Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
Monte Carlo Filtering of
Piecewise-Deterministic Processes
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Adam M. Johansen
with Nick Whiteley and Simon J. Godsill
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
University of Warwick, Dept. of Statistics
a.m.johansen@warwick.ac.uk
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
EPSRC Workshop on MCMC and Related Methods
1
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
Outline
A. M. Johansen
Background
I
Background
I
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Methodology
I
I
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Piecewise Deterministic Processes (PDPs)
Sequential Monte Carlo (Samplers)
“Particle Filtering” of PDPs
“Auxiliary Particle Filtering” of PDPs
Examples
I
I
Object Tracking
Rate Estimation: Shot Noise Cox Processes
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
2
Motivation:
Observing a Manoeuvering Object
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
I
For t ∈ R+
0 , consider object with position st , velocity vt
and acceleration at
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Summarise state by ζt = (st , vt , at )
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From initial condition ζ0 , state evolves until random
time τ1 , at which acceleration jumps to a new random
value, yielding ζτ1
Examples
I
From ζτ1 , evolution until τ2 , state becomes ζτ2 , etc.
Summary
I
Observation times, (tn )n∈N , at each tn a noisy
measurement of the object’s position is made
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
3
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
Example Trajectory
30
A. M. Johansen
Background
25
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
20
Methodology
sy/km
15
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
10
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
5
Summary
0
−5
−10
0
5
10
15
20
25
30
35
sx/km
4
An Abstract Formulation
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
I
Pair Markov chain (τj , θj )j∈N , τj ∈ R+ , θj ∈ Θ
Background
I
p(d(τj , θj )|τj−1 , θj−1 ) = q(dθj |θj−1 , τj , τj−1 )f (dτj |τj−1 ),
P
Count the jumps νt := j I[τj ≤t]
Deterministic evolution function F : R+
0 × Θ → Θ, s.t.
∀θ ∈ Θ,
F (0, θ) = θ
Methodology
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Signal process (ζt )t∈R+ ,
0
A. M. Johansen
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
ζt := F (t − τνt , θνt )
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This describes a piecewise deterministic process.
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It’s partially observed via observations (Yn )n∈N , with
likelihood function gn (yn |ζtn )
5
Sequential Importance Resampling
For a sequence, πn , of distributions over spaces E n :
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Sample X1i ∼ q1 .
Calculate W1i ∝ π1 (X1i )/q1 (X1i ).
Iterate: n ← n + 1
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i
i
i
Resample (Wn−1
, Xn−1
) to obtain (1/N, Xn,1:n−1
).
For i = 1 : N
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I
A. M. Johansen
Background
Set n = 1
For i = 1 : N
I
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
i
Xn,n
Sample
Calculate
Wni ∝
∼
i
qn (·|Xn,1:n−1
).
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
i
πn (Xn,1:n
)
i
i
i |X i
πn−1
(Xn,1:n−1
)qn (Xn,n
n,1:n−1 )
In a filtering context, can use πn (x1:n ) = p(x1:n |y1:n ).
6
The Variable Rate Particle Filter
The VRPF iteration:
I
A. M. Johansen
i
i
i
Resample (Wn−1
, (kn−1
, τ1:k
i
n−1
I
i
, θ1:k
i
)).
n−1
For i = 1 : N
I
i
While τn,
< tn
kn
I
I
I
i
i
Sample τn,k
∼ f (·|τn,k
).
n +1
n
kni ← kni + 1.
i
i
Sample θn,k
).
i ∼ h(·|τn , θn,ki
n
I
n−1
Calculate
Wni ∝
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
i
g(yn |xin )f (θkn−1
)
i |θk i
+1:kn
n−1
Summary
i
i
h(θki i +1:ki |τn,1:k
, θn,k
)
i
i
n
n−1
n−1
n−1
Something like the bootstrap filter.
7
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
SMC Samplers
SMC can be used to sample from any sequence of
distributions (Del Moral et al., 2006).
