The Chopthin Algorithm for Resampling

Din-Houn Lau

Joint work with Axel GandyImperial College London

Sir David Cox Celebration Event7th December 2016

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 1

Particle Filter: Example: Hidden Markov Model

Observations

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π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Initial Guesses

Observations

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(i)1 )

f (x2|x1)

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π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Update Weights using Likelihood

Observations

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(i)1 )

f (x2|x1)

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π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Resample

Observations

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(i)1 )

f (x2|x1)

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π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Resample

Observations

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(i)1 )

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π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Transition

Observations

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(i)1 )

f (x2|x1)

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1

π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Update Weight

Observations

Hidden

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23π(x1|y1) ≈

likelihoodg(y1|x

(i)1 )

f (x2|x1)

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1

1

π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Resample

Observations

Hidden

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23π(x1|y1) ≈

likelihoodg(y1|x

(i)1 )

f (x2|x1)

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x

x

x

x x2

1

2

22

2

2

1

1

π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Particle Filter: Resample

Observations

Hidden

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1121

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23π(x1|y1) ≈

likelihoodg(y1|x

(i)1 )

f (x2|x1)

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x

x

x

x x2

1

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22

2

2

1

1

π(x2|y1, y2) ≈

particle system{x (i)

t ,w (i)t

}N

i=1

resample step : necessary→ avoids weight degeneracy.Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 2

Available Resampling Methods

I Multinomial: sample with replacement using weights.

I Residual (Liu and Chen, 1998)

I Branching (Bain and Crisan, 2009, p. 278)

I Systematic (Carpenter et al., 1999)

I Stratified (Kitagawa, 1996)

General consensus — possible to outperform multinomialresampling; the other resamplers are comparable in terms oftheir performance in particle filters. (Douc and Cappe, 2005;Hol et al., 2006)

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 3

Resampling: Common ConstraintsLet N i = number of offspring from particle i and G a σ-field generatedby the particle system {x (i),w (i)}N

i=1 with∑N

i=1 w (i) = 1.

1.∑N

i=1 N i = N ←− returns N particles

2. E(N i |G) = Nw (i) ←− unbiasedness

3. Equal weights after resampling = 1/N.

Objective: Construct resampler that makes

N∑i=1

N i

Nf (x (i))

small for any function of interest f .

Chopthin resampling: returns unequal weights.

Estimator of interest afterresamplingE

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

=N∑

i=1

w (i)f (x (i))Var

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 4

Resampling: Common ConstraintsLet N i = number of offspring from particle i and G a σ-field generatedby the particle system {x (i),w (i)}N

i=1 with∑N

i=1 w (i) = 1.

1.∑N

i=1 N i = N ←− returns N particles

2. E(N i |G) = Nw (i) ←− unbiasedness

3. Equal weights after resampling = 1/N.

Objective: Construct resampler that makes

N∑i=1

N i

Nf (x (i))

small for any function of interest f .

Chopthin resampling: returns unequal weights.

Estimator of interest afterresampling

E

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

=N∑

i=1

w (i)f (x (i))Var

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 4

Resampling: Common ConstraintsLet N i = number of offspring from particle i and G a σ-field generatedby the particle system {x (i),w (i)}N

i=1 with∑N

i=1 w (i) = 1.

1.∑N

i=1 N i = N ←− returns N particles

2. E(N i |G) = Nw (i) ←− unbiasedness

3. Equal weights after resampling = 1/N.

Objective: Construct resampler that makes

N∑i=1

N i

Nf (x (i))

small for any function of interest f .

Chopthin resampling: returns unequal weights.

Estimator of interest afterresampling

E

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

=N∑

i=1

w (i)f (x (i))

Var

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 4

Resampling: Common ConstraintsLet N i = number of offspring from particle i and G a σ-field generatedby the particle system {x (i),w (i)}N

i=1 with∑N

i=1 w (i) = 1.

1.∑N

i=1 N i = N ←− returns N particles

2. E(N i |G) = Nw (i) ←− unbiasedness

3. Equal weights after resampling = 1/N.

Objective: Construct resampler that makes

N∑i=1

N i

Nf (x (i))

small for any function of interest f .

Chopthin resampling: returns unequal weights.

Estimator of interest afterresamplingE

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

=N∑

i=1

w (i)f (x (i))

Var

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 4

Resampling: Common ConstraintsLet N i = number of offspring from particle i and G a σ-field generatedby the particle system {x (i),w (i)}N

i=1 with∑N

i=1 w (i) = 1.

1.∑N

i=1 N i = N ←− returns N particles

2. E(N i |G) = Nw (i) ←− unbiasedness

3. Equal weights after resampling = 1/N.

Objective: Construct resampler that makes

N∑i=1

N i

Nf (x (i))

small for any function of interest f .

Chopthin resampling: returns unequal weights.

