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Extended Kalman filter –linearize the measurements and evolution models using 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
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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