tutorial on particle filters

NCAF January Meeting, Aston University, Birmingham.

Tutorial on Particle filters

Keith Copsey

Pattern and Information Processing Group

DERA Malvern

[email protected]


Outline

Introduction to particle filters

– Recursive Bayesian estimation Bayesian Importance sampling

– Sequential Importance sampling (SIS)

– Sampling Importance resampling (SIR) Improvements to SIR

– On-line Markov chain Monte Carlo Basic Particle Filter algorithm Examples Conclusions Demonstration


Particle Filters

Sequential Monte Carlo methods for on-line learning within a Bayesian framework.

Known as

– Particle filters

– Sequential sampling-importance resampling (SIR)

– Bootstrap filters

– Condensation trackers

– Interacting particle approximations

– Survival of the fittest


Recursive Bayesian estimation (I)

Recursive filter:

– System model:

– Measurement model:

– Information available:

)|( ),( 11 kkkkkk xxpxfx

)|( ),( kkkkkk xypxhy

),,( 1 kk yyD

)( 0xp


Seek:

– i = 0: filtering.

– i > 0: prediction.

– i<0: smoothing.

Prediction:

– since:

)|( kik Dxp

1111 )|,()|( kkkkkk dxDxxpDxp

11111 )|()|()|( kkkkkkk dxDxpxxpDxp

Recursive Bayesian estimation (II)


Update:

where:

– since:

kkkkkk dxDxypDyp )|,()|( 11

kkkkkkk dxDxpxypDyp )|()|()|( 11

)|(

)|()|()|(

1

1

kk

kkkkkk Dyp

DxpxypDxp

Recursive Bayesian estimation (III)


Classical approximations

Analytical methods:

– Extended Kalman filter,

– Gaussian sums… (Alspach et al. 1971)

• Perform poorly in numerous cases of interest

Numerical methods:

– point masses approximations,

– splines. (Bucy 1971, de Figueiro 1974…)

• Very complex to implement, not flexible.


Perfect Monte Carlo simulation (I)

Introduce the notation

Represent posterior distribution using a set of samples or particles.

Random samples are drawn from the posterior distribution.

),,( 0:0 kk xxx

N

ikxkk dx

NDxp i

k1:0:0 )(

1)|(

:0

ikx :0


Easy to approximate expectations of the form:

– by:

kkkkk dxDxpxgxgE :0:0:0:0 )|()())((

N

i

ikk xg

NxgE

1:0:0 )(

1))((

Perfect Monte Carlo simulation (II)


Random samples and the pdf (I)

Take p(x)=Gamma(4,1) Generate some random samples Plot histogram and basic approximation to pdf

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

200 samples


Random samples and the pdf (II)

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

500 samples 1000 samples


Random samples and the pdf (III)

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

200000 samples5000 samples


Unfortunately it is often not possible to sample directly from the posterior distribution.

Circumvent by drawing from a known easy to sample proposal distribution giving:

Bayesian Importance Sampling (I)

)|( :0 kk Dxq

kkkk

kkk

kkkkkk

kkkk

kkkkk

kkkk

dxDxqDp

xwxg

dxDxqDxqDp

xpxDpxg

dxDxqDxq

DxpxgxgE

:0:0:0

:0

:0:0:0

:0:0:0

:0:0:0

:0:0:0

)|()(

)()(

)|()|()(

)()|()(

)|()|(

)|()())((


Bayesian Importance Sampling (II)

where are unnormalised importance weights:

Now:

)( :0 kk xw

)|(

)()|()(

:0

:0:0:0

kk

kkkkk Dxq

xpxDpxw

kkkkk

kkk

kkkkk

kkkk

dxDxqxw

dxDxq

DxqxpxDpdxxDpDp

:0:0:0

:0:0

:0:0:0

:0:0

)|()(

)|(

)|()()|( ),()(


Bayesian Importance Sampling (III)