Estimator of interest afterresamplingE

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

=N∑

i=1

w (i)f (x (i))

Var

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 4

Resampling: Common ConstraintsLet N i = number of offspring from particle i and G a σ-field generatedby the particle system {x (i),w (i)}N

i=1 with∑N

i=1 w (i) = 1.

1.∑N

i=1 N i = N ←− returns N particles

2. E(N i |G) = Nw (i) ←− unbiasedness

3. Equal weights after resampling = 1/N.

Objective: Construct resampler that makes

N∑i=1

N i

Nf (x (i))

small for any function of interest f .

Chopthin resampling: returns unequal weights.

Estimator of interest afterresamplingE

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

=N∑

i=1

w (i)f (x (i))

Var

(N∑

i=1

N i

Nf (x (i))

∣∣∣∣∣G)

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 4

Motivation for Chopthin

I Resampling introduces variance. But is necessary. . .

I Particle filter: resampling not performed at every iteration e.g.if ESS ≤ β: Resample

if ESS > β: No Resampling

where ESS =(∑n

i=1 w (i))2∑ni=1(w (i))2

, typically β = 0.5N

I Can some “in-between” resampling be constructed?

I For particle filters to work one essentially needs to ensure thatweights

• do not get too large and • do not get too small.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 5

Aims for the Chopthin Algorithm

1.∑N

i=1 N i = N.

2. Unbiasedness: E[N iw (i)

new

∣∣∣G] = w (i)old ∀i

3. Equal weights after resampling = 1/N.

? Bound ratio of weights:w (i)

new

w (j)new

≤ η ∀i , j

? Conserve total weight:N∑

i=1

w (i)old =

N∑i=1

w (i)new

Also: Efficient Implementation

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 6

Aims for the Chopthin Algorithm

1.∑N

i=1 N i = N.

2. Unbiasedness: E[N iw (i)

new

∣∣∣G] = w (i)old ∀i

3. Equal weights after resampling = 1/N.

? Bound ratio of weights:w (i)

new

w (j)new

≤ η ∀i , j

? Conserve total weight:N∑

i=1

w (i)old =

N∑i=1

w (i)new

Also: Efficient Implementation

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 6

Aims for the Chopthin Algorithm

1.∑N

i=1 N i = N.

2. Unbiasedness: E[N iw (i)

new

∣∣∣G] = w (i)old ∀i

3. Equal weights after resampling = 1/N.

? Bound ratio of weights:w (i)

new

w (j)new

≤ η ∀i , j

? Conserve total weight:N∑

i=1

w (i)old =

N∑i=1

w (i)new

Also: Efficient Implementation

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 6

Chopthin

I New weights: w (i)new ∈ [a, ηa] =⇒ ratio bounded X.

(a > 0 to be determined)I Weights below a get thinned:

0 or 1 offspring with weight a.I Weights above a get chopped:

replicated, weighted divided among offspring.

I To ensure that N particles are returned (in expectation), needto find a such that

N∑i=1

E(N i |G) :=N∑

i=1

hηa(w

(i)) = N

I Need to choose hηa such that all the aims are satisfied.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 7

Chopthin: Key Steps

1. Find a: that solves∑N

i=1 hηa(w (i)) = N.

2. Thinning step: Offspring→ Systematic resampling on smallweights (on hη

a(wi) s.t. wi < a).

3. Chopping step: Offspring→ Systematic resampling on largeweights (on fractional parts of hη

a(wi) s.t. wi ≥ a).

Systematic resampling – ensures exactly N particles arereturned =⇒

∑Ni=1 N i = N X

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 8

Our Choice of hηa

hηa(w) =

w/a if w < a1 if a ≤ w < ηa/22w/ηa if w ≥ ηa/2

.0.

01.

02.

0

weight, w

h aη (w)

0 a ηa 2 ηa

THIN CHOPNothing

Choice:∑N

i=1 hηa(w (i)) = N can be solved for a efficiently (expected

linear effort).

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 9

Our Choice of hηa

hηa(w) =

w/a if w < a1 if a ≤ w < ηa/22w/ηa if w ≥ ηa/2

.0.

01.

02.

0

weight, w

h aη (w)

0 a ηa 2 ηa

THIN CHOPNothing

Choice:∑N

i=1 hηa(w (i)) = N can be solved for a efficiently (expected

linear effort).

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 9

Our Choice of hηa

hηa(w) =

w/a if w < a1 if a ≤ w < ηa/22w/ηa if w ≥ ηa/2

.0.

01.

02.

0

weight, w

h aη (w)

0 a ηa 2 ηa

THIN CHOPNothing

Choice:∑N

i=1 hηa(w (i)) = N can be solved for a efficiently (expected

linear effort).Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 9

ExampleI Model: For t = 1, . . . ,T = 1000

Xt ∼ N(Xt−1,1) , with X0 ∼ N(0,1), (hidden)

Yt ∼ N(Xt , σ2Y ) (observed)

I Targets: Xt |Y1, . . .Yt .

