## Particle Filters

author: Simon Godsill, University of Cambridge
published: Nov. 2, 2009,   recorded: September 2009,   views: 8868
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# Slides

0:00 Slides Sequential Monte Carlo Methods Overview In many applications When the system is linear Contents Bayes' Theorem and inference Monte Carlo Methods -1 Monte Carlo Methods -2 In cases -1 In cases -2 In cases -3 Alternatively -1 Alternatively -2 Alternatively -3 There are numerous versions of Monte Cralo samplers -1 There are numerous versions of Monte Cralo samplers -2 Important trick -1 Important trick -2 Important trick -3 state space models, filtering and smoothing Note -1 Note -2 Note -3 Summarise as a "state space" or "dynamical" model -1 Summarise as a "state space" or "dynamical" model -2 Picture -1 Example: linear AR model observed in noise Form state vector as: Picture -2 Form state vector as: Picture -2 Picture -3 Alternatively, in terms of state evolution and observation densities -1 Alternatively, in terms of state evolution and observation densities -2 Example: Non-linear model Estimation tasks Specifically: -1 Specifically: -2 Specifically: -3 Picture -4 Filtering -1 Filtering -2 The sequential scheme is as follows: Linear Gaussian models We can write this equivalenly as: -1 We can write this equivalenly as: -2 We first require -2 Filtering -2 We first require -2 We first require -3 Now from 11 we have -1 Now from 11 we have -2 Now from 11 we have -3 The correction step -1 Now from 11 we have -3 The correction step -1 The correction step -2 where -1 where -2 Hence the whole kalman filtering recursion can be summarised as: -1 Things you can do with a Kalman filter Likelihood evaluation -1 Likelihood evaluation -2 Likelihood evaluation -3 Likelihood evaluation -4 Numerical methods The extended Kalman filter Perform a 1st order Taylor expansion Monte Carlo Filtering -1 Example: Non-linear model Monte Carlo Filtering -1 Monte Carlo Filtering -2 Monte Carlo Filtering -3 Monte Carlo Filtering -4 Monte Carlo Filtering -5 Monte Carlo Filtering -6 Resampling -1 Monte Carlo Filtering -6 Resampling -1 Resampling -2 Resampling -3 Monte Carlo Filtering -5 Resampling -3 Sequential Monte Carlo -1 Sequential Monte Carlo -2 Sequential Monte Carlo -3 Sequential Monte Carlo -4 Sequential Monte Carlo -5 Sequential Monte Carlo -4 Sequential Monte Carlo -5 Sequential Monte Carlo -6 Sequential Monte Carlo -4 Sequential Monte Carlo -5 Sequential Monte Carlo -4 Sequential Monte Carlo -5 Sequential Monte Carlo -6 Sequential Monte Carlo -7 Three steps -1 Three steps -2 Three steps -3 Three steps -4 Step 0 -1 Step 0 -2 Step 0 -3 Step 2 Three steps -1 Step 2 Options -1 Options -2 A basic algorithm Example: standard nonlinear model Picture -5 General Sequential Impportance Sampling A basic algorithm The importance function -1 The importance function -2 Repeated application -1 Conclusions

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1 Janjua, February 3, 2010 at 7:33 a.m.:

An excellent presentation on the basic concepts of Particle Filters.

2 Jim Smith, March 9, 2010 at 10:02 a.m.:

Part 2 of video: missing video section, skips from slide 69 to 89 (skips over important parts).

3 Jan Galkowski, April 25, 2011 at 12:24 a.m.:

I'm assuming the rightmost term of the last expression on page 11 which reads "P(x|theta|y)" is meant to read "P(x|theta,y)".

4 Zhiyuan, July 2, 2012 at 10:15 p.m.:

I believe in part 1 slide 32, the first equation (below "where"), the last term for P_{t+1} inside the inverse, should be the inverse of P_{t+1|t}.

5 ymezali, February 20, 2013 at 11:15 a.m.:

Thank you for this excellent introduction to particle filtering