Sampling strategies for Sequential Monte Carlo (SMC) methods

Sampling strategies forSequential Monte Carlo methods

Arnaud Doucet1, Stephane Senecal2

1Department of Engineering, University of Cambridge

2The Institute of Statistical Mathematics

2004

thanks to the Japanese Ministry of Education and the Japan Society for

the Promotion of Science

1

Overview

– Introduction : state space models, Monte Carlo methods

– Sequential Importance Sampling/Resampling

– Strategies for sampling

– Examples, applications

– References

2

Estimation of state space models

xt = ft(xt−1, ut) yt = gt(xt, vt)

p(x0:t|y1:t) → p(xt|y1:t) =

∫p(x0:t|y1:t)dx0:t−1

distribution of x0:t ⇒ computation of estimate x0:t :

x0:t =

∫x0:tp(x0:t|y1:t)dx0:t → Ep(.|y1:t){f(x0:t)}

x0:t = arg maxx0:t

p(x0:t|y1:t)

3

Computation of the estimates

p(x0:t|y1:t) ⇒ multidimensionnal, non-standard distributions :

→ analytical, numerical approximations

→ integration, optimisation methods

⇒ Monte Carlo techniques

4

Monte Carlo approach

compute estimates for distribution π(.) → samples x1, . . . , xN ∼ π

x

\pi(x)

x_1 x_N

⇒ distribution πN = 1N

∑Ni=1 δxi approximates π(.)

5

Monte Carlo estimates

SN (f) =1

N

N∑

i=1

f(xi) −→∫f(x)π(x)dx = Eπ{f(x)}

arg max(xi)1≤i≤N πN (xi) approximates arg maxx π(x)

⇒ sampling xi ∼ π difficult

→ importance sampling techniques

6

Importance Sampling

xi ∼ π → candidate/proposal distribution xi ∼ g

x

g(x)

\pi(x)

x_Nx_1

7

Importance Sampling

xi ∼ g 6= π → (xi, wi) weighted sample

⇒ weight wi =π(xi)

g(xi)

x

g(x)

\pi(x)

x_Nx_1

8

Estimation

importance sampling → computation of Monte Carlo estimates

e. g. expectations Eπ{f(x)} :∫f(x)

π(x)

g(x)g(x)dx =

∫f(x)π(x)dx

N∑

i=1

wif(xi) →∫f(x)π(x)dx = Eπ{f(x)}

dynamic model (xt, yt) ⇒ recursive estimation x0:t−1 → x0:t

Monte Carlo techniques ⇒ sampling sequences x(i)0:t−1 → x

(i)0:t

9

Sequential simulation

sampling sequences x(i)0:t ∼ πt(x0:t) recursively :

time

variablestate

x

p(x,t) target distribution:

t

t2

t1

p(x,t2)

x_t1

x_t2

p(x_t1)

p(x_t2)

p(x,t1)

10

Sequential simulation : importance sampling

samples x(i)0:t ∼ πt(x0:t) approximated by weighted particles

(x(i)0:t, w

(i)t )1≤i≤N

time


p(x,t2)

t

t2

t1

x

p(x,t1)

11

Sequential importance sampling

diffusing particles x(i)0:t1→ x

(i)0:t2

time


p(x,t2)

t

x

p(x,t1)

t2

t1

⇒ sampling scheme x(i)0:t−1 → x

(i)0:t

12

Sequential importance sampling

updating weights w(i)t1 → w

(i)t2

time


p(x,t2)

t

p(x,t1)

x

t2

t1

⇒ updating rule w(i)t−1 → w

(i)t

13

Sequential Importance Sampling

x0:t ∼ πt(x0:t)⇒ (x(i)0:t, w

(i)t )1≤i≤N

Simulation scheme t− 1 → t :

– Sampling step x(i)t ∼ qt(xt|x(i)

0:t−1)

