The Unscented Particle Filter

2000/09/2

9이 시은

Introduction

• Filtering– estimate the states(parameters or hidden variable) a

s a set of observations becomes available on-line• To solve it

– modeling the evolution of the system and noise• Resulting models

– non-linearity and non-Gaussian distribution

• Extended Kalman filter– linearize the measurements and evolution models u

sing Taylor series• Unscented Kalman Filter

– not apply to general non Gaussian distribution• Seq. Monte Carlo Methods : Particle filters

– represent posterior distribution of states.– any statistical estimates can be computed.– deal with nonlinearities distribution

• Particle Filter– rely on importance sampling– design of proposal distribution

• Proposal for Particle Filter– EKF Gaussian approximation– UKF proposal

• control rate at which tails go to zero• heavy tailed distribution

Dynamic State Space Model

• Transition equation and a measurement’s equation

• Goal – approximate the posterior – one of marginals, filtering density recursively

)|( 1tt xxp)|( tt xyp

)|( :1:0 tt yxp)|( :1 tt yxp

Extended Kalman Filter

• MMSE estimator based on Taylor expansion of nonlinear f and g around estimate of state tx1| ttx

Unscented Kalman Filter

• Not approximate non-linear process and observation models

• Use true nonlinear models and approximate distribution of the state random variable

• Unscented transformation

Particle Filtering

• Not require Gaussian approximation • Many variations, but based on sequential im

portance sampling– degenerate with time

• Include resampling stage

Perfect Monte Carlo Simulation

• A set of weighted particles(samples) drawn from the posterior

• Expectation

)(1)|(ˆ :01

:1:0 :0)( t

N

ixtt dx

Nyxp

ti

ttttttt dxyxpxgxgE :0:1:0:0:0 )|()())((

)(1))(( )(:0

1:0

it

N

ittt xg

NxgE

))(())(( :0.

:0 ttsa

tt xgExgE

)))((var,0()))(())((( :0)|(:0:0 :1 ttypNtttt xgNxgExgENt

Bayesian Importance Sampling

• Impossible to sample directly from the posterior

• sample from easy-to-sample, proposal distribution )|( :1:0 tt yxq

tttt

tttt

tttttt

ttttt

ttttt

tttttt

dxyxqypxwxg

dxyxqyxqypxpxypxg

dxyxqyxqyxpxgxgE

:0:1:0:1

:0:0

:0:1:0:1:0:1

:0:0:1:0

:0:1:0:1:0

:1:0:0:0

)|()()()(

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

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

))(())()((

)|()(

)|()()(

)|()|()()|(

)|()()(

)|()()()(

1))((

:0)|(

:0:0)|(

:0:1:0:0

:0:1:0:0:0

:0:1:0

:1:0:0:0:1

:0:1:0:0:0

:0:1:0:0:0:1

:0

:1

:1

ttyq

ttttyq

ttttt

ttttttt

ttt

ttttt

ttttttt

tttttttt

tt

xwExgxwE

dxyxqxw

dxyxqxwxg

dxyxqyxqxpxyp

dxyxqxwxg

dxyxqxwxgyp

xgE

t

t

)(~)(

)(/1

)()(/1))((

:0)(

1:0

)(

1 :0)(

1:0

)(:0

)(

:0

ti

N

itt

it

N

i ti

t

N

it

itt

it

tt

xwxg

xwN

xwxgNxgE

• Asymptotic convergence and a central theorem for under the following assumptions– i.i.d samples drawn from the proposal, su

pport of the proposal include support of posterior and finite exists.

– Expectation of , exist and are finite.

))(( :0 tt xgE

tix :0)(

))(( :0 tt xgE)( :0

2ttt xgwtw

) ( ~ 1 ) | ( ˆ: 01

) (:1 : 0: 0

) (t

N

ix

it t tdx w

Ny x pt

i

Sequential Importance Sampling

• Proposal distribution

• assumption – state: Markov process– observations: independent given states

),|()|()|( :11

1:1:10:1:0 j

t

jjjttt yxxqyxqyxq

– we can sample from the proposal and evaluate likelihood and transition probability, generate a prior set of samples and iteratively compute the importance weights

),|()|()|(

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

:11:0

11

1:11:01:11:0

:0:0:1

ttt

ttttt

ttttt

tttt

yxxqxxpxypw

yxxqyxqxpxypw

Choice of proposal distribution

• Minimize variance of the importance weights

• popular choice

• move particle towards the region of high likelihood

)|()|( :1,1:0:1,1:0 tttttt yxxpyxxq

)|()|( 1:1,1:0 ttttt xxpyxxq

Degeneracy of SIS algorithm

• Variance of importance ratios increases stochastically over time

Selection(Resampling)• Eliminate samples with low importance ratios

and multiply samples with high importance ratios.

• Associate to each particle a number of children

tix :0)(

NNN N

i ii 1,

SIR and Multinomial sampling

• Mapping Dirac random measure onto an equally weighted random measure

• Multinomial distribution

}~{ ,:0)(

tti wx

}/1{ ,:0)( Nx t

i

Residual resampling

• Set• perform an SIR procedure to select

remaining samples with new weights• add the results to the current

)(~~ iti wNN

iN~

Minimum variance samplingWhen to sample

Generic Particle Filter

1. Initialization t=02. For t=1,2, …

(a) Importance sampling stepfor I=1, …N, sample:

evaluate importance weightnormalize the importance weights

(b) Selection (resampling)(c) output

~ˆ )(itx

Improving Particle Filters

• Monte Carlo(MC) assumption – Dirac point-mass approx. provides an adequate

representaion of posterior • Importance sampling(IS) assumption

– obtain samples from posterior by sampling from a suitable proposal and apply importance sampling corrections.

MCMC Move Step• Introduce MCMC steps of invariant distribution• If particles are distributed according to the poste

rior then applying a Markov chain transition kernel

)|~( :1:0 tt yxp)~|( :0:0 tt xxK

Designing Better Importance Proposals

• Move samples to regions of high likelihood• prior editing

– ad-hoc acceptance test of proposing particles• Local linearization

– Taylor series expansion of likelihood and transition prior

– ex)– improved simulated annealed sampling algorithm

)ˆ,()|( )()(:1

)(,1:0

)( it

itt

it

it PxNyxxq

Rejection methods

• If likelihood is bounded, sample from optimal importance distribution

ttt Mxyp )|(

Auxiliary Particle Filters

• Obtain approximate samples from the optimal importance distribution by an auxiliary variable k.

• draw samples from joint distribution

Unscented Particle Filter

• Using UKF for proposal distribution generation within a particle filter framework

Theoretical Convergence

• Theorem1If importance weight is upper bounded for any and if one o

f selection schemes, then for all , there exists independent of N s.t. for any

Nf

cydxpxfxfN

E tt

N

itttt

itt

22

1:1:0:0

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

)|()|()|(

:1,1:0

1

ttt

ttttt yxxq

xxpxypw

),( 1 tt yx