Particle Filters M. Sami Fadali Professor of EE University of Nevada Outline Monte Carlo integration. Particle filter. Importance sampling. Degeneracy Resampling Example. 1 2 Monte Carlo Integration Numerical integration using randomly generated points. No curse of dimensionality Use to calculate moments (integrals) Integral of continuous MC Integration Procedure Generate Calculate Calculate the estimate of Calculate the estimate of 3 4
Use Unbiased Properties of Consistent (shown earlier) Example (Hogg et al., p. 289) Estimate Use Generate Calculate using Monte Carlo integration consistent estimate of (estimate improves as increases) 5 6 MC Integration Factorization Generate Calculate pdf (integral is unity, nonnegative) Calculate the estimate of Importance Sampling Not always possible to generate samples from Generate samples from a similar pdf (same support) 7 8
Use MC Integration Generate Calculate Relation to Estimation Fundamental theorem of estimation theory Minimum mean-square error estimator:. all measurements up to and including time 9 10 Particle Filter View random samples as particles Can handle non-gaussian statistics and multimodal distributions. Approximate a continuous pdf with appropriately weighted random samples. A posteriori conditional density Approximation of PDF Using Weighted Random Samples 11 12
Model Nonlinear process model Measurement model Noise processes are not necessarily Gaussian. measurements up to and including time Recursive Bayesian Filter Predictor Corrector Initial condition for recursion 13 14 Corrector Modified Rewrite using Markov property Introduce Importance Density 15 16
Choose Importance Density Choose to allow the factorization Observe that Substitute Back to Integral 17 18 Weights Monte Carlo Estimate (normalize) Substitute 19 20
Choice of Importance Density Critical step in particle filter design. Optimal importance density: minimizes the variance of importance weights conditioned upon and Choice of Importance Density Popular choice: bootstrap filter Assume that the weights are resampled every cycle such that they become uniformly distributed. 21 22 Importance Sampling Unlike Monte Carlo and ensemble filters, does not use a simple distribution to generate the random samples. Generate points that fit a general pdf. Importance Density: proposal density that can easily generate samples. Algorithm used: Sequential Importance Sampling (SIS) Sequential Importance Sampling Weight recursion likelihood function transition prior importance density 23 24
Estimates Based on the approximate discretize pdf State Estimate Error Covariance 25 Input Output for Draw SIS Algorithm from for 26 Degeneracy Phenomenon Normalized weights t to concentrate into one particle after a number of recursions (all other particles degenerate) Measure of degeneracy Degeneracy After 2 Cycles 27 28
Resampling Resampling Required if Particles get more meaningful weights (uniformly distributed weights) Eliminates samples with low weights and produces a set with uniform weights. 29 30 Resampling Algorithms Idea Generate a random number Add the weights until their sum exceeds at which point choose the corresponding particle as a resampled point and save the index of its parent Can show that the resampled particles pdf converges to 31 Resampling Algorithm Input,, 1,, Output,,, 1,, for 2: Draw a starting point ~ 0, for 1: 1 While 1 32
Particle Filter Algorithm (Simon) Generate initial particles using the pdf Predictor step to obtain Compute the weights then scale them so that their sum is unity. Corrector step: SIR Particle Filter Algorithm Input,, 1,,, Output, 1,, 0 for 1: Draw ~ Calculate End for 1: / Resample using SIR 33 34 References R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering, 4ed, J. Wiley, NY, 2012. R. V. Hogg, J. W. McKean, A. T. Craig, Introduction to Mathematical Statistics, 6 th Ed., Pearson/Prentice-Hall, Upper Saddle River, NJ, 2005. B. Ristic, S. Arulampalam, N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Applications, Artech House, Boston, 2004. 35