tutorial on particle filterstutorial on particle filters 4 pf suppl 1... · 2015-11-24 · tutorial...

Tutorial on Particle filtersTutorial on Particle filters

Keith CopseyPattern and Information

Processing GroupProcessing GroupDERA Malvern

[email protected]

NCAF January Meeting, Aston University, Birmingham.

Outline

Introduction to particle filters

– Recursive Bayesian estimationBayesian Importance sampling

– Sequential Importance sampling (SIS)Sequential Importance sampling (SIS)

– Sampling Importance resampling (SIR)Improvements to SIR

– On-line Markov chain Monte CarloBasic Particle Filter algorithmExamplesExamplesConclusionsDemonstration


Particle Filters

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

Known as

– Particle filters

– Sequential sampling-importance resampling (SIR)

– Bootstrap filters

– Condensation trackersCo de sat o t ac e s

– Interacting particle approximations

Survival of the fittest– Survival of the fittest


Recursive Bayesian estimation (I)y ( )

Recursive filter:

– System model:

)|( ),( 11 −− ↔= kkkkkk xxpxfx ω

– Measurement model:

)|( ),( kkkkkk xypxhy ↔= υ

– Information available:

)( 1 kk yyD = ),,( 1 kk yyD …=

)( 0xp


Recursive Bayesian estimation (II)

Seek: )|( kik Dxp +

y ( )

– i = 0: filtering.

– i > 0: prediction.

– i<0: smoothing.

P di tiPrediction:

∫ −−−−− = 11111 )|()|()|( kkkkkkk dxDxpxxpDxp– since:

∫ −−−− = 1111 )|,()|( kkkkkk dxDxxpDxp ∫


Recursive Bayesian estimation (III)

Update:

y ( )

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

1

1

−

−=kk

kkkkkk Dyp

DxpxypDxp

where:

∫ dDD )|()|()|(

– since:

∫ −− = kkkkkkk dxDxpxypDyp )|()|()|( 11

– since:

∫ −− = kkkkkk dxDxypDyp )|,()|( 11


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

),,( 0:0 kk xxx …=

particles.

∑=N

kxkk dxN

Dxp ik

:0:0 )(1)|(0

δ

Random samples are drawn from the posterior distribution

=ixN k1 :0

ikx :0

distribution.


Perfect Monte Carlo simulation (II)

Easy to approximate expectations of the form:

( )

∫= kkkkk dxDxpxgxgE :0:0:0:0 )|()())((

– by:

∑N

i1 ∑=

=i

ikk xg

NxgE

1:0:0 )(1))((


Random samples and the pdf (I)( )

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

0.412

0 25

0.3

0.35

8

10

0.15

0.2

0.25

4

6

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0 20 40 60 80 100 120 140 160 180 2000

2

0 20 40 60 80 100 120 140 160 180 200

200 samples


Random samples and the pdf (II)( )

0 45 0.35

0 3

0.35

0.4

0.45

0.25

0.3

0.35

0.15

0.2

0.25

0.3

0 1

0.15

0.2

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0 2 4 6 8 10 12 14 16 18 20

500 samples 1000 samples


Random samples and the pdf (III)( )

0.25 0.25

0.15

0.2

0 15

0.2

0.1

0.15

0.1

0.15

0 5 10 15 20 250

0.05

0 5 10 15 20 250

0.05

200000 samples5000 samples


Bayesian Importance Sampling (I)Unfortunately it is often not possible to sample directly from the posterior distribution.

y g ( )

Circumvent by drawing from a known easy to sample proposal distribution giving:)|( :0 kk Dxq

∫= kkkkk

kkkk dxDxq

DxqDxpxgxgE :0:0

:0

:0:0:0 )|(

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

∫= kkkkkk

kkkk dxDxq

DxqDpxpxDpxg :0:0

:0

:0:0:0

)(

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

∫= kkkk

kkk dxDxq

Dpxwxg :0:0

:0:0 )|(

)()()(


Bayesian Importance Sampling (II)y g ( )

where are unnormalised importance weights:)( :0 kk xw

)|()()|()(

0

:0:0:0

kk

kkkkk Dxq

xpxDpxw =

Now:

