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Filter-WorkshopBucuresti 2003

Particle Filtersan overview

Matthias Muhlich

Institut fur Angewandte PhysikJ.W.Goethe-Universitat Frankfurt

[email protected]

mailto:[email protected]

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Particle Filtersa tutorial

Matthias Muhlich

Institut fur Angewandte PhysikJ.W.Goethe-Universitat Frankfurt

[email protected]

mailto:[email protected]

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1 Introduction

An increasing number of researchers is using a family oftechniques and algorithms called

� condensation algorithms

� bootstrap filtering

� particle filters

� interacting particle approximations

� sequential Monte Carlo methods

� SIS, SIR, ASIR, RPF, . . .

Time scale: last 10 years [e.g. Isard & Blake 1996; Kita-gawa 1996; Gordon, Salmond & Smith 1993]

The question of this talk is: What is behind all that?

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General Classification of Filter Strategies

Gaussian models:

� Kalman filter

� extended Kalman filter

� linear-update filter / linear regression filter /statistical linearization filter

– unscented filter

– central difference filter

– divided difference filter

� assumed density filter / moment matching

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Mixture of Gaussian models:

� assumed density filter / pseudo-Bayes

� Gaussian-sum filter

Nonparametric models:

� particle filter class

� histogram filter

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Some Basic Remarks

� various applications: computer vision (i.e. tracking),control theory, econometrics (stock markets, mone-tary flow, interest rates), . . .

� we deal with discrete time systems only

� no out-of-sequence measurements

� we are mainly interested in estimating the state attime k from measurements up to time k′ = k (op-posite: smoothing (k′ > k) and prediction (k′ < k);furthermore k′ need not be fixed. . . )

� no restrictions to linear processes or Gaussian noise!

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Overview of this Talk

� The Dynamic System Model

� Bayesian Filter Approach

� Optimal and Suboptimal Solutions

� The Particle Filter

� Experiments and Summary

– states of a system and state transition equation

– measurement equation

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– estimation of the state

– probabilistic modelling

– Bayesian filter

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– filtered pdf can be written down easily, but it is notalways tractable (→ ugly integrals . . . )

– conditions under which optimal solutions exist: Kalmanfilter and grid-based filter

– what can be done in other cases: suboptimal approaches

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– standard particle filter

– various improved versions

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– some experimental data and conclusion

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2 Dynamic System

A dynamic system can be modelled with two equations:

State Transition or Evolution Equation

xk = fk(xk−1,uk−1,vk−1)

f (·, ·, ·): evolution function (possible non-linear)xk,xk−1 ∈ IRnx: current and previous statevk−1 ∈ IRnv: state noise (usually not Gaussian)uk−1 ∈ IRnu: known input

Note: state only depends on previous state, i.e. first orderMarkov process

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Measurement Equation

zk = hk(xk,uk,nk)

h(·, ·, ·): measurement function (possible non-linear)zk ∈ IRnz: measurementxk ∈ IRnx: statenk ∈ IRnn: measurement noise (usually not Gaussian)uk ∈ IRnu: known input

(dimensionality of state, measurement, input, state noise,and measurement noise can all be different!)

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measurements(observed)

states(cannot be observed and have to be estimated)

timek-1 k k+1

xk-1

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z k-1

timek-1 k k+1

xk-1

measurement process

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z k-1

timek-1 k k+1

xk-1 xk

state transition

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z k-1 z k

timek-1 k k+1

xk-1 xk

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z k-1 z k z k+1

timek-1 k k+1

xk-1 xk xk+1

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z k-1 z k z k+1

timek-1 k k+1

xk-1 xk xk+1

p(x |x ) (first order Markov process)

k k-1

p(z |x ) (measurementsonly depend onstate

k k

Assumptions:

The observations are conditionally independent given thestate: p(zk|xk).

Hidden Markov Model (HMM):p(x0) given and p(xk|xk−1) defines state transition prob-ability for k≥ 1.

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3 Bayesian Filters

Estimating the Posterior

Bayesian approach: We attempt to construct the posteriorpdf of the state given all measurements.

⇒ can be termed a complete solution to the estimationproblem because all available information is used; fromthe pdf, an optimal estimate can theoretically be foundfor any criterion.

in detail: We seek estimates of xk based on all availablemeasurements up to time k (abbreviated as z1:k) by con-structing the posterior p(xk|z1:k).