Giving:

so that:

where are normalised importance weights

– and are independent random samples from

N

i

ikk

ikN

i

ikk

N

i

ikk

ik

k xwxg

xwN

xwxgN

xgE1

:0:0

1:0

1:0:0

:0 )(~)(

)(1

)()(1

))((

)(~~:0ikk

ik xww

kkkkk

kkkkkkk dxDxqxw

dxDxqxwxgxgE

:0:0:0

:0:0:0:0:0 )|()(

)|())()(())((

ikx :0 )|( :0 kk Dxq


Sequential Importance Sampling (I)

Factorising the proposal distribution:

and remembering that the state evolution is modelled as a Markov process

obtain a recursive estimate of the importance weights:

k

jjjjkk DxxqxqDxq

11:00:0 ),|()()|(

),|(

)|()|(

:0

11

kkk

kkkkkk Dxxq

xxpxypww


Derivation of SIR weights

Since:

We have:

k

jjjk xxpxpxp

110:0 )|()()( and

k

jjjkk xypxDp

1:0 )|()|(

),|(

)|()|(),|(

1

)(

)(

)|(

)|()|(),|(

)()|(

1:0

11

1:01:0

:0

1:01

:01

11:01:0

:0:0

kkk

kkkkk

kkkk

k

kk

kkk

kkkkk

kkkk

Dxxq

xxpxypw

Dxxqxp

xp

xDp

xDpw

DxqDxxq

xpxDpw


Sequential Importance Sampling (II)

Choice of the proposal distribution:

Choose proposal function to minimise variance of (Doucet et al. 1999):

Although Common choice is the prior distribution:

),|( 1:0 kkk Dxxq

kw

),|(),|( 1:01:0 kkkkkk DxxpDxxq

)|(),|( 11:0 kkkkk xxpDxxq


Illustration of SIS:

Degeneracy problems:

– variance of importance ratios

increases stochastically over time (Kong et al. 1994; Doucet

et al. 1999).

Sequential Importance Sampling (III)

w

Time 19

w

Time 10

w

Time 1

)|(/)|( :0:0 kkkk DxqDxp


SIS - why variance increase is bad

Suppose we want to sample from the posterior

– choose a proposal density to be very close to the posterior

density

• Then

• and

So we expect the variance to be close to 0 to obtain reasonable estimates

– thus a variance increase has a harmful effect on accuracy

1)|(

)|(

:0

:0

kk

kkq Dxq

DxpE

01)|(

)|(

)|(

)|(var

2

:0

:0

:0

:0

kk

kkq

kk

kkq Dxq

DxpE

Dxq

Dxp


Sequential Importance Sampling (IV)

Illustration of degeneracy:

w

Time 19

w

Time 10

w

Time 1


Sampling-Importance Resampling

SIS suffers from degeneracy problems so we don’t want to do that!

Introduce a selection (resampling) step to eliminate samples with low importance ratios and multiply samples with high importance ratios.

Resampling maps the weighted random measure on to the equally weighted random measure

– by sampling uniformly with replacement from

with probabilities

Scheme generates children such that and satisfies:

)}(~,{ :0:0ikk

ik xwx} { 1-

:0 Nx j k},,1;{ :0 Nixi k

},,1;~{ Niwik

NNN

ii

1iN

iki wNNE ~)(

)~1(~)var( ik

iki wwNN


Improvements to SIR (I)

Variety of resampling schemes with varying performance in terms of the variance of the particles :

– Residual sampling (Liu & Chen, 1998).

– Systematic sampling (Carpenter et al., 1999).

– Mixture of SIS and SIR, only resample when necessary (Liu &

Chen, 1995; Doucet et al., 1999).

Degeneracy may still be a problem:

– During resampling a sample with high importance weight may

be duplicated many times.

– Samples may eventually collapse to a single point.

)var( iN


Improvements to SIR (II)

To alleviate numerical degeneracy problems, sample smoothing methods may be adopted.

– Roughening (Gordon et al., 1993).

• Adds an independent jitter to the resampled particles

– Prior boosting (Gordon et al., 1993).

• Increase the number of samples from the proposal distribution to M>N,

• but in the resampling stage only draw N particles.


