Introduction to particle filters: a trajectory tracking ... · Particle ltering algorithms Real...

Definitions and issuesParticle filtering algorithms

Real applications

Introduction to particle filters:a trajectory tracking example

Alexis Huet

23 May 2014

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Real applications

Outline

1 Definitions and issuesMarkov chainsHidden Markov modelsIssues

2 Particle filtering algorithmsSequential Importance Sampling algorithmSequential Importance Sampling Resampling algorithmComparison of algorithms

3 Real applications

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Real applications

Markov chainsHidden Markov modelsIssues

Outline



3 Real applications

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Real applications


Markov chains

We consider the following state space : (Rn,L(Rn)).

Definition

A sequence of random variables (Xk)k∈N taking values in Rn is aMarkov chain if for all k ≥ 1, x0, . . . xk−1 ∈ Rn and A ∈ L(Rn) :

P(Xk ∈ A|Xk−1 = xk−1, . . . ,X0 = x0) = P(Xk ∈ A|Xk−1 = xk−1).

Hypothesis

In the sequel, the Markov chain is homogeneous and admits afamily of density functions (p(.|x))x∈Rn such that :

P(Xk ∈ A|Xk−1 = xk−1) =

∫Ap(xk |xk−1)dxk .

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Real applications


Example

The state space is here R2. For all x ∈ R2, let B(x , 1) be the unitball centered in x .

Definition

For all x ∈ R2, the distribution Unif (B(x , 1)) is defined by itsdensity :

1

π1(. ∈ B(x , 1)).

For the initial condition, we take :

X0 Unif (B(0, 1)).

And for transition distributions :

Xk |(Xk−1 = xk−1) Unif (B(xk−1, 1)).

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Real applications


Example

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Hidden Markov models

Definition

(Xk ,Yk)k∈0:m−1 is a hidden Markov model if : (Xk)k∈0:m−1 is aMarkov chain, (Yk)k∈0:m−1 are independent conditionally to(Xk)k∈0:m−1 and for all k , Yk depends only on Xk .

Schematically, we have :

X0 −−−−→ X1 −−−−→ X2 −−−−→ . . . −−−−→ Xm−1y y y y yY0 Y1 Y2 . . . Ym−1

.

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Example

We choose (Yk) taking values in ]− π, π]. Conditionally toXk = xk , we define :

Yk = Arg(xk) + ε

where ε N (0, σ2).

Related density : yk 7−→ p(yk |xk).

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Real applications


Example (σ = 0.1)

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Example (σ = 0.1)

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Issues

Now, the hidden chain (xk) is unknown and we only have theobservations (yk).

Aim : reconstitute x0:m−1 := (x0, . . . , xm−1) conditionally to theobservations.

Specifically, to generate a sample according to the followingdensity :

p(x0:m−1|y0:m−1)dx0:m−1.

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Real applications

Sequential Importance Sampling algorithmSequential Importance Sampling Resampling algorithmComparison of algorithms

Outline



3 Real applications

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Real applications


Sequential Importance Sampling (SIS) algorithm

We try to simulate a sample from the density p(x0|y0)dx0.

p(x0|y0) =p(x0, y0)

p(y0)=

1

p(y0)p(y0|x0)p(x0).

We can generate a sample from p(x0)dx0.

We can compute x0 7→ p(y0|x0).

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Real applications


p(x0|y0)dx0 =1

p(y0)p(y0|x0)p(x0)dx0.

SIS algorithm (first step) :

Generate a sample of length N from p(x0)dx0 :

x(1)0 , . . . , x

(N)0 .

Compute for each particle j :

w(j)0 =

1

p(y0)p(y0|x (j)

0 ).

The sample (x(j)0 )j∈1:N which approximates p(x0|y0)dx0 is :

N∑j=1

w(j)0∑N

j ′=1 w(j ′)0

1x

(j)0

(dx0).

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Real applications


p(x0|y0)dx0 =1


SIS algorithm (first step) :

Generate a sample of length N from p(x0)dx0 :

x(1)0 , . . . , x

(N)0 .


w(j)0 = p(y0|x (j)

0 ).

The sample (x(j)0 )j∈1:N which approximates p(x0|y0)dx0 is :

N∑j=1

w(j)0∑N

j ′=1 w(j ′)0

1x

(j)0

(dx0).