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Given target distributions, ηn , on En . . . ,
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construct a synthetic sequence ηen on spaces
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n
N
A. M. Johansen
Background
Ep
p=1
by introducing Markov kernels, Lp from Ep+1 to Ep :
ηen (x1:n ) = ηn (xn )
n−1
Y
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Lp (xp+1 , xp ) ,
p=1
I
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
These distributions
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have the target distributions as time marginals,
have the correct structure to employ SMC techniques.
8
Inference and PDPs
I
Consider the spaces (En )n∈N ,
En =
∞
]
A. M. Johansen
Background
{k} × Tn,k × Θk+1
k=0
Tn,k ={τ1:k : 0 < τ1 < τ2 < ... < τk ≤ tn }.
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Define kn := νtn and Xn = (ζ0 , kn , τ1:kn , θ1:kn ) ∈ En
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Sequence of posterior distributions (ηn )n∈N
ηn (xn ) ∝q(ζ0 )
×
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
kn
Y
f (τj |τj−1 )q(θj |θj−1 , τj , τj−1 )
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
j=1
n
Y
gp (yp |ζtp )S(τkn , tn )
p=1
9
Particle Filtering of PDPs
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
Background
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Can use SMC sampler to target ηn for n = 1, . . . .
This allows more sophisticated proposals:
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Use of observations.
Explicit dimension-changing proposals.
No sampling into the future.
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Often interested only in p(ζtn |y1:n ).
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Consequently, need only keep track of (kn , τkn , θkn ).
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Many techniques from particle filtering can be used.
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
10
Algorithmic Details
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
I
Use a mixture of moves:
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Augment with MCMC moves if required:
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Birth moves.
Refinement moves.
Can include RJMCMC moves.
Use observations to obtain good proposals.
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
I
Recover the VRPF under certain circumstances.
11
Auxiliary Particle Filtering of PDPs
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
Background
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Can construct an analogue of discrete time APF:
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Use n + 1st observation at time n.
Prevents resampling eliminating promising particles.
Pre-weighting is then corrected for by later weights.
Interpretation:
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Approximate p̂(·|y1:n+1 ) at time n.
Correct with weights p(·|y1:n )/p̂(·|y1:n+1 ).
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
12
Algorithmic Details
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
I
An auxiliary PDP particle filter comprises:
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SMC sampler for auxiliary sequence(µn )n≥1 ,
fn ) between
Sequence of importance weights, (W
(µn )n≥1 and (ηn )n≥1 .
We use auxiliary distributions of the following form:
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
µn (xn ) ∝ Vn (τn,kn , θn,kn , yn+1 )ηn (xn )
with Vn : R+ × Ξ → (0, ∞).
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Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
Vn approximates predictive likelihood.
13
Back to the motivating example. . .
A. M. Johansen
I
Observation interval, ∆ = 5 s.
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Changepoint intervals are a priori Gamma(a, b) with:
a =10
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
b =5∆/a
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
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Used a mixture of birth and adjustment moves (in
proportion 2:1).
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Observations are noisy range/bearing pairs.
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Model linearisation used to approximate optimal kernels.
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Stratified resampling when ESS < N/2.
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Can also consider a Rao-Blackwellised version.
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
14
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
The final sample set
30
A. M. Johansen
25
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
20
15
sy/km
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
10
5
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
0
Summary
−5
−10
−15
0
5
10
15
20
25
30
35
sx/km
15
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
N
50
100
250
500
1000
2500
5000
Godsill et al. 2007
RMSE / km CPU / s
42.62
0.24
33.49
0.49
22.89
1.23
17.26
2.42
12.68
5.00
6.18
13.20
3.52
28.79
Whiteley et al. 2007
RMSE / km CPU / s
0.88
1.32
0.66
2.62
0.54
6.56
0.51
12.98
0.50
26.07
0.49
67.32
0.48
142.84
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
Root mean square filtering error and
CPU time — over 200 runs.
16
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
4
A. M. Johansen
log RMSE
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
2
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
0
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
−2
0
20
40
60
CPU/s
80
100
120
17
Estimation of Insurance Event Rates
The model is specified by the following distributions:
f (τj |τj−1 ) =λτ exp(−λτ (τj − τj−1 )) × I[τj−1 ,∞) (τj ),
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
Background
q(ζ0 ) = exp(−λθ ζ0 ) × I[0,∞) (ζ0 ),
q(θj |θj−1 , τj , τj−1 ) =λθ exp(−λθ (θj − ζτ−j )) × I[ζτ− ,∞) (θj ),
j
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
where
ζτ−j =θj−1 exp(−κ(τj − τj−1 )), and
F (t − τ, θ) =θ exp(−κ(t − τ )).
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
The likelihood function is given by:
Z
g(yn |ζ(tn−1,tn ] ) = exp −
!
tn
ζs ds
tn−1
Y
ζyn,i
i
where yn,i is the time of the ith event observed in (tn−1 , tn ].
18
Implementation Details
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
Background
I
Proposal step: simple birth proposal.
Kn (xn−1 , dxn ) = δkn−1 +1 (kn )δτn−1,1:kn−1 (dτn,1:kn −1 )
× δθn−1,1:kn−1 (dθn,1:kn −1 )hn (dτn,kn )
× πn (dθn,kn |xn \ θn,kn ),
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Systematic resampling applied when ESS < 0.4N .
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RJMCMC move applied.
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
19
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
Unique Particles
500
400
A. M. Johansen
300
n
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
200
100
Methodology
0
1000
1200
1400
1600
1800
2000
t
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
500
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
400
Summary
n
300
200
100
0
1000
1200
1400
1600
1800
2000
t
20
SMCTC: A C++ Template Class
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
I
Implementing SMC algorithms in C/C++ isn’t hard.
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
I
Software for implementing general SMC algorithms.
I
C++ element largely confined to the library.
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Available (under a GPL-3 license from)
www2.warwick.ac.uk/fac/sci/statistics/staff/
academic/johansen/smctc/
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
or type “smctc” into google.
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Particle filters can also be implemented easily.
21
Conclusions
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
I
SMC sampler-type approaches allow efficient online
algorithms to be developed.
I
The proposed approach can dramatically outperform
existing algorithms.
I
This approach allows a class of piecewise deterministic
models to be used when online inference is required.
I
The VRPF can be interpreted within this framework,
allowing existing convergence results to be applied to it.
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
I
Possible extensions:
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Parameter estimation via PMCMC.
22
References
P. Del Moral, A. Doucet, and A. Jasra.
Sequential Monte Carlo samplers.
Journal of the Royal Statistical Society B, 63(3):411–436, 2006.
S. J. Godsill, J. Vermaak, K.-F. Ng, and J.-F. Li.
Models and algorithms for tracking of manoeuvring objects using
variable rate particle filters.
Proceedings of IEEE, 95(5):925–952, 2007.
A. M. Johansen.
SMCTC: Sequential Monte Carlo in C++.
Research Report 08:16, University of Bristol, Department of Mathematics
– Statistics Group, University Walk, Bristol, BS8 1TW, UK, July 2008.
N. Whiteley, A. M. Johansen, and S. Godsill.
Efficient Monte Carlo filtering for discretely observed jumping processes.
In Proceedings of IEEE Statistical Signal Processing Workshop, pages
89–93, Madison, WI, USA, August 26th–29th 2007. IEEE.
Monte Carlo
Filtering of
PiecewiseDeterministic
Processes
A. M. Johansen
Background
Piecewise
Deterministic
Processes
Sequential Monte
Carlo (Samplers)
Methodology
Particle Filtering of
PDPs
Auxiliary Particle
Filtering of PDPs
Examples
Object Tracking
Rate Estimation: Shot
Noise Cox Processes
Summary
N. Whiteley, A. M. Johansen, and S. Godsill.
Monte Carlo filtering of piecewise-deterministic processes.
In revision., 2008.
23