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An introduction to particle filters
Andreas Svensson
Department of Information Technology
Uppsala University
June 10, 2014
June 10, 2014, 1 / 16 Andreas Svensson - An introduction to particle filters
OutlineMotivation and ideas
Algorithm
High-level
Matlab code
Practical aspects
Resampling
Computational complexity
Software
Terminology
Advanced topics
Convergence
Extensions
References
June 10, 2014, 2 / 16 Andreas Svensson - An introduction to particle filters
State Space Models
I Linear Gaussian state space model
xt+1 = Axt + But + wt
yt = Cxt + Dut + et
with wt and et Gaussian and E[wtwT
t]= Q, E
[eteT
t]= R.
I A more general state space model in different notation
xt+1 ∼ f (xt+1|xt , ut)
yt ∼ g(yt |xt , ut)
June 10, 2014, 3 / 16 Andreas Svensson - An introduction to particle filters
State Space Models
I Linear Gaussian state space model
xt+1 = Axt + But + wt
yt = Cxt + Dut + et
with wt and et Gaussian and E[wtwT
t]= Q, E
[eteT
t]= R.
I A more general state space model in different notation
xt+1 ∼ f (xt+1|xt , ut)
yt ∼ g(yt |xt , ut)
June 10, 2014, 3 / 16 Andreas Svensson - An introduction to particle filters
State Space Models
I Linear Gaussian state space model
xt+1 = Axt + But + wt
yt = Cxt + Dut + et
with wt and et Gaussian and E[wtwT
t]= Q, E
[eteT
t]= R.
I A more general state space model in different notation
xt+1 ∼ f (xt+1|xt , ut)
yt ∼ g(yt |xt , ut)
June 10, 2014, 3 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter! ''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, … ''Exact solution to the wrong problem''I Particle filter ''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter! ''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, … ''Exact solution to the wrong problem''I Particle filter ''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter!
''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, … ''Exact solution to the wrong problem''I Particle filter ''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter!
''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, … ''Exact solution to the wrong problem''I Particle filter ''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter!
''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, …
''Exact solution to the wrong problem''
I Particle filter
''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter! ''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, …
''Exact solution to the wrong problem''
I Particle filter
''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter! ''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, … ''Exact solution to the wrong problem''I Particle filter
''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Filtering - Find p(xt|y1:t)
Linear Gaussian problems:
I Kalman filter! ''Exact solution to the right problem''
Nonlinear problems:
I EKF, UKF, … ''Exact solution to the wrong problem''I Particle filter ''Approximate solution to the right problem''
June 10, 2014, 4 / 16 Andreas Svensson - An introduction to particle filters
Idea
Time
Sta
te
True state tracejctory
A linear system - Find p(xt|y1:t) (true xt shown)!
June 10, 2014, 5 / 16 Andreas Svensson - An introduction to particle filters
Idea
Time
Sta
te
True state tracejctory
Kalman filter mean
Kalman filter mean
June 10, 2014, 5 / 16 Andreas Svensson - An introduction to particle filters
Idea
Time
Sta
te
True state tracejctory
Kalman filter mean
Kalman filter covariance
Kalman filter mean and covariance
June 10, 2014, 5 / 16 Andreas Svensson - An introduction to particle filters
Idea
Time
Sta
te
True state tracejctory
Kalman filter mean
Kalman filter covariance
Kalman filter mean and covariance defines a Gaussian distribution at each t
June 10, 2014, 5 / 16 Andreas Svensson - An introduction to particle filters
Idea
Time
Sta
te
True state tracejctory
A numerical approximation can be used to describe the distribution - Particle Filter
June 10, 2014, 5 / 16 Andreas Svensson - An introduction to particle filters
Idea
Time
Sta
te
True state tracejctory
The Particle Filter can easily handle, e.g., non-gaussian multimodal hypotheses
June 10, 2014, 5 / 16 Andreas Svensson - An introduction to particle filters
Idea
Generate a lot of hypotheses about xt , keep the most likely ones andpropagate them further to xt+1. Keep the likely hypotheses about
xt+1, propagate them again to xt+2, etc.