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Real applications


Example (SIS) N = 20

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p(x0:i |y0:i )dx0:i =

1

p(y0:i )p(y0|x0) . . . p(yi |xi )p(x0)p(x1|x0) . . . p(xi |xi−1)dx0:i .

SIS algorithm (step i). For all particle j ∈ 1 : N :

Conditionally to x(j)i−1, generate a sample according to

p(xi |xi−1)dxi :

x(1)i , . . . , x

(N)i .


w(j)i = w

(j)i−1 × p(yi |x

(j)i ).

The sample (x(j)1:i )j∈1:N which approximates p(x1:i |yi )dx1:i is :

N∑j=1

w(j)i∑N

j ′=1 w(j ′)i

1x

(j)1:i

(dx1:i ).

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Real applications



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Real applications


Convergence and degeneracy problem (SIS)

Convergence :

for m the length of the chain fixed, central limit theoremwhen the number of particles N → +∞.

Degeneracy problem :

many particles have a relative weight close to 0%,

for N fixed, from a certain number of sites, only one particlehas an important relative weight.

→ Resampling of the particles.

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Real applications


p(x0|y0)dx0 =1


SISR algorithm (first step)

Generate a sample of length N according to p(x0)dx0 :

x(1)0 , . . . , x

(N)0 .


w(j)0 = p(y0|x (j)

0 ).

Generate (x(j)0 )j∈1:N from :

N∑j=1

w(j)0∑N

j ′=1 w(j ′)0

1x

(j)0

(dx0)

and let (w(j)0 )j∈1:N ≡ 1.

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Real applications


SISR algorithm (step i). For all particle j ∈ 1 : N :

Conditionally to x(j)i−1, generate a sample according to

p(xi |xi−1)dxi :

x(1)i , . . . , x

(N)i .


w(j)i = w

(j)i−1 × p(yi |x

(j)i ) = p(yi |x

(j)i ).

Generate (x(j)i )j∈1:N from :

N∑j=1

w(j)i∑N

j ′=1 w(j ′)i

1x

(j)i

(dxi )

and let (w(j)i )j∈1:N ≡ 1.

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Real applications


Example (SISR)

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Convergence (SISR)

Elimination of the weights degeneracy problem.

Convergence :

for m the length of the chain fixed, central limit theoremwhen the number of particles N → +∞.

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Real applications


Example

Comparison with m = 100 sites and N = 10000 particles.

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Real applications


Example (SIS)

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Real applications


Example (SISR)

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Real applications


Example (SIS)

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Real applications


Example (SISR)

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Real applications

Outline



3 Real applications

26/31 Alexis Huet


Real applications

Path tracking

Observations : noisy position.

Hidden states : real position.

Parameters : behavior of the moving body.

27/31 Alexis Huet


Real applications

Voice recognition system

Observations : a word is pronounced, cut every 15ms.

Hidden states : phonemes that led to this pronounced word.

Parameters : the set of all dictionary words.

28/31 Alexis Huet


Real applications

Phylogenetic analysis

Observations : DNA sequences of several species at the leafsof a tree graph.

Hidden states : all DNA sequences from the common ancestrysequence to the present time.

Parameters : mutation parameters, lengths of the treebranches.

29/31 Alexis Huet


Real applications


t = −1 ACGAGGTGA

ACAAGGTGA

ACAGGGTGA

ACAGGGCGA

ACAGGGCAA

t = 0 ACAGGGCAA

30/31 Alexis Huet


Real applications


i 1 2 3 4 5 6 7 8 9t = −1 A C G A G G T G A

A C A A G G T G A

A C A G G G T G A

A C A G G G C G A

A C A G G G C A A

t = 0 A C A G G G C A A

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Real applications


i 1 2 3 4 5 6 7 8 9t = −1 ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ?

t = 0 A C A G G G C A A

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Real applications

Olivier Cappe, Eric Moulines, and Tobias Ryden. Inference inhidden Markov models. Springer, 2005.

Neil Gordon, David Salmond, and Adrian Smith. Novelapproach to nonlinear/non-Gaussian Bayesian stateestimation. IEEE Proceedings F (Radar and SignalProcessing), 1993.

Lawrence Rabiner. A tutorial on hidden Markov models andselected applications in speech recognition. Proceedings of theIEEE, 1989.

31/31 Alexis Huet

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