Bayesian Methods in Positioning Applications
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1 Bayesian Methods in Positioning Applications Vedran Dizdarević Graz University of Technology, Austria 24. May 2006 Bayesian Methods in Positioning Applications p.1/21
2 Outline Problem Statement Solution Alternatives Introduction to Bayesian Estimation Bayes Rule Recursive Bayesian Estimator Particle Filtering MATLAB Demo Bayesian Estimation in Positioning Applications Bayesian Methods in Positioning Applications p.2/21
3 Problem Statement (1/2) System description in two steps Find system model (modeling) Find the particular realization of the model (identification; parameter estimation ˆθ) Time variant system parameters require recursive estimation on-line vs. off-line data processing Bayesian Methods in Positioning Applications p.3/21
4 Problem Statement (2/2) Basic notation Time-discrete data record z 0:k = {z 0,..., z k } With N-dimensional observation z k = {z 1 k,..., zn k } Inference of the model parameter z 0:k ˆθ(k; z 0:k ) Recursive estimation ˆθ k = F (ˆθ k 1, z k, S k 1 ) Initial condition(s) - selection of θ 0 Bayesian Methods in Positioning Applications p.4/21
5 Solution Alternatives Modification of off-line estimation methods RLS Stochastic approximation Gradient-based methods Model reference techniques and pseudolinear regression Adaptive adjustment through comparison of two systems Extension of linear regression (LS) methods Bayesian inference Estimation of distribution of θ Bayesian Methods in Positioning Applications p.5/21
6 Introduction to Bayesian Estimation Bayesian approach Probabilistic framework for recursive state estimation In Bayesian estimation θ is treated as a random variable with p(θ) By defining p(θ) estimationuncertaintiess are implied The formalism allows introduction of prior knowledge of p(θ) Relying on a Bayes rule updated p(θ) isinferredd from prior (predicted) p(θ) combined with incoming observations Bayesian Methods in Positioning Applications p.6/21
7 Bayes Rule and Contemporary Science Published in T. Bayes Essay Towards Solving a Problem in the Doctrine of Chances (1764) Steam engine of J. Watt (1769) B. Franklin defines the concept of positive and negative polarity (1752) L. Galvani and frogs (1770) K.F. Gauss was born (1777) Bayesian Methods in Positioning Applications p.7/21
8 Bayes Rule Assume a disjoint set of n discrete events {E i } n i=1 Probabilities P (E i ) n i=1 are assigned to each event If events are not independent observing that event E k hasoccurredd, probabilities of all other events {E i } n i=1 will change This is described using conditional probability P (E i E k ) P (E i, E k ) = P (E i E k )P (E k ) = P (E k E i )P (E i ) P (E i E k ) = P (E k E i )P (E i ) P (E k ) Bayesian Methods in Positioning Applications p.8/21
9 Bayesian Estimator (1/4) Two equations describe the discrete-time, time-variant, d-dimensional system Motion model θ k+1 = f(θ k ) + w k θ k, w k R d, f : R d R d Measurement model z k = h k (θ k ) + v k z k, v k R m k, h k : R d R m k Bayesian Methods in Positioning Applications p.9/21
10 Bayesian Estimator (2/4) Notation Conditional pdf p k l (θ k ) = p(θ k z 1,..., z l ) Likelihood function L k (θ k ) = p(z k θ k ) = p vk (z k h k (θ k )) Transition pdf φ k (θ k+1 θ k ) = p wk (θ k+1 f(θ k )) Bayesian Methods in Positioning Applications p.10/21
11 Bayesian Estimator (3/4) Step 0 Select initial distribution p 0 0 and set k = 1 Step 1 Compute the predictive pdf p k k 1 (θ k ) = R d p(θ k θ k 1 )p(θ k 1 z 1,..., z k 1 )dθ Step 2 Compute the posterior pdf p k k (θ k ) = p(z k θ k )p(θ k z 1,...,z k 1 ) p(z k z 1,...,z k 1 ) Step 3 Output ˆθ k and variance V (p k k (θ k )) Step 4 Increment k and repeat from Step 1 p k k 1 (θ k )L k (θ) Bayesian Methods in Positioning Applications p.11/21