June 10, 2014, 6 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
xit+1 from f (·|xi
t , ut)for i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
0 x(:,1) = random(i_dist,N,1);w(:,1) = ones(N,1)/N;
for t = 1:T
1 w(:,t) =pdf(m_dist,y(t)-g(x(:,t)));w(:,t) =w(:,t)/sum(w(:,t));
2 Resample x(:,t)3 x(:,t+1) = f(x(:,t),u(t))
+ random(t_dist,N,1);
end
June 10, 2014, 7 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
xit+1 from f (·|xi
t , ut)for i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
0 x(:,1) = random(i_dist,N,1);w(:,1) = ones(N,1)/N;
for t = 1:T
1 w(:,t) =pdf(m_dist,y(t)-g(x(:,t)));w(:,t) =w(:,t)/sum(w(:,t));
2 Resample x(:,t)3 x(:,t+1) = f(x(:,t),u(t))
+ random(t_dist,N,1);
end
June 10, 2014, 7 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
→0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
→1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
→2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
→3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
→1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
→2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
→3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
→1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
→2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
→3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
→1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
→2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
→3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
→1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
→2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
→3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Algorithm
0 Initialize xi1 ∼ p(x1) and
wi =1N for i = 1, . . . ,N
for t = 1 to T
1 Evaluate wit = g(yt|xi
t , ut)for i = 1, . . . ,N
2 Resample {xit}N
i=1 from
{xit ,wi
t}Ni=1
3 Propagate xit by sampling
from f (·|xit−1, ut−1) for
i = 1, . . . ,N
end
N Number of particles,
xit Particles,
wit Particle weights
5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
15
Time
Sta
te x
June 10, 2014, 8 / 16 Andreas Svensson - An introduction to particle filters
Resampling
I Represent the information contained in the N black dots (of
different sizes) …
with the N red dots (of equal sizes)
I Can be seen as sampling from a cathegorical distribution
I To avoid particle depletion (''a lot of 0-weights particles'')
I Some Matlab code:
v = rand(N,1); wc = cumsum(w(:,t)); [ ,ind1] = sort([v;wc]);ind=find(ind1<=N)-(0:N-1)'; x(:,t)=x(ind,t);w(:,t)=ones(N,1)./N;
June 10, 2014, 9 / 16 Andreas Svensson - An introduction to particle filters
Resampling
I Represent the information contained in the N black dots (of
different sizes) …
with the N red dots (of equal sizes)
state
I Can be seen as sampling from a cathegorical distribution
I To avoid particle depletion (''a lot of 0-weights particles'')
I Some Matlab code:
v = rand(N,1); wc = cumsum(w(:,t)); [ ,ind1] = sort([v;wc]);ind=find(ind1<=N)-(0:N-1)'; x(:,t)=x(ind,t);w(:,t)=ones(N,1)./N;
June 10, 2014, 9 / 16 Andreas Svensson - An introduction to particle filters
Resampling
I Represent the information contained in the N black dots (of
different sizes) … with the N red dots (of equal sizes)
state
I Can be seen as sampling from a cathegorical distribution
I To avoid particle depletion (''a lot of 0-weights particles'')
I Some Matlab code:
v = rand(N,1); wc = cumsum(w(:,t)); [ ,ind1] = sort([v;wc]);ind=find(ind1<=N)-(0:N-1)'; x(:,t)=x(ind,t);w(:,t)=ones(N,1)./N;
June 10, 2014, 9 / 16 Andreas Svensson - An introduction to particle filters
Resampling
I Represent the information contained in the N black dots (of
different sizes) … with the N red dots (of equal sizes)
state
I Can be seen as sampling from a cathegorical distribution
I To avoid particle depletion (''a lot of 0-weights particles'')
I Some Matlab code:
v = rand(N,1); wc = cumsum(w(:,t)); [ ,ind1] = sort([v;wc]);ind=find(ind1<=N)-(0:N-1)'; x(:,t)=x(ind,t);w(:,t)=ones(N,1)./N;
June 10, 2014, 9 / 16 Andreas Svensson - An introduction to particle filters
Resampling
I Represent the information contained in the N black dots (of
different sizes) … with the N red dots (of equal sizes)
state
I Can be seen as sampling from a cathegorical distribution
I To avoid particle depletion (''a lot of 0-weights particles'')
I Some Matlab code:
v = rand(N,1); wc = cumsum(w(:,t)); [ ,ind1] = sort([v;wc]);ind=find(ind1<=N)-(0:N-1)'; x(:,t)=x(ind,t);w(:,t)=ones(N,1)./N;
June 10, 2014, 9 / 16 Andreas Svensson - An introduction to particle filters
Resampling
I Represent the information contained in the N black dots (of
different sizes) … with the N red dots (of equal sizes)
state
I Can be seen as sampling from a cathegorical distribution
I To avoid particle depletion (''a lot of 0-weights particles'')
I Some Matlab code:
v = rand(N,1); wc = cumsum(w(:,t)); [ ,ind1] = sort([v;wc]);ind=find(ind1<=N)-(0:N-1)'; x(:,t)=x(ind,t);w(:,t)=ones(N,1)./N;
June 10, 2014, 9 / 16 Andreas Svensson - An introduction to particle filters
Resampling cont'd
I Crucial step in the development in the 90's for making particle
filters useful in practice
I Many different techniques exists with different properties and
effiency (the Matlab code shown was only one way of doing it)
I Computationally heavy. Not necessary to do in every iteration -
a ''depletion measure'' can be introduced, and the resampling
only performed when it reaches a certain threshold.