– Updating weights

w(i)t ∝ w(i)

t−1 ×πt(x

(i)0:t−1, x

(i)t )

πt−1(x(i)0:t−1)qt(x

(i)t |x(i)

0:t−1)︸︷︷︸incremental weight (iw)

normalizing∑Ni=1 w

(i)t = 1

14


x0:t ∼ πt(x0:t)⇒ (x(i)0:t, w

(i)t )1≤i≤N

proposal + reweighting →

\pi(x_t)

x_t

15


proposal + reweighting → var{(w(i)t )1≤i≤N} ↗ with t

x_t

\pi(x_t)

→ w(i)t ≈ 0 for all i except one

16

⇒ Resampling

x_t

\pi(x_t)

0 x_t^(1)

x_t^(j)1x_t^(i)2 x_t^(k)3

x_t^(N)0

→ draw N particles paths from the set (x(i)0:t)1≤i≤N

with probability (w(i)t )1≤i≤N

17

Sequential Importance Sampling/Resampling

Simulation scheme t− 1 → t :

– Sampling step x,(i)t ∼ qt(x,t|x(i)

0:t−1)

– Updating weights w(i)t ∝ w(i)

t−1 ×πt(x

(i)0:t−1,x

,(i)t )


,(i)t |x

(i)0:t−1)

→ parallel computing

– ⇒ Resampling step : sample N paths from (x(i)0:t−1, x

,(i)t )1≤i≤N

→ particles interacting : computation at least O(N)

18

SISR for recursive estimation of state space models

xt = ft(xt−1, ut) → p(xt|xt−1)

yt = gt(xt, vt) → p(yt|xt)

Usual SISR : Bootstrap filter (Gordon et al. 93, Kitagawa 96) :

– Sampling step x(i)t ∼ p(xt|x(i)

t−1)

– Updating weights : incremental weight w(i)t ∝ w(i)

t−1 × iw

iw ∝ p(yt|x(i)t )

– Stratified/Deterministic resampling

efficient, easy, fast for a wide class of models

→ tracking, time series

19

Overview - Break ,

– Introduction :

→ state space models

→ estimation, computating estimates via Monte Carlo methods

→ importance sampling

– recursive estimation → sequential simulation

⇒ Sequential Importance Sampling/Resampling

– ⇒ Strategies for sampling :

→ designing/sampling “optimal” candidate distribution

→ considering blocks of variables : reweighting, → sampling

– Examples and applications

20

Improving simulation

sampling multimodal, multidimensional distributions

model with informative observation → peaky likelihood

→ prior dynamics to diffuse particles : poor approximation results

→ efficient propagation for a finite number of particles N

⇒ need for good sampling proposals

21


Optimal proposal distribution qt(xt|x(i)0:t−1)

→ mimimizing variance of incremental weight (w(i)t ∝ w(i)

t−1 × iw)

iw =πt(x

(i)0:t−1, x

(i)t )


(i)t |x(i)

0:t−1)

⇒ 1-step ahead predictive :

πt(xt|x0:t−1) = p(xt|xt−1, yt)

⇒ incremental weight :

iw → πt(x0:t−1)

πt−1(x0:t−1)=

p(x0:t−1|y1:t)

p(x0:t−1|y1:t−1)

∝ p(yt|xt−1) =

∫p(yt|xt)p(xt|xt−1)dxt

22

Approximations

Sampling the predictive distribution πt(xt|x0:t−1) = p(xt|xt−1, yt) :

– expansions of the p.d.f. or log(p.d.f.), Taylor

– mixture models : Gaussian∑i πiN (µi, σ

2i )

– Accept/Reject schemes

– Markov chain schemes : Metropolis-Hastings, Gibbs sampler

– dynamic stochastic simulation (Hybrid Monte Carlo)

– augmented sampling spaces :

→ slice samplers

→ auxiliary variables

23

Auxiliary variables

Pitt and Shephard 99 : approximating predictive p(xt|x(k)t−1, yt)

via augmented sampling space → p(xt, k|x(k)t−1, yt)

x_t

p(x_t/y_t)

x_t^(j)

x_t−1^(1)0

x_t−1

p(x_t−1/y_t−1)

1 x_t−1^(j) 3 x_t−1^(k)

x_t−1(N)0

x_t−1^(i)2

x_t^(i2)x_t^(i1) x_t^(k1) x_t^(k3)

x_t^(k2)

index of particle k (→ number of offsping(s) of particle x(k)t−1) ∼ .|yt

⇒ boost particles with high likelihood

24

Auxiliary variables

→ importance sampling for p(xt, k|x(k)t−1, yt) :

candidate distribution :

g(xt, k|xt−1, yt) ∝ p(yt|µ(k)t )p(xt|x(k)

t−1)

where µ(k)t = mean, mode, draw from xt|x(k)

t−1

x_t

p(x_t/x_t−1^(k))

mean maximummu_t^(k)= ,

25

Auxiliary variables

– sample (x(j)t , kj)1≤j≤R from g(xt, k|x(k)

t−1, yt) :

k ∼ g(k|xt−1, yt) ∝∫p(yt|µ(k)

t )p(xt|x(k)t−1)dxt = p(yt|µ(k)

t )

xt ∼ p(xt|x(k)t−1)

– reweighting (x(j)t , kj) with

wj ∝p(yt|x(j)

t )

p(yt|µ(kj)t )

– resample N paths from (x(kj)0:t−1, x

(j)t )1≤j≤R with second stage

weights wj

26


sampling/approximating predictive πt(xt|x0:t−1) may not be sufficient

for diffusing particles efficiently : e.g. discrepancy (πt)t>0 high :

⇒ consider a block of variables xt−L:t for a fixed lag L

27

Approaches using a block of variables

– discrete distributions : Meirovitch 85

– reweighting before resampling :

auxiliary variables Pitt and Shephard 99,

Wang et al. 02

⇒ discrete distribution → analytical form for proposal

xt ∼ πt+L(xt|x0:t−1) =

∫πt+L(xt:t+L|x0:t−1)dxt+1:t+L

Meirovitch 85 : growing a polymer, random walk in discrete space

→ complexity ]XL for lag L

28

Reweighting idea

29

Reweighting technique

30

Reweighting + resampling

21

010

0

0

0

1 1

31


→ auxiliary variables : Pitt and Shephard 99

proposal distribution :

p(xt, k|x(k)t−1, yt:t+L) ∝

∫p(xt:t+L, k|x(k)

t−1, yt:t+L)dxt+1:t+L

approximated with importance sampling :

g(xt, k|x(k)t−1, yt:t+L) = p(yt+L|µ(k)

t+L) . . . p(yt|µ(k)t )p(xt|x(k)

t−1)

→ sample (x(j)t , kj)1≤j≤R

kj ∼ g(k|yt:t+L) ∝ p(yt+L|µ(k)t+L) . . . p(yt|µ(k)

t )

x(j)t ∼ p(xt|x

(kj)t−1 )

32


auxiliary variables → resampling from (x(j)t )1≤j≤R :

→ propagate/sample x(j)t+1 → x

(j)t+L with prior transitions p(xt|xt−1)

→ use second stage weights : w(j)t ∝ w(j)

t−1 × iw

iw ∝ p(yt+L|x(j)t+L) . . . p(yt|x(j)

t )

p(yt+L|µ(kj)t+L) . . . p(yt|µ(kj)

t )

for resampling N paths (x(i)0:t)1≤i≤N

33


→ reweighting before resampling : Wang et al. 02

0

x_t^(i)

x_t

t

w_t^(i)

a_t^(i)

t t+L

x_t+L^(i_j)

a_t^(i_j)

propagate particles x(i)t → x

(ij)t+1:t+L for j = 1, . . . , R

compute weights a(ij)t , particle path x

(i)0:t reweighted with e.g.

a(i)t =

(∑Rj=1 a

(ij)t

)α, resampling from the set (x

(i)0:t, a

(i)t )i=1,...,N

34

Reweighting

→ need to sample/propagate xt from a/by block of variables :

πt+L(xt|x0:t−1) =

∫πt+L(xt:t+L|x0:t−1)dxt+1:t+L

⇒ sampling a block of variables

→ design a proposal/candidate distribution

35

Sampling recursively a block of variables

t−L t−L+1 tt−1

xt−L:t−1 → xt−L+1:t : imputing xt and re-imputing xt−L+1:t−1

36

Sampling a block of variables


t−L+1x’(

0

:0 t−1x( )

: t)

direct sampling :

x′t−L+1:t ∼ qt(x′t−L+1:t|x0:t−1)

37



t−L+1x’(

t−L+1x(

0

:0 t−1x(

:0 t−Lx(

: t)

)

)

)t−1:

proposal/candidate distribution for the block :

(x0:t−L, x′t−L+1:t) ∼

∫πt−1(x0:t−1)qt(x

′t−L+1:t|x0:t−1)dxt−L+1:t−1

38


⇒ Idea : consider extended block of variables

(x0:t−L, x′t−L+1:t)→ (x0:t−L, xt−L+1:t−1, x

′t−L+1:t) = (x0:t−1, x

′t−L+1:t)


t−L+1x’(

t−L+1x(

0

:0 t−Lx(

: t)

)

)t−1:

39


candidate distribution for extended block

(x0:t−L, xt−L+1:t−1, x′t−L+1:t) = (x0:t−1, x

′t−L+1:t) :

(x0:t−1, x′t−L+1:t) ∼ πt−1(x0:t−1)qt(x

′t−L+1:t|x0:t−1)

direct sampling :

(x0:t−L, x′t−L+1:t) ∼

∫πt−1(x0:t−1)qt(x

′t−L+1:t|x0:t−1)dxt−L+1:t−1

40


target distribution for the block (x0:t−L, x′t−L+1:t) :

πt(x0:t−L, x′t−L+1:t)

⇒ auxiliary target distribution for the extended block

(x0:t−1, x′t−L+1:t) = (x0:t−L, xt−L+1:t−1, x

′t−L+1:t) :

πt(x0:t−L, x′t−L+1:t)rt(xt−L+1:t−1|x0:t−L, x

′t−L+1:t)

with rt = any conditional distribution

⇒ proposal + target distributions → importance sampling

41

Sequential Importance Block Sampling/Resampling

Simulation scheme t− 1 → t (index (i) dropped) :

– Proposal sampling step

x′t−L+1:t ∼ qt(x′t−L+1:t|x0:t−1)

– Updating weights

wt ∝ wt−1 ×πt(x0:t−L, x′t−L+1:t)rt(xt−L+1:t−1|x0:t−L, x′t−L+1:t)

πt−1(x0:t−1)qt(x′t−L+1:t|x0:t−1)

– Resampling step

42

Sampling techniques for a block of variables

sampling the block xt−L+1:t ∼ qt(xt−L+1:t|x0:t−1) :

→ forward-backward recursion : e. g. Carter and Kohn 94



43

Sampling techniques for a block of variables

→ forward-backward recursion :



→ approximations : expansions, mixture models, MCMC, . . .

44


Optimal proposal qt(x′t−L+1:t|x0:t−1) distribution :

→ mimimizing variance of incremental weight w(i)t ∝ w(i)

t−1 × iw :

iw =πt(x0:t−L, x′t−L+1:t)rt(xt−L+1:t−1|x0:t−L, x′t−L+1:t)

πt−1(x0:t−1)qt(x′t−L+1:t|x0:t−1)

⇒ qt = L-step ahead predictive

πt(x′t−L+1:t|x0:t−L) = p(x′t−L+1:t|xt−L, yt−L+1:t)

For one variable : optimal qt = 1-step ahead predictive

πt(xt|x0:t−1) = p(xt|xt−1, yt)

45


→ block of variables ⇒ optimal proposal and target distribution

mimimizing variance of incremental weight w(i)t ∝ w(i)

t−1 × iw

iw =πt(x0:t−L, x′t−L+1:t)rt(xt−L+1:t−1|x0:t−L, x′t−L+1:t)

πt−1(x0:t−1)qt(x′t−L+1:t|x0:t−1)

→ optimal conditional distribution rt(xt−L+1:t−1|x0:t−L, x′t−L+1:t)

⇒ rt = (L− 1)-step ahead predictive

πt−1(xt−L+1:t−1|x0:t−L) = p(xt−L+1:t−1|xt−L, yt−L+1:t−1)

46


For optimal qt and rt, incremental weight w(i)t ∝ w(i)

t−1 × iw :

iw → πt(x0:t−L)

πt−1(x0:t−L)=

p(x0:t−L|y1:t)

p(x0:t−L|y1:t−1)∝ p(yt|xt−L, yt−L+1:t−1)

∝∫p(yt, xt−L+1:t|xt−L, yt−L+1:t−1)dxt−L+1:t

SISR for one variable with optimal proposal qt :

iw → πt(x0:t−1)

πt−1(x0:t−1)= p(yt|xt−1) =

∫p(yt|xt)p(xt|xt−1)dxt

Bootstrap filter : iw = p(yt|xt)

47

Approximations for block sampling

Sampling (sub-)optimal qt(x′t−L+1:t|x0:t−1) :

→ exact/approximated forward-backward recursions

→ approximations : expansions, mixture models, MCMC, . . .

For approximated optimal qt and rt, incremental weight :

πt(x0:t−L, x′t−L+1:t)qt−1(xt−L+1:t−1|x0:t−L)

πt−1(x0:t−1)qt(x′t−L+1:t|x0:t−L)

p(x0:t−L, x′t−L+1:t|y1:t)q(xt−L+1:t−1|xt−L, yt−L+1:t−1)

p(x0:t−1|y1:t−1)q(x′t−L+1:t|xt−L, yt−L+1:t)

48

Overview - Break ,– Introduction : state space models, Monte Carlo methods


– Strategies for sampling :

→ “optimal” candidate distribution

sampling with e.g. auxiliary variables

→ considering a block of variables : reweighting

⇒ sampling a block of variables :

definition of importance sampling for a block

performing sampling → “optimal” candidate distribution

– ⇒ Examples, applications :

→ simple, complex models

→ why the sampling strategy for particles can be crucial ?

49

Example

Linear and Gaussian state space model :

xt = αxt−1 + ut x0, ut ∼ N (0, 1)

yt = xt + vt vt ∼ N (0, σ2)

Sequential Monte Carlo methods :

– Bootstrap filter, proposal p(xt|xt−1)

– SISR with optimal proposal p(xt|xt−1, yt)

– SISR for blocks with optimal proposal p(xt−L+1:t|xt−L, yt−L+1:t)

computed by forward-backward exact recursions

⇒ estimates compared with Kalman filter results

⇒ approximation of target distribution p(xt|y1:t)

50

Estimation

0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3

4

time index

x(t)

x(t) and estimates

model (α, σ) = (0.9, 0.1) xt =∑Ni=1 w

(i)t x

(i)t N=100

51

Approximation of the target distribution

⇒ Effective Sample Size :

ESS =1

∑Ni=1[w

(i)t ]2

w(i) = 1N : ESS = N

\pi(x_t)

x_t

w(i) ≈ 0 ∀i except one : ESS = 1

x_t

\pi(x_t)

⇒ Resampling performed for ESS ≤ N2 ,

N10

52


Resampling for ESS ≤ N2 , N = 100

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

time index

ESS

Efficient Sample Size (ESS)