Improvements to SIR (III)

Local Monte Carlo methods for alleviating degeneracy:

– Local linearisation - using an EKF (Doucet, 1999; Pitt &

Shephard, 1999) or UKF (Doucet et al, 2000) to estimate the

importance distribution.

– Rejection methods (Müller, 1991; Doucet, 1999; Pitt & Shephard,

1999).

– Auxiliary particle filters (Pitt & Shephard, 1999)

– Kernel smoothing (Gordon, 1994; Hürzeler & Künsch, 1998; Liu &

West, 2000; Musso et al., 2000).

– MCMC methods (Müller, 1992; Gordon & Whitby, 1995; Berzuini et

al., 1997; Gilks & Berzuini, 1998; Andrieu et al., 1999).


Improvements to SIR (IV)

Illustration of SIR with sample smoothing:

w

Time 19

w

Time 10

w

Time 1


MCMC move step Improve results by introducing MCMC steps with invariant

distribution .

– By applying a Markov transition kernel, the total variation of

the current distribution w.r.t. the invariant distribution can only

decrease.

Introduces possibility of variable dimension state space through the use of reversible jump MCMC (de Freitas et al., 1999; Gilks & Berzuini, 2001)

)|( :0 kk Dxp


Ingredients for SMC

Importance sampling function

– Gordon et al

– Optimal

– UKF pdf from UKF at Redistribution scheme

– Gordon et al SIR

– Liu & Chen Residual

– Carpenter et al Systematic

– Liu & Chen, Doucet et al Resample when necessary

Careful initialisation procedure (for efficiency)

)|( 1ikk xxp

),|( 1:0 kikk Dxxp

ikx 1


Basic Particle Filter - Schematic

Initialisation

Importancesampling step

Resamplingstep

0k

1 kk

)}(~,{ :0:0ikk

ik xwx

},{ 1:0

Nxi k

measurement

ky

Extract estimate, kx :0ˆ


Basic Particle Filter algorithm (I)

Initialisation

–

– For sample

– and set

In practice, to avoid having to take too many samples, for the first step we may want to ensure that we have a reasonable number of particles in the region of high likelihood

– perhaps use MCMC techniques

0k

Ni ,,1 )(~ 00 xpxi

1k


Basic Particle Filter algorithm (II)

Importance Sampling step

– For sample

– For evaluate the importance weights

– Normalise the importance weights,

N

j

jk

ik

ik www

1

/~

Ni ,,1 )|(~~1

ikk

ik xxpx

),(~1:0:0

ik

ik

ik xxx and set

Ni ,,1

)~|( ikk

ik xypw


Basic Particle Filter algorithm (III)

Resampling step

– Resample with replacement particles:

– from the set:

– according to the normalised importance weights,

Set

– proceed to the Importance Sampling step, as the next

measurement arrives.

N

),,1;( :0 Nixi k

),,1;~( :0 Nix i k ikw

~

1 kk


Example

On-line Data Fusion (Marrs, 2000).


Example - Sensor Deployment

Aim to reduce target sd below some threshold...

… and keep it there

… by placing the minimum number of sensors possible

Sensor positions chosen according to particle distribution.


Example - In-situ monitoring of growing semiconductor crystal composition

Si1-xGex

substrate


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Conclusions

On-line Bayesian learning a realistic proposition for many applications.

Appropriate for complex non-linear/non-Gaussian models

– don’t bother if KF based solution adequate. Representation of full posterior pdf leading to

– estimation of moments.

– estimation of HPD regions.

– multi-modality easy to deal with. Model order can be included in unknowns. Can mix SMC and KF based solutions


Tracking Demo

Illustrate a running particle filter

– compare with Kalman Filter

Running as we watch - not pre-recorded

Pre-defined scenarios, or design your own

– available to play with at coffee and lunch breaks.

Tracking Demo


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Contents:

– Introduction to SPR, Estimation, Density estimation, Linear

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networks, Nonlinear discriminant analysis - statistical methods,

Classification trees, Feature selction and extraction, Clustering,

Additional topics, Measures of dissimilarity, Parameter estimation,

Linear algebra, Data, Probability theory.

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