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Published on January 2, 2008

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Introducing the Marginalized Particle Filter - Exploiting Structures in State-Space Models:  Thomas Schön Division of Automatic Control, Linköping University, Sweden Introducing the Marginalized Particle Filter - Exploiting Structures in State-Space Models Introduction – The Problem:  Introduction – The Problem The marginalized particle fitler is applicable to the following model structure, which is a conditionally linear-Gaussian structure. Compute The goal is to infer the infomation in the measurements into the current state Introduction – The Underlying Idea:  Introduction – The Underlying Idea The marginalized particle filter is all about exploiting any linear-Gaussian sub-structure present in the model. A.k.a. the Rao-Blackwellized particle filter. Introduction – The Underlying Idea II:  Introduction – The Underlying Idea II The marginalized particle filter estimates the probability density function by a clever combination of un-parametric and parametric density functions. The estimated density function is represented by a weighted sum of Gaussians, where each particle has a Gaussian distribution attached to it. Introduction – Representation of the PDF:  Introduction – Representation of the PDF Schematic illustration of the relation to other known estimation approaches. MPF The Standard Particle Filter:  The Standard Particle Filter The particle filter provides an estimate of a general filter density function, not necessarily a Gaussian. The Kalman filter assumes linear transformations and Gaussian densities. In that case, all densities can be parametrized using two parameters (mean and covariance). The Basic Particle Filter Algorithm:  The Basic Particle Filter Algorithm A 5 minute implementation available in my thesis. The Marginalized Particle Filter Algorithm:  The Marginalized Particle Filter Algorithm The marginalized particle filter algorithm: Initialize the particles. Particle filter measurement update: Evaluate weights. Resample. Particle filter time update. 4.a Kalman filter measurement update. 4.b Predict new particles. 4.c Kalman filter ”measurement” and time update. Iterate from step 2. Major problem with the standard particle filter: Cannot handle large state dimensions. Handwaving Explanation of the Algorithm:  Handwaving Explanation of the Algorithm The MPF is a combination of the particle filter and the Kalman filter, where each particle has a Kalman filter associated to it. The Marginalization I:  The Marginalization I Theorem: Using the model given before the conditional PDF’s for the linear state is given by, Use the information present in yt. 4 (a) The Marginalization II:  The Marginalization II Particle filter time update 4 (b) Theorem continued: Conditioned on the nonlinear states this is a measurement equation. 4 (c) Where is the Marginalization?:  Where is the Marginalization? where the linear states have been marginalized out according to Rao-Blackwellized particle filter, see e.g., Chen, R. and Liu, J. S. (2000). Mixture Kalman filters. Journal of the Royal Statistical Society, 62(3):493–508. Doucet, A., Godsill, S. J., and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3):197–208. Academic Example - Illustration:  Academic Example - Illustration To illustrate the marginalized particle filter I have borrowed an academic system from a paper by Torsten Söderström et.al in Automatica this summer. The nonlinear process is a first-order AR process, where the linear process is the time-varying parameter. Academic Example - Illustration:  Academic Example - Illustration MPF estimate of the filter PDF at time 10. Zoomed version Visualization of the fact that the filter PDF is a combination of an un-parametric PDF and a parametric PDF. Academic Example – Illustration II:  Academic Example – Illustration II Another illustration of the same filter PDF. Application Example – Fighter Aircraft Navigation:  Application Example – Fighter Aircraft Navigation Digital Terrain Elevation Database: 200 000 000 grid points 50 meter between points 2.5 meters uncertainty Ground Cover Database: 14 types of vegetation Obstacle Database: All man made obstacles above 40 m With P-J Nordlund (SAAB Aerosystems) Sensors: Barometric altitude Radar altitude Terrain elevation DB Used as an application example in our IEEE SP paper. Application Example – Dynamic Model:  Application Example – Dynamic Model Dynamic model This can be used to model the noise depending on position. Probability of obtaining an echo from the ground. Probability of obtaining an echo from the tree tops. There are 27 states, but there are only 3 ”truly” nonlinear states (horizontal position and heading). Application Example – Some Results:  Application Example – Some Results Conclusion: A very good result can be obtained with only 5000 particles. This can readily be implemented in the computer used in the aircraft. Properties – Quality:  Properties – Quality If the same number of particles are used in the standard particle filter and the marginalized particle filter the latter will produce estimates of the same of better quality. 1. Intuition 2. Theory The dimension of is smaller than the dimension of The linear states are estimated using an optimal algorithm Let g(U,V) be an estimtor depending on two r.v. U, V. Properties – Computational Complexity:  Properties – Computational Complexity Rickard Karlsson, Thomas Schön and Fredrik Gustafsson. Complexity Analysis of the Marginalized Particle Filter. IEEE Transactions on Signal Processing, 53(11):4408-4411, Nov. 2005. Each particle is assocciated with a Kalman filter In the general case this implies that we have to perform M Riccati recursions at each iteration of the algorithm. A detailed study of the computational complexity of the MPF is performed in Important (several reasons for its importance) special case: Here, we only need 1 Riccati recursion, not M! Conclusions:  Conclusions MPF = Exploit any linear-Gaussian sub-structure present in the problem. Provides a clever combination of un-parametric and parametric density functions. Provides a way to handle high-dimensional problems. Provides estimates of better or the same quality as the standard particle filter. The algorithm has found many applications. Thomas B. Schön, (2006), Estimation of Nonlinear Dynamic Systems – Theory and Applications, PhD thesis, Linköping University, Linköping, Sweden. Thomas Schön, Fredrik Gustafsson, and Per-Johan Nordlund. Marginalized Particle Filters for Mixed Linear/Nonlinear State-Space Models. IEEE Transactions on Signal Processing, 53(7):2279-2289, Jul. 2005. If you want to know more:

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