Algorithm: Bootstrap Particle Filter

Sample x (1), . . . , x (N) ∼ N(0, 1); Let w (1) = · · · = w (N) = 1/Nfor t = 1, . . . ,T do

Sample x (i) ∼ N(x (i), 1), i = 1, . . . ,N

w (i) ← w (i)φ(yt − x (i)/σY ), i = 1, . . . ,N

(potentially) resample (x (i),w (i))

I Kalman filter gives exact posterior distribution in this case —but not in more general filtering problems.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 10

Simulation ResultsI 1000 repetitions – MSE of posterior mean divided by MSE

using systematic resampling.N 103 103 103 103 104 104 104 104

β η σY 1/3 1 3 9 1/3 1 3 9chopthin N 3+

√8 1.00 0.89 0.86 0.87 0.95 0.90 0.89 0.90

chopthin 0.5N 3+√

8 0.97 0.98 0.96 0.94 0.96 0.98 0.96 0.94chopthin N 4 0.98 0.90 0.89 0.92 0.97 0.91 0.90 0.94chopthin N 10 0.98 0.91 0.87 0.86 0.94 0.93 0.86 0.85multinomial 0.5N - 0.99 1.04 1.15 1.24 0.98 1.04 1.15 1.22branching 0.5N - 1.00 0.99 1.00 1.01 0.95 1.01 1.00 1.00residual 0.5N - 0.99 1.00 1.00 1.01 1.00 1.00 1.00 1.01stratified 0.5N - 0.99 1.01 1.02 1.04 0.96 1.02 1.01 1.01residual-stratified 0.5N - 0.97 1.00 1.01 1.01 0.96 1.03 1.00 1.05systematic 0.5N - 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

I consistently lower MSE for the posterior mean.I For η = 3 +

√8 and η = 10, ESS is above theoretical lower bound of

0.5N and 0.33N, respectively.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 11

What about the Variance

Var

(N iw (i)

new

∣∣∣∣∣G)

var. of total weight of offspring from particle i

0.000 0.005 0.010 0.015 0.0200.00

000

0.00

010

0.00

020

weight

Var

ianc

e of

tota

l wei

ght o

f offs

prin

g

MultinomialResidualSystematic

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 12

What about the Variance

Var

(N iw (i)

new

∣∣∣∣∣G)

var. of total weight of offspring from particle i

0.000 0.005 0.010 0.015 0.0200.00

000

0.00

010

0.00

020

weight

Var

ianc

e of

tota

l wei

ght o

f offs

prin

g

MultinomialResidualSystematicChopthin

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 12

Summary

I Chopthin is a new resampler that returns unequal weightswhose ratio is bounded.

I Chopthin can be performed at every resampling step in aparticle filter.

I Simulations: chopthin consistently outperforms otherresamplers.

I Chopthin can be implemented efficiently. Implementations forC++, R (on CRAN), Python and for Matlab are available.

I Next: Central Limit type theorem (as N →∞).

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 13

ReferencesBain, A. and D. Crisan (2009). Fundamentals of Stochastic Filtering.

Springer.

Carpenter, J., P. Clifford, and P. Fearnhead (1999). Improved particle filter fornonlinear problems. IEE Proceedings - Radar, Sonar and Navigation 146,2–7.

Douc, R. and O. Cappe (2005). Comparison of resampling schemes forparticle filtering. In Proceedings of the 4th International Symposium onImage and Signal Processing and Analysis, pp. 64–69. IEEE.

Gandy, A. and F. D.-H. Lau (2016). The chopthin algorithm for resampling.IEEE Transactions on Signal Processing 64(16), 4273–4281.

Hol, J. D., T. B. Schon, and F. Gustafsson (2006). On resampling algorithmsfor particle filters. In Nonlinear Statistical Signal Processing Workshop,2006 IEEE, pp. 79–82.

Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussiannonlinear state space models. Journal of Computational and GraphicalStatistics 5(1), 1–25.

Liu, J. S. and R. Chen (1998). Sequential Monte Carlo methods for dynamicsystems. Journal of the American statistical association 93(443),1032–1044.

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 14

Chopthin Variance Explained: IllustrationBEFORE CHOPTHIN

AFTER CHOPTHIN

0

0

a ηa/2 ηa

weight

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 15

Chopthin Variance Explained: IllustrationBEFORE CHOPTHIN

AFTER CHOPTHIN

0

0

a ηa/2 ηa

weight

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 15

Chopthin Variance Explained: IllustrationBEFORE CHOPTHIN

AFTER CHOPTHIN

0

0

a ηa/2 ηa

weight

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 15

Chopthin Variance Explained: IllustrationBEFORE CHOPTHIN

AFTER CHOPTHIN

0

0

a ηa/2 ηa

weight

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 15

Chopthin Variance Explained: IllustrationBEFORE CHOPTHIN

AFTER CHOPTHIN

0

0

a ηa/2 ηa

weight

Imperial College London Din-Houn Lau The Chopthin Algorithm for Resampling 15