)|( :0 kk Dxq

∫ dDD )()(

∫∫

=

=

kkkkkk

kkkk

dxD

DxqxpxDpdxxDpDp

:0:0:0:0

:0:0

)|(

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

∫

∫= kkkkk

kkk

dxDxqxwDxq

:0:0:0

:0:0

)|()()|(


Bayesian Importance Sampling (III)y g ( )

Giving:

∫ dxDxqxwxg )|())()((

∫∫=

kkkkk

kkkkkkk dxDxqxw

dxDxqxwxgxgE

:0:0:0

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

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

so that:

∑N

ii xwxg )()(1

∑∑

∑

=

= ==N

i

ikk

ikN

i

ikkk

k xwxgxw

xwxgN

xgE1

:0:01

:0:0

:0 )(~)()(1

)()())((

where are normalised importance weights

∑=

i

ikk xw

N1

1:0 )(

)(~~0i

kkik xww =where are normalised importance weights

– and are independent random samples from

)( :0 kkk xww =i

kx :0 )|( :0 kk Dxq


Sequential Importance Sampling (I)g ( )

Factorising the proposal distribution:k∏=

−=k

jjjjkk DxxqxqDxq

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

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

obtain a recursive estimate of the importance weights:

)|()|( 1kkkk xxpxyp),|(

)|()|(:0

11

kkk

kkkkkk Dxxq

xxpxypww −−=


Derivation of SIR weightsg

Since:k k∏=

−=k

jjjk xxpxpxp

110:0 )|()()( and ∏

==

k

jjjkk xypxDp

1:0 )|()|(

We have:

)()|( :0:0 kkk xpxDpw

1)()|()|(),|(

)()|(

:0:011:01:0

:0:0

kkkkkkkk

kkkk

xpxDpw

DxqDxxqppw

−−−

=

=

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

11

1:01:01:011

kkkkk

kkkkkkk

xxpxypw

DxxqxpxDpw

−−−−−

−

=

=

),|( 1:01

kkkk Dxxq

w−

−=


Sequential Importance Sampling (II)g ( )

Choice of the proposal distribution:

Choose proposal function to minimise variance of

),|( 1:0 kkk Dxxq −

kw(Doucet et al. 1999):

),|(),|( 1:01:0 kkkkkk DxxpDxxq −− =

Although Common choice is the prior distribution:

1:01:0 kkkkkk

)|(),|( 11:0 −− = kkkkk xxpDxxq


Sequential Importance Sampling (III)

Illustration of SIS:

g ( )

w

Time 1

Time 10

w

Time 19

ww

Degeneracy problems:

– variance of importance ratios i t h ti ll ti (K t l 1994 D t

)|(/)|( :0:0 kkkk DxqDxpincreases stochastically over time (Kong et al. 1994; Doucet et al. 1999).


SIS - why variance increase is bady

Suppose we want to sample from the posterior

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

Th• Then

d

1)|()|(

:0

:0 =⎟⎟⎠

⎞⎜⎜⎝

⎛

kk

kkq Dxq

DxpE

• and01

)|()|(

)|()|(var

2

:0

:0

:0

:0 =⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

kk

kkq

kk

kkq Dxq

DxpEDxqDxp

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

)|()|( :0:0 ⎠⎝ ⎠⎝⎠⎝ kkkk qq

– thus a variance increase has a harmful effect on accuracy


Sequential Importance Sampling (IV)g ( )

Illustration of degeneracy: Time 1

w

Time 10

w

Time 19

w


Sampling-Importance Resamplingg g

SIS suffers from degeneracy problems so we don’t want to d th t!do that!Introduce a selection (resampling) step to eliminate samples with low importance ratios and multiply samples

ith hi h i t tiwith high importance ratios. Resampling maps the weighted random measure on to the equally weighted random measure

)}(~,{ :0:0i

kki

k xwx}{ 1-

:0 Nx jk

– by sampling uniformly with replacement from with probabilities

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

k …=},,1;~{ Niwi

k …=

Scheme generates children such that and satisfies:

NNN

ii =∑

=1iN

i 1iki wNNE ~)( =

)~1(~)var( ik

iki wwNN −=


kki

Improvements to SIR (I)( )

Variety of resampling schemes with varying performance in terms of the variance of the particles :)var( iNterms of the variance of the particles :

– Residual sampling (Liu & Chen, 1998).