I It is sometimes preferred to use the logarithms of the weights
for numerical reasons.
June 10, 2014, 10 / 16 Andreas Svensson - An introduction to particle filters
Resampling cont'd
I Crucial step in the development in the 90's for making particle
filters useful in practice
I Many different techniques exists with different properties and
effiency (the Matlab code shown was only one way of doing it)
I Computationally heavy. Not necessary to do in every iteration -
a ''depletion measure'' can be introduced, and the resampling
only performed when it reaches a certain threshold.
I It is sometimes preferred to use the logarithms of the weights
for numerical reasons.
June 10, 2014, 10 / 16 Andreas Svensson - An introduction to particle filters
Resampling cont'd
I Crucial step in the development in the 90's for making particle
filters useful in practice
I Many different techniques exists with different properties and
effiency (the Matlab code shown was only one way of doing it)
I Computationally heavy. Not necessary to do in every iteration -
a ''depletion measure'' can be introduced, and the resampling
only performed when it reaches a certain threshold.
I It is sometimes preferred to use the logarithms of the weights
for numerical reasons.
June 10, 2014, 10 / 16 Andreas Svensson - An introduction to particle filters
Resampling cont'd
I Crucial step in the development in the 90's for making particle
filters useful in practice
I Many different techniques exists with different properties and
effiency (the Matlab code shown was only one way of doing it)
I Computationally heavy. Not necessary to do in every iteration -
a ''depletion measure'' can be introduced, and the resampling
only performed when it reaches a certain threshold.
I It is sometimes preferred to use the logarithms of the weights
for numerical reasons.
June 10, 2014, 10 / 16 Andreas Svensson - An introduction to particle filters
Resampling cont'd
I Crucial step in the development in the 90's for making particle
filters useful in practice
I Many different techniques exists with different properties and
effiency (the Matlab code shown was only one way of doing it)
I Computationally heavy. Not necessary to do in every iteration -
a ''depletion measure'' can be introduced, and the resampling
only performed when it reaches a certain threshold.
I It is sometimes preferred to use the logarithms of the weights
for numerical reasons.
June 10, 2014, 10 / 16 Andreas Svensson - An introduction to particle filters
Computational complexity
In theory O(NTn2x) (nx is the number of states)
(Kalman filter is approx. of order O(Tn3x))
Possible bottlenecks:
I Resampling stepI Likelihood evaluation (for the weight evaluation)I Sampling from f for the propagation (trick: use proposaldistributions!)
June 10, 2014, 11 / 16 Andreas Svensson - An introduction to particle filters
Computational complexity
In theory O(NTn2x) (nx is the number of states)
(Kalman filter is approx. of order O(Tn3x))
Possible bottlenecks:
I Resampling stepI Likelihood evaluation (for the weight evaluation)I Sampling from f for the propagation (trick: use proposaldistributions!)
June 10, 2014, 11 / 16 Andreas Svensson - An introduction to particle filters
Computational complexity
In theory O(NTn2x) (nx is the number of states)
(Kalman filter is approx. of order O(Tn3x))
Possible bottlenecks:
I Resampling stepI Likelihood evaluation (for the weight evaluation)I Sampling from f for the propagation (trick: use proposaldistributions!)