Assumption: initial state pdf (prior) p(x0) is given

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The Use of Knowing the Posterior

Let fk : IR(k+1)×nx→ IR be any arbitrary (integrable) func-tion that can depend

� on all components of the state x

� on the whole trajectory in state-space

Examples: This function can be an estimator for the cur-rent state or for future observations.

Then we can compute its expectation using

E [ fk(x0:k)] =∫

f (x0:k)p(x0:k|z1:k)dx0:k

MMSE estimate of state: x = E [xk]. Other estimates thatcan be computed: median, modes, confidence intervals,kurtosis, . . .

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Recursive Filters

recursive filters (i.e. sequential update of previous esti-mate) ↔ batch processing (computation with all data inone step)

not only faster: allow on-line processing of data (lowerstorage costs, rapid adaption to changing signals charac-teristics)

essentially consist of two steps:

prediction step: p(xk−1|z1:k−1)→ p(xk|z1:k−1)(usually deforms / translates / spreads state pdf dueto noise)

update step: p(xk|z1:k−1),zk→ p(xk|z1:k)(combines likelihood of current measurement withpredicted state; usually concentrates state pdf)

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General Prediction-Update Framework

Assume that pdf p(xk−1|z1:k−1) is available at time k−1.

Prediction step: (using Chapman-Kolmogoroff equation)

p(xk|z1:k−1) =∫

p(xk|xk−1)p(xk−1|z1:k−1)dxk−1 (1)

This is the prior of the state xk at time k without knowl-edge of the measurement zk, i.e. the probability given onlyprevious measurements.

Update step: (compute posterior pdf from predicted priorpdf and new measurement)

p(xk|z1:k) =p(zk|xk)p(xk|z1:k−1)

p(zk|z1:k−1)(2)

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Let us prove formula (2) (just in order to train calculationswith joint and conditional probabilities. . . )

p(xk|z1:k)

=p(z1:k|xk)p(xk)

p(z1:k)

(Bayes rule)

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p(xk|z1:k)

=p(z1:k|xk)p(xk)

p(z1:k)

=p(zk,z1:k−1|xk)p(xk)

p(zk,z1:k−1)

(separate p(z1:k) into p(zk,z1:k−1))

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p(xk|z1:k)

=p(z1:k|xk)p(xk)

p(z1:k)


p(zk,z1:k−1)


p(zk|z1:k−1)p(z1:(k−1))

=p(zk|z1:k−1,xk)p(xk|z1:k−1)p(z1:k−1)p(xk)

p(zk|z1:k−1)p(z1:k−1)p(xk)

(Bayes rule)

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p(xk|z1:k)

=p(z1:k|xk)p(xk)

p(z1:k)


p(zk,z1:k−1)


p(zk|z1:k−1)p(z1:(k−1))

=p(zk|z1:k−1,xk)p(xk|z1:k−1)p(z1:k−1)p(xk)

p(zk|z1:k−1)p(z1:k−1)p(xk)

=p(zk|xk)p(xk|z1:k−1)

p(zk|z1:k−1)

(independence of observations; cancelling out terms)

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The Structure of the Update Equation

p(xk|z1:k) =p(zk|xk) · p(xk|z1:k−1)

p(zk|z1:k−1)

posterior =likelihood ·prior

evidence

prior: given by prediction equation

likelihood: given by observation model

evidence: the normalizing constant in the denominator

p(zk|z1:k−1) =∫

p(zk|xk)p(xk|z1:k−1)dxk

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This theoretically allows an optimal Bayesian solution (inthe sense of computing the posterior pdf).

Problem: only a conceptual solution; integrals arenot tractable.

But: in some restricted cases, an optimal solution is pos-sible. Two optimal solutions (under restrictive assump-tions):

� (standard) Kalman filter

� grid-based filter

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4 Kalman Filter

Introduction

Assumptions:

� posterior at time k−1, i.e. p(xk−1|zk−1), is Gaussian

� dynamic system characterized by

xk = Fkxk−1 +Gkvk−1

zk = Hkxk +Jknk

� both noise vectors Gaussian (covariance matrices areQk−1 and Rk)

Then new posterior p(xk|zk) is Gaussian, too, and can becomputed using simple linear equations.

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5 Grid-Based Filter

Introduction

Assumptions:

� state space is discrete

� number of different states (Ns) is limited(Note: implicitly includes discreteness)

Suppose at time k−1 we have states xi with i = 1, . . . ,Ns.Conditional probability of these states:

Pr(xk−1 = xi |z1:k−1) = wik−1|k−1

Then the (old) posterior at time k−1 is given by:

p(xk−1|z1:k−1) =Ns

∑i=1

wik−1|k−1 δ(xk−1−xi)

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Results (Summary)

Both the (new) prior and the (new) posterior have thesame structure: a sum of weighted Dirac peaks:

p(xk|z1:k−1) =Ns

∑i=1

wik|k−1 δ(xk−1−xi)

p(xk|z1:k) =Ns

∑i=1

wik|k δ(xk−1−xi)

Note: extension to different sets of states for each timestep

{xi} : i = 1, . . . ,Ns −→ {xik} : i = 1, . . . ,Ns,k

with time-varying index k is easily possible; the ‘allowed’states need not be constant.

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Prediction Equation (in Detail)


p(xk|z1:k−1) =Ns

∑i=1

Ns

∑j=1

w jk−1|k−1p(xi |x j)δ(xk−1−xi)

=Ns

∑i=1

wik|k−1δ(xk−1−xi)

where wik|k−1 = ∑Ns

j=1w jk−1|k−1p(xi |x j)

(new) prior weights = old posterior weights,reweighted using state transition probabilities

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Update Equation (in Detail)


p(xk|z1:k) =Ns

∑i

wik|kδ(xk−1−xi)

where wik|k =

wik|k−1p(zk|xi)

∑Nsj w j

k|k−1p(zk|x j ).

Note: denominator only needed for normalization

posterior weights = prior weights, reweighted usinglikelihoods

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6 Particle Filter

Suboptimal Approximations

If we want to preserve Kalman filter principle. . .

� Extended Kalman Filter (EKF)

� Unscented Kalman Filter (UKF)

...we get better results,

BUT: we cannot get rid off Gaussian approximations

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EKF / UKF:

All these approaches fail if we have

� bimodal / multimodal pdfs

� heavily skewed pdfs

We need a more general scheme to tackle these problems.

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Particle Filter – General Concept

Many different names (do you remember the introduc-tion?) but the general concept is rather simple:

PARTICLE FILTER:If we cannot solve the integrals required for aBayesian recursive filter analytically . . . we representthe posterior probabilities by a set of randomly cho-sen weighted samples.

Note: “randomly chosen” ≡ “Monte Carlo”(we are playing roulette / throwing the dice)

Increasing number of samples ⇒ (almost sure) conver-gence to true pdf

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Sequential Importance Sampling (SIS)

SIS is the basic framework for most particle filter algo-rithms. Let

{xi0:k} : set of support points (samples, particles)

i = 1, . . . ,Ns

(whole trajectory for each particle!)

wik : associated weights, normalized to ∑i w

ik = 1

Then:

p(xk|z1:k)≈Ns

∑i=1

wikδ(x0:k−xi

0:k)

(discrete weighted approximation to the true posterior)

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SIS (continued)

Usually we cannot draw samples xik from p(·) directly. As-

sume we sample directly from a (different) importancefunction q(·). Our approximation is still correct (up tonormalization) if

wik ∝

p(xi0:k|z1:k)

q(xi0:k|z1:k)

The trick: we can choose q(·) freely!

If the importance function is chosen to factorize such that

q(x0:k|z1:k) = q(xk|x0:k−1,z1:k) q(x0:k−1|z1:k−1)

then one can augment old particles xi0:k−1 by xk ∼

q(xk|x0:k−1,z1:k) to get new particles xi0:k.

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SIS (continued)

Weight update (after some lengthy computations. . . ):

wik = wi

k−1

p(zk|xik) p(xi

k|xik−1)

q(xik|xi

0:k−1,z1:k)(3)

Furthermore, if q(xk|x0:k−1,z1:k) = q(xk|xk−1,z1:k)(only dependent on last state and observations):

p(x|z1:k)≈Ns

∑i=1

wik δ(xk−xi

k)

(and we need not preserve trajectories xi0:k−1 and history

of observations z1:k−1)

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SIS Algorithm – Pseudo Code

[ {xik,w

ik}

Nsi=1] = SIS( {xi

k−1,wik−1}

Nsi=1, zk)

FOR i = 1 : Ns

draw xik ∼ q(xk|xi

k−1,zk)update weights according to (3)

END FORnormalize weights to ∑Ns

i=1wik = 1

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PROBLEM: Degeneracy Problem

Problem with SIS approach: after a few iterations, mostparticles have negligible weight (the weight is concentratedon a few particles only)

Counter measures:

� brute force: many, many samples Ns

� good choice of importance density

� resampling

Note: amount of degeneracy can be estimated based onvariance of weights [Liu 1996].

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Optimal Impotance Density:

It can be shown that the optimal importance density isgiven by

q(xk|xk−1,zk)opt = p(xk|xk−1,zk)

Then

wik = wi

k−1

∫p(zk|x′k)p(x′k|xi

k−1)dx′k

Two major drawbacks: usually neither sampling from qoptnor solving the integral in wi

k is possible. . . (but in somespecial cases, it works)

Other alternative which is often convenient:q(·) = p(xk|xk−1) (prior). Easy to implement, but does nottake measurements into account.

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Resampling Approaches

Basic idea of resampling:

Whenever degeneracy rises above threshold: replaceold set of samples (+ weights) with new set of sam-ples (+ weights), such that sample density betterreflects posterior pdf.

This eliminates particles with low weight and chooses moreparticles in more probable regions.

Complexity: possible in O(Ns) operations

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The resampling principle:Particle Filters

(graphics taken from Van der Merwe et al.)

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General Particle Filter – Pseudo Code

[ {xik,w

ik}

Nsi=1] = PF( {xi

k−1,wik−1}

Nsi=1, zk)

FOR i = 1 : Ns

draw xik ∼ q(xk|xi

k−1,zk)update weights according to (3)

END FORnormalize weights to ∑Ns

i=1wik = 1

IF degeneracy too highresample {xi

k,wik}

Nsi=1

END IF

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PROBLEM: Loss of Diversity

No degeneracy problem but new problem arises:

Particles with high weight are selected more andmore often, others die out slowly⇒ loss of diversity or sample impoverishment

For small process noise, all particles can collapse into asingle point within a few iterations.

Other problem: resampling limits the ability to parallelizealgorithm.

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Other Particle Filter Variants

Methods to counteract loss of diversity and degeneracyproblem:

� resample-move algorithm

� regularization

� Rao-Blackwellisation

� multiple Monte-Carlo

Other particle filter variants found in the literature:

� sampling importance resampling (SIR)

� auxiliary sampling importance resampling (ASIR)

� regularized particle filter (RPF)

� . . .

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7 Experiments

see videos...

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8 Summary

First of all: what I did not talk about. . .

� speed of convergence

� number of samples needed

� complexity issues / tricks for speed-up of algorithms

� advanced particle filter variants in detail

⇒ refer to the literature if you want to know more

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Advantages of particle filters (PFs):

� can deal with non-linearities

� can deal with non-Gaussian noise

� can be implemented in O(Ns)

� mostly parallelizable

� easy to implement

� in contrast to HMM filters (state-space discretizedto N fixed states) : PFs focus adaptively on probableregions of state-space

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Thesis:

If you want to solve a filtering problem, then particlefilters are the best filters you can use, much betterthan e.g. Kalman filters.

Right or wrong?

Title Page

Introduction

Dynamic System

Bayesian Filters

Kalman Filter

Grid-Based Filter

Particle Filter

Experiments

Summary

Page 43 of 45

JJ II

J I


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WRONG!

Particle filters include a random element; they only conver-gence to the true posterior pdf (almost surely) if Ns→∞.

Therefore: If the assumptions for Kalman filters or grid-based filters are valid, no PF can outperform them!

Additionally: depending on the dynamic model, Gaussiansum filters, unscented Kalman filters or extended Kalmanfilters may produce satisfactory results at lower computa-tional cost.

(But you should at least try a PF; it is usually better thanother suboptimal methods!)

Title Page

Introduction

Dynamic System

Bayesian Filters

Kalman Filter

Grid-Based Filter

Particle Filter

Experiments

Summary

Page 44 of 45

JJ II

J I


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PF approaches proved their usefulness in a variety of ap-plications.

But:

� choice of importance function q(·) is crucial in PFdesign

� large sample number Ns increases computational ef-fort

� potential problems: degeneracy and loss of diversity

If these points are taken into account, then particlefilters are an extremely powerful tool for filtering /estimation.

(“black box usage” vs “know what you’re doing!”)

Title Page

Introduction

Dynamic System

Bayesian Filters

Kalman Filter

Grid-Based Filter

Particle Filter

Experiments

Summary

Page 45 of 45

JJ II

J I


Search

Close


Thank you!

This presentation was made with LATEX.(try to write Bucuresti in Powerpoint. . . )

particle filters - snu 4 pf suppl 2... · title page introduction dynamic system bayesian filters...

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