Systematic sampling (Carpenter et al 1999)

)var( iN

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

– Mixture of SIS and SIR, only resample when necessary (Liu & Chen 1995; Doucet et al 1999)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.


Improvements to SIR (II)( )

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

– Roughening (Gordon et al., 1993).

• Adds an independent jitter to the resampled• Adds an independent jitter to the resampled particles

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

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

• but in the resampling stage only draw N particles.p


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 distributionimportance distribution.

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

– Auxiliary particle filters (Pitt & Shephard, 1999)

– Kernel smoothing (Gordon 1994; Hürzeler & Künsch 1998; Liu &Kernel smoothing (Gordon, 1994; Hürzeler & Künsch, 1998; Liu & West, 2000; Musso et al., 2000).

– MCMC methods (Müller, 1992; Gordon & Whitby, 1995; Berzuini et ( , ; y, ;al., 1997; Gilks & Berzuini, 1998; Andrieu et al., 1999).


Improvements to SIR (IV)( )

Illustration of SIR with sample smoothing:

w

Time 1

Time 10

Time 19

ww


MCMC move stepImprove results by introducing MCMC steps with invariant distribution .)|( :0 kk Dxp

– By applying a Markov transition kernel, the total variation of the current distribution w.r.t. the invariant distribution can only decreasedecrease.

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


Ingredients for SMCg

Importance sampling functioni– Gordon et al →

– Optimal →

)|( 1ikk xxp −

),|( 1:0 ki

kk Dxxp −– UKF → pdf from UKF at

Redistribution scheme

ikx 1−

– Gordon et al → SIR

– Liu & Chen → Residual

– Carpenter et al → Systematic

– Liu & Chen, Doucet et al → Resample when necessary

Careful initialisation procedure (for efficiency)


Basic Particle Filter - SchematicInitialisation

0k t0=k

1+→ kk

measurement

ky

I tR li

},{ 1:0

−Nxik

Importancesampling step

Resamplingstep

)}(~,{ :0:0i

kki

k xwx

Extract estimate, kx :0ˆ


Basic Particle Filter algorithm (I)g ( )

Initialisation

–

– For sample

0=kNi ,,1…= )(~ 00 xpxi

– and set

I ti t id h i t t k t l f

1=k

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 likelihoodlikelihood

– perhaps use MCMC techniques


Basic Particle Filter algorithm (II)g ( )

Importance Sampling step

– For sample Ni ,,1…= )|(~~1

ikk

ik xxpx −

),(~1:0:0

ik

ik

ik xxx −=and set

– For evaluate the importance weightsNi ,,1…=

)~|( ikk

ik xypw =

– Normalise the importance weights, ∑=N

jk

ik

ik www /~

)|( kkk yp

∑=j

kkk1


Basic Particle Filter algorithm (III)g ( )

Resampling step

– Resample with replacement particles:N),,1;( :0 Nixi

k …=– from the set:

),,1;~( :0 Nix ik …=

– according to the normalised importance weights, ikw~

Set

– proceed to the Importance Sampling step, as the next

1+→ kk

measurement arrives.


Example

On-line Data Fusion (Marrs, 2000).


Example - Sensor Deploymenty

Aim to reduce target sd below some thresholdthreshold...

… and keep it there

… by placing the minimum number of sensors possible

Sensor positions chosen according to particle distribution.


E l I it it i f i i d tExample - 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 applicationsapplications.Appropriate for complex non-linear/non-Gaussian models

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

– estimation of moments.

– estimation of HPD regions.

– multi-modality easy to deal with.u t oda ty easy to dea tModel order can be included in unknowns.Can mix SMC and KF based solutions


Tracking Demog

Illustrate a running particle filter

– compare with Kalman Filter

Running as we watch - not pre-recordedRunning 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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– Introduction to SPR, Estimation, Density estimation, Linear discriminant analysis, Nonlinear discriminant analysis - neural 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.


tutorial on particle filterstutorial on particle filters 4 pf suppl 1... · 2015-11-24 · tutorial...

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