June 10, 2014, 11 / 16 Andreas Svensson - An introduction to particle filters
Computational complexity
In theory O(NTn2x) (nx is the number of states)
(Kalman filter is approx. of order O(Tn3x))
Possible bottlenecks:
I Resampling stepI Likelihood evaluation (for the weight evaluation)I Sampling from f for the propagation (trick: use proposaldistributions!)
June 10, 2014, 11 / 16 Andreas Svensson - An introduction to particle filters
Software
Lot of software packages exists. However, to my knowledge, no
package that ''everybody'' uses. (In our research, we write our own
code.)
A few relevant existing packages:
I Python: pyParticleEst
I Matlab: PFToolbox, PFLib
I C++: Particle++
Disclaimer: I have not used any of these packages myself
June 10, 2014, 12 / 16 Andreas Svensson - An introduction to particle filters
Software
Lot of software packages exists. However, to my knowledge, no
package that ''everybody'' uses. (In our research, we write our own
code.)
A few relevant existing packages:
I Python: pyParticleEst
I Matlab: PFToolbox, PFLib
I C++: Particle++
Disclaimer: I have not used any of these packages myself
June 10, 2014, 12 / 16 Andreas Svensson - An introduction to particle filters
Software
Lot of software packages exists. However, to my knowledge, no
package that ''everybody'' uses. (In our research, we write our own
code.)
A few relevant existing packages:
I Python: pyParticleEst
I Matlab: PFToolbox, PFLib
I C++: Particle++
Disclaimer: I have not used any of these packages myself
June 10, 2014, 12 / 16 Andreas Svensson - An introduction to particle filters
Terminology
Bootstrap particle filter ≈ ''Standard'' particle filter
Sequential Monte Carlo (SMC) ≈ Particle filter
June 10, 2014, 13 / 16 Andreas Svensson - An introduction to particle filters
Terminology
Bootstrap particle filter ≈ ''Standard'' particle filter
Sequential Monte Carlo (SMC) ≈ Particle filter
June 10, 2014, 13 / 16 Andreas Svensson - An introduction to particle filters
Terminology
Bootstrap particle filter ≈ ''Standard'' particle filter
Sequential Monte Carlo (SMC) ≈ Particle filter
June 10, 2014, 13 / 16 Andreas Svensson - An introduction to particle filters
Convergence
The particle filter is ''exact for N = ∞''
Consider a function g(xt) of interest. How well can it be
approximated as g(xt) when xt is estimated in a particle filter?
E [g(xt)− E [g(xt)]]2 ≤ pt‖g(xt)‖sup
NIf the system ''forgets exponentially fast'' (e.g. linear systems),
and some additional weak assumptions, pt = p < ∞, i.e.,
E [g(xt)− E [g(xt)]]2 ≤ C
N
June 10, 2014, 14 / 16 Andreas Svensson - An introduction to particle filters
Convergence
The particle filter is ''exact for N = ∞''
Consider a function g(xt) of interest. How well can it be
approximated as g(xt) when xt is estimated in a particle filter?
E [g(xt)− E [g(xt)]]2 ≤ pt‖g(xt)‖sup
NIf the system ''forgets exponentially fast'' (e.g. linear systems),
and some additional weak assumptions, pt = p < ∞, i.e.,
E [g(xt)− E [g(xt)]]2 ≤ C
N
June 10, 2014, 14 / 16 Andreas Svensson - An introduction to particle filters
Convergence
The particle filter is ''exact for N = ∞''
Consider a function g(xt) of interest. How well can it be
approximated as g(xt) when xt is estimated in a particle filter?
E [g(xt)− E [g(xt)]]2 ≤ pt‖g(xt)‖sup
NIf the system ''forgets exponentially fast'' (e.g. linear systems),
and some additional weak assumptions, pt = p < ∞, i.e.,
E [g(xt)− E [g(xt)]]2 ≤ C
N
June 10, 2014, 14 / 16 Andreas Svensson - An introduction to particle filters
Convergence
The particle filter is ''exact for N = ∞''
Consider a function g(xt) of interest. How well can it be
approximated as g(xt) when xt is estimated in a particle filter?
