0

gk(Xk) ] = 〈γn−1, 1〉

c–a–d que la constante de normalisation est conservee


en utilisant la propriete de Markov, on a

〈γ−n , φ〉 = E[φ(Xn)

n−1∏

k=0

gk(Xk) ]

= E[E[φ(Xn) | X0:n−1]n−1∏

k=0

gk(Xk) ]

= E[E[φ(Xn) | Xn−1]

n−1∏

k=0

gk(Xk) ]

= E[Qn φ(Xn−1)

n−1∏

k=0

gk(Xk)] = 〈γn−1, Qn φ〉 = 〈γn−1Qn, φ〉

pour toute fonction φ definie sur E


la derniere egalite exprime simplement que

〈γn−1, Qn φ〉 =

∫

E

γn−1(dx) [

∫

E

Qn(x, dx′)φ(x′) ]

=

∫

E

[

∫

E

γn−1(dx)Qn(x, dx′) ] φ(x′)

=

∫

E

γn−1Qn(dx′) φ(x′) = 〈γn−1Qn, φ〉

comme la fonction φ est quelconque, on en deduit que

γ−n = γn−1Qn

et en normalisant, on obtient

µ−n =

γ−n

〈γ−n , 1〉=γn−1Qn

〈γn−1, 1〉= µn−1Qn


◮ etape de correction expression de µn en fonction de µ−n

on a simplement

〈γn, φ〉 = E[φ(Xn)

n∏

k=0

gk(Xk) ]

= E[φ(Xn) gn(Xn)n−1∏

k=0

gk(Xk) ] = 〈γ−n , gn φ〉 = 〈gn γ−n , φ〉

pour toute fonction φ definie sur E, ou la derniere egalite exprime simplement que

〈γ−n , gn φ〉 =

∫

E

[gn(x)φ(x)] γ−n (dx) =

∫

E

φ(x) [gn(x) γ−n (dx)] = 〈gn γ

−n , φ〉

comme la fonction φ est quelconque, on en deduit que

γn = gn γ−n

et en normalisant, on obtient

µn =γn

〈γn, 1〉=

gn γ−n

〈γ−n , gn〉=

gn µ−n

〈µ−n , gn〉

2

29 (a suivre : 1)

a suivre : approximation particulaire du filtre bayesien

objectif : approximation des distributions de probabilite definies par la relation de

recurrence

µk−1mutation

−−−−−−−−−−→ ηk = µk−1Qk

ponderation−−−−−−−−−−−−→ µk = gk · ηk

avec la condition initiale µ0 = g0 · η0

idee : rechercher une approximation sous la forme de distributions de probabilite

empiriques (eventuellement ponderees)

ηk ≈ ηNk =1

N

N∑

i=1

δξik

et µk ≈ µNk =

N∑

i=1

wik δξik

avec

N∑

i=1

wik = 1

associees a une population de N particules caracterisee par

• les positions (ξ1k, · · · , ξNk ) dans E

• et les poids positifs (w1k, · · · , w

Nk )

31 (a suivre : 3)

etape de prediction : a partir de la definition

〈µNk−1Qk, φ〉 =

∫µNk−1(dx)

∫Qk(x, dx

′)φ(x′)

=

N∑

i=1

wik−1

∫Qk(ξ

ik−1, dx

′)φ(x′)

=

∫[

N∑

i=1

wik−1 Qk(ξ

ik−1, dx

′) ]φ(x′)

pour toute fonction φ, de sorte que

µNk−1Qk =

N∑

i=1

wik−1 m

ik

s’exprime comme un melange fini, avec

mik(dx

′) = Qk(ξik−1, dx

′) pour tout i = 1 · · ·N

qu’il s’agit d’approximer / echantillonner (nombreuses methodes possibles)

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