ESS for Bootstrap (→), SISR (→) and SISR for blocks

of 2 variables (→) with optimal proposals

53


Resampling for ESS ≤ N2 , N = 100, 100 time steps

algorithm ESS resampling steps CPU time

Bootstrap 11.19 99 0.84

optimal SISR 77.1 2 0.12

Block-SISR L = 2 99.1 1 0.23

54


Resampling for ESS ≤ N2 , N = 100, ∞ time steps

algorithm ESS resampling steps CPU time

Bootstrap 10 100% ∝0.84

optimal SISR 75 0.04% ∝0.12

Block-SISR L = 2 99 0% ∝0.23

55


Resampling for ESS ≤ N2 , various N

algorithm ESS resampling steps

Bootstrap 10%N 100%

optimal SISR 75%N 0.04%

Block-SISR L = 2 99%N 0%

computational complexity : resampling O(N) → CPU time

56

CPU time / number of particles N

Resampling for ESS ≤ N2 , 1,000 time steps

100 150 200 250 300 350 400 450 5000

5

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Number of particles N

CPU t

ime

Bootstrap (→), SISR (→) and SISR for blocks


57



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CPU t

ime

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SISR (→) and SISR for blocks


58

Sequential Monte Carlo methods

for this model :

– Same estimation results as Kalman filtering

– sampling ⇒ 6= approximation for the target distribution

– N ≤ 500 : computational complexity, CPU time

→ SISR with optimal proposal p(xt|xt−1, yt)

– N ≥ 500 : → block SISR with optimal proposal

p(xt−L+1:t|xt−L, yt−L+1:t)

59

Sampling strategies

xt = αxt−1 + ut x0, ut ∼ N (0, σ2u)

yt = xt + vt vt ∼ N (0, σ2v)

– σv=0.1 → observation yt very informative/prior σu=1.0

⇒ take into account yt for diffusing particles

p(xt|xt−1)→ p(xt|xt−1, yt) ⇒ ESS ↗

– α=0.9 → variables (xt)t correlated

⇒ sampling by block xt−L+1:t

block of observations yt−L+1:t more informative/single one yt

p(xt|xt−1, yt)→ p(xt−L+1:t|xt−L, yt−L+1:t) ⇒ ESS ↗

60



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ESS for Bootstrap (→), SISR with optimal proposals for 1 (→),

2 (→) and 10 variables (→)

σu=1.0, left/right : σv=0.1/1.0, top/bottom : α=0.9/0.5

61



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SISR with optimal proposals for 1 (→), 2 (→) and 10 variables (→)


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Overview- Break ,

– Introduction : state space models, Monte Carlo methods


– Strategies for sampling

– Applications : Linear and Gaussian model

⇒ sampling strategy :

→ approximation of the target distribution, CPU time

→ information in observation, dynamic of the state variable

→ nonlinear non-Gaussian models

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Example

Nonlinear state space model :

xt = α(xt−1 + βx3t−1) + ut x0, ut ∼ N (0, σ2

u)

yt = xt + vt vt ∼ N (0, σ2v)

Sequential Monte Carlo methods :

– Bootstrap filter, proposal p(xt|xt−1)

– SISR with optimal proposal p(xt|xt−1, yt) approximated by

KF/EKF

– SISR for blocks with optimal proposal p(xt−L+1:t|xt−L, yt−L+1:t)

approximated by forward-backward recursions with KF/EKF

Parameters values α=0.9, β=0.2, σu=0.1 and σv=0.05

⇒ approximation of target distribution p(xt|y1:t)

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Simulation results

algorithm MSE ESS RS CPU

Bootstrap 0.0021 36.8 70.3 % 0.68

SISR-KF 0.0019 64.7 19.3% 0.44

SISR-EKF 0.0019 65.8 19.2% 0.48

BSISR-KF 0.0018 72.3 0.9% 0.21

BSISR-EKF 0.0018 73.5 0.8% 0.24

N = 100 particles, 100 runs of particle filters for a single and for a

block of L = 2 variables (MSE from KF/EKF = 0.0034).