E [g(xt)− E [g(xt)]]2 ≤ pt‖g(xt)‖sup
N
If the system ''forgets exponentially fast'' (e.g. linear systems),
and some additional weak assumptions, pt = p < ∞, i.e.,
E [g(xt)− E [g(xt)]]2 ≤ C
N
June 10, 2014, 14 / 16 Andreas Svensson - An introduction to particle filters
Convergence
The particle filter is ''exact for N = ∞''
Consider a function g(xt) of interest. How well can it be
approximated as g(xt) when xt is estimated in a particle filter?
E [g(xt)− E [g(xt)]]2 ≤ pt‖g(xt)‖sup
NIf the system ''forgets exponentially fast'' (e.g. linear systems),
and some additional weak assumptions, pt = p < ∞, i.e.,
E [g(xt)− E [g(xt)]]2 ≤ C
N
June 10, 2014, 14 / 16 Andreas Svensson - An introduction to particle filters
Extensions
I Smoothing: Finding p(xt |y1:T ) (marginal smoothing) orp(x1:t |y1:T ) (joint smoothing) instead of p(xt |y1:t) (filtering).Offline (y1:T has to be available). Increased computational load.
I If f (xt+1|xt , ut) is not suitable to sample from, proposaldistributions can be used. In fact, an optimal proposal (w.r.t.
reduced variance) exists.
I Rao-Blackwellization for mixed linear/nonlinear models.
I System identification: PMCMC, SMC2.
June 10, 2014, 15 / 16 Andreas Svensson - An introduction to particle filters
Extensions
I Smoothing: Finding p(xt |y1:T ) (marginal smoothing) orp(x1:t |y1:T ) (joint smoothing) instead of p(xt |y1:t) (filtering).Offline (y1:T has to be available). Increased computational load.
I If f (xt+1|xt , ut) is not suitable to sample from, proposaldistributions can be used. In fact, an optimal proposal (w.r.t.
reduced variance) exists.
I Rao-Blackwellization for mixed linear/nonlinear models.
I System identification: PMCMC, SMC2.
June 10, 2014, 15 / 16 Andreas Svensson - An introduction to particle filters
Extensions
I Smoothing: Finding p(xt |y1:T ) (marginal smoothing) orp(x1:t |y1:T ) (joint smoothing) instead of p(xt |y1:t) (filtering).Offline (y1:T has to be available). Increased computational load.
I If f (xt+1|xt , ut) is not suitable to sample from, proposaldistributions can be used. In fact, an optimal proposal (w.r.t.
reduced variance) exists.
I Rao-Blackwellization for mixed linear/nonlinear models.
I System identification: PMCMC, SMC2.
June 10, 2014, 15 / 16 Andreas Svensson - An introduction to particle filters
Extensions
I Smoothing: Finding p(xt |y1:T ) (marginal smoothing) orp(x1:t |y1:T ) (joint smoothing) instead of p(xt |y1:t) (filtering).Offline (y1:T has to be available). Increased computational load.
I If f (xt+1|xt , ut) is not suitable to sample from, proposaldistributions can be used. In fact, an optimal proposal (w.r.t.
reduced variance) exists.
I Rao-Blackwellization for mixed linear/nonlinear models.
I System identification: PMCMC, SMC2.
June 10, 2014, 15 / 16 Andreas Svensson - An introduction to particle filters
Extensions
I Smoothing: Finding p(xt |y1:T ) (marginal smoothing) orp(x1:t |y1:T ) (joint smoothing) instead of p(xt |y1:t) (filtering).Offline (y1:T has to be available). Increased computational load.
I If f (xt+1|xt , ut) is not suitable to sample from, proposaldistributions can be used. In fact, an optimal proposal (w.r.t.
reduced variance) exists.
I Rao-Blackwellization for mixed linear/nonlinear models.
I System identification: PMCMC, SMC2.
June 10, 2014, 15 / 16 Andreas Svensson - An introduction to particle filters
References
I Gustafsson, F. (2010). Particle filter theory and practice with
positioning applications. Aerospace and Electronic Systems Magazine, IEEE,
25(7), 53-82.
I Schön, T.B., & Lindsten, F. Learning of dynamical systems - Particle
Filters and Markov chain methods. Draft available.
I Doucet, A., & Johansen, A. M. (2009). A tutorial on particle filtering and
smoothing: Fifteen years later. Handbook of Nonlinear Filtering, 12,
656-704.
Homework: Implement your own Particle Filter for any (simple) problem of your
choice!
June 10, 2014, 16 / 16 Andreas Svensson - An introduction to particle filters