12 Bayesian Estimator (4/4) Calculation of ˆθ k ˆθ k = E(p k k (θ k )) ˆθ k = argmax(p k k (θ k )) (MAP) Problem: it is difficult to find an analytical solution to this problem! If the noise pdfs are independent, white and zero-mean Gaussian and the measurement equation linear in θ, it can be derived that the Bayesian approach reduces to the Kalman Filter A general algorithm which approximates the pdfs with arbitrary accuracy is the particle filter Bayesian Methods in Positioning Applications p.12/21
13 Particle Filter (1/4) Two steps require intractable integration Posterior pdf computation Expectation of the posterior pdf Numerical means for the tracking the evolution of the pdfs The pdf is approximated as weighted sum of samples in the space state (particles) p(θ) M m=1 w(m) δ(x x (m) ) Recursive update Prediction: Calculate the predictive pdf using the motion model Update: Weights update with normalization Bayesian Methods in Positioning Applications p.13/21
14 Particle Filter (2/4) Initialization Select an importance function π and create a set of particles x (m) 0 π Measurement update Weight update according to the likelihood w (m) k = w (m) k 1 π(θ z) Normalize the weights w (m) k Calculate the estimate ˆθ k M Prediction θ k+1 = h(θ k ) + v k := w (m) k / m w(m) k m=1 w(m) k θ (m) k Bayesian Methods in Positioning Applications p.14/21
15 Measurement update Resampling Prediction Particle Filter (3/4) Weights tend to concentrate in few particles Draw P particles from the current particle with probabilities proportional to the corresponding weight; set weights for the new particle as w k = P 1 Optionally resampling only if a given criterion is fulfilled 1 N eff = < N th Pi (w(i) k )2 Bayesian Methods in Positioning Applications p.15/21
16 Particle Filter (4/4) Bayesian Methods in Positioning Applications p.16/21
17 MATLAB example System equations θ k = 0.5θ k θ k 1 1+θ 2 k 1 + 8cos(1.2k) + w k z k = θ2 k 20 + v k Simulation parameters σ 2 w = 10 σ 2 v = 1 σ 2 π = 5 M = 500 Bayesian Methods in Positioning Applications p.17/21
18 Pro and Contra This algorithm is known under many names! Pro: Non-linear sate model needs no approximations Non-Gaussian distributions θ Combines different observations in a single model Variable dimension of the observation vector no problem Contra: Computationally expensive Selection of prior distribution Bayesian Methods in Positioning Applications p.18/21
19 Bayesian Estimation in Positioning Applications (1/2) Application: Positioning, Navigation, Tracking Type of object: Persons, Robots, Vehicles Supplementary information: Road map, terrain profile State space: θ = [p, φ, v, v,...] T Observations: z = [p ext, φ ins, v ins, γ image, γ radar, S ss,...] T Bayesian Methods in Positioning Applications p.19/21
20 Bayesian Estimation in GRAZ UNIVERSITY OF TECHNOLOGY Positioning Applications (2/2) Current research directions Motion models Prediction of pedestrian movements Complexity reduction Employ KF for linear states Parallelization Distributed PF in a sensor network Bayesian Methods in Positioning Applications p.20/21
21 References L. Jung, T. Söderström Theory and Practice of Recursive Identification, MIT Press, S. Kay Fundamentals of Statistical Signal Processing, Estimation Theory, Prentice Hall PTR, P. Djurić et al., Particle Filtering, IEEE Signal Processing Magazine, Vol.20, No.5, September F. Gustafsson et al., Particle Filters for Positioning, Navigation and Tracking, IEEE Transactions on Signal Processing, Vol.50, Issue 2, February N. Sirola et al., Local Positioning with Parallelepiped Moving Grid, Proceedings of The 3rd Workshop on Positioning, Navigation and Communication (WPