Nonlinear and Non-Gaussian Estimation with A Focus on Particle Filters

Prasanth JeevanMary Knox

May 12, 2006

Background

• Optimal linear filters Wiener Stationary Kalman Gaussian Posterior, p(x|y)

• Filters for nonlinear systems Extended Kalman Particle

Extended Kalman Filter (EKF)

• Locally linearize the non-linear functions

• Assume p(xk|y1,…,k) is Gaussian

Particle Filter (PF)

• Weighted point mass or “particle” representation of possibly intractable posterior probability density functions, p(x|y)

• Estimates recursively in time allowing for online calculations

• Attempts to place particles in important regions of the posterior pdf

• O(N) complexity on number of particles

Particle Filter Background [Ristic et. al. 2004]

• Monte Carlo Estimation

• Pick N>>1 “particles” with distribution p(x)Assumption: xi is independent

Importance Sampling

• Cannot sample directly from p(x)• Instead sample from known importance

density, q(x), where:

• Estimate I from samples and importance weights

where

Sequential Importance Sampling (SIS)

• Iteratively represent posterior density function by random samples with associated weightsAssumptions: xk Hidden Markov process, yk conditionally independent given xk

Degeneracy

• Variance of sample weights increases with time if importance density not optimal [Doucet 2000]

• In a few cycles all but one particle will have negligible weights PF will updating particles that contribute little in

approximating the posterior

• Neff, estimate of effective sample size [Kong et. al. 1994]:

Optimal Importance Density [Doucet et. al. 2000]

• Minimizes variance of importance weights to prevent degeneracy

• Rarely possible to obtain, instead often use

Resampling

• Generate new set of samples from:

• Weights are equal after i.i.d. sampling

• O(N) complexity• Coupled with SIS,

these are the two key components of a PF

Sample Impoverishment

• Set of particles with low diversity Particles with high

weights are selected more often

Sampling Importance Resampling (SIR)

[Gordon et. al. 1993]

• Importance density is the transitional prior

• Resampling at every time step

SIR Pros and Cons

• Pro: importance density and weight updates are easy to evaluate

• Con: Observations not used when transitioning state to next time step

A Cycle of SIR

Auxiliary SIR - Motivation[Pitt and Shephard 1999]

• Want to use observation when exploring the state space ( ’s) To have particles in regions

of high likelihood

• Incorporate into resampling at time k-1 Looking one step ahead to

choose particles

ASIR - from SIR

• From SIR we had

• If we move the likelihood inside we get:

• We don’t have though

• Use , a characterization of given such as

ASIR continued

• So then we get:

• And the new importance weight becomes:

ASIR Pros & Cons

• Pro Can be less sensitive to peaked likelihoods and outliers by using observation Outliers - Model-improbable states that can result in a

dramatic loss of high-weight particles

• Cons Added computation per cycle If is a bad characterization of (ie.

large process noise), then resampling suffers, and performance can degrade

Simulation Linear

• System Equations:

where v ~ N(0,6) and w ~ N(0,5)

Simulation Linear10 Samples

Simulation Linear50 Samples

Simulation Linear

Table 1: Mean Squared Error Per Time Step

  Number of Particles

Filter 10 50 100 1000

KF 0.0349 0.0351 0.0350 0.0352

ASIR 0.7792 0.0886 0.0417 0.0350

SIR 0.9053 0.0977 0.0496 0.0354

Simulation Nonlinear

• System Equations:

where v ~ N(0,6) and w ~ N(0,5)

Simulation Nonlinear10 Samples

Simulation Nonlinear50 Samples

Simulation Nonlinear100 Samples

Simulation Nonlinear1000 Samples

Simulation Nonlinear

Table 2: Mean Squared Error Per Time Step

  Number of Particles

Filter 10 50 100 1000

EKF 812.08 826.20 827.94 838.75

ASIR 30.14 20.15 18.81 17.86

SIR 37.97 22.62 21.49 19.78

Conclusion

• PF approaches KF optimal estimates as N

• PF better than EKF for nonlinear systems• ASIR generates ‘better particles’ in certain

conditions by incorporating the observation

• PF is applicable to a broad class of system dynamics Simulation approaches have their own

limitations Degeneracy and sample impoverishment

Conclusion (2)

• Particle filters composed of SIS and resampling Many variations to improve efficiency

(both computationally and for getting ‘better’ particles)

• Other PFs: Regularized PF, (EKF/UKF)+PF, etc.