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0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

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100

time index

Effect

ive Sa

mple S

ize

Approximated ESS vs. time index for a realization of the Bootstrap

filter (dotted), the SISR with Kalman filter proposal for a single

variable (dashdotted) and for a block of L=2 variables (straight).

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Simulation results

block size L N=100 N=500 N=1000 RS

2 74 370 715 0.9%

3 96 493 985 0.9%

4 99 496 989 1%

5 98 494 988 1%

10 97 486 972 2.5%

Approximated ESS averaged over 100 runs of particle filters for

blocks of L variables, considering N particles.

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100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

Number of Particles

Comp

uting T

ime

CPU time vs. N for bootstrap filter (dotted), SISR with KF proposal

for a single variable (KF : dashed, EKF : dashdotted) and for a block

of L=2 variables (straight), 100 realizations.

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2000 4000 6000 8000 100000

2

4

6

8

10

12

14

Number of Particles

Comp

uting T

ime

Computational time vs. N for block sampling scheme with lags from

L=2 (bottom), 3, 4, 5, 10 (top), 100 realizations.

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Sequential Monte Carlo methods

for this model :

– Good approximations of the target distribution

– sampling ⇒ 6= approximation for the target distribution

– even for small N , block SISR with approximated optimal proposal

p(xt−L+1:t|xt−L, yt−L+1:t) is efficient for L=3, 4, 5

– → information in observation : σu=0.1, σv=0.05

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Conclusion

⇒ Importance of proposal/candidate distribution for

Sequential Monte Carlo simulation methods

design of proposal :

→ information in observation, dynamic of the state variable :

p(xt|xt−1)←→ p(xt|yt, xt−1)←→ p(xt|yt)

→ sampling a block/fixed lag of variables can be useful :

– for intermittent/informative observation, correlated variables

– applications ⇒ tracking, radar, navigation, positioning . . .

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References - SISR, Sequential Monte Carlo

– N. Gordon, D. Salmond, and A. F. M. Smith, “Novel approach to

nonlinear and non-Gaussian Bayesian state estimation,”

Proceedings IEE-F, vol. 140, pp. 107–113, 1993.

– G. Kitagawa, “Monte carlo filter and smoother for non-Gaussian

nonlinear state space models,” J. Comput. Graph. Statist., vol. 5,

pp. 1–25, 1996.

– A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte

Carlo methods in practice, Statistics for engineering and

information science. Springer, 2001.

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References - block/fixed lag approaches

– H. Meirovitch, “Scanning method as an unbiased simulation

technique and its application to the study of self-avoiding random

walks,” Phys. Rev. A, vol. 32, pp. 3699–3708, 1985.

– M. K. Pitt and N. Shephard, “Filtering via simulation : auxiliary

particle filter,” J. Am. Stat. Assoc., vol. 94, pp. 590–599, 1999.

– X. Wang, R. Chen, and D. Guo, “Delayed-pilot sampling for

mixture Kalman filter with application in fading channels,” IEEE

Trans. Sig. Proc., vol. 50, pp. 241–253, 2002.

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References - block/fixed lag sampling methods

– A. Doucet and S. Senecal, “Fixed-Lag Sequential Monte Carlo”,

accepted at EUSIPCO 2004.

– S. Senecal and A. Doucet, “An example of sequential Monte Carlo

block sampling method,” AIC2003 Science of Modeling,

pp. 418-419, 2003.

– C. K. Carter and R. Kohn, “On the Gibbs sampling for state space

models,” Biometrika, vol. 81, pp. 541–553, 1994.

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Documents

Sampling strategies for Sequential Monte Carlo (SMC) methods