The Unscented Particle Filter 2000/09/29 이 시은

Introduction Filtering –estimate the states(parameters or hidden variable) as 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 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

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

Extended Kalman Filter MMSE estimator based on Taylor expansion of nonlinear f and g around estimate of state

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 importance sampling –degenerate with time Include resampling stage

Perfect Monte Carlo Simulation A set of weighted particles(samples) drawn from the posterior Expectation

Bayesian Importance Sampling Impossible to sample directly from the posterior sample from easy-to-sample, proposal distribution

Asymptotic convergence and a central theorem for under the following assumptions – i.i.d samples drawn from the proposal, support of the proposal include support of posterior and finite exists. –Expectation of, exist and are finite.

Sequential Importance Sampling Proposal distribution assumption –state: Markov process –observations: independent given states

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

Choice of proposal distribution Minimize variance of the importance weights popular choice move particle towards the region of high likelihood

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

SIR and Multinomial sampling Mapping Dirac random measure onto an equally weighted random measure Multinomial distribution

Residual resampling Set perform an SIR procedure to select remaining samples with new weights add the results to the current

Minimum variance sampling When to sample

Generic Particle Filter 1. Initialization t=0 2. For t=1,2, … (a) Importance sampling step for I=1, …N, sample: evaluate importance weight normalize the importance weights (b) Selection (resampling) (c) output

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 posterior then applying a Markov chain transition kernel

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

Rejection methods If likelihood is bounded, sample from optimal importance distribution

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 Theorem1 If importance weight is upper bounded for any and if one of selection schemes, then for all, there exists independent of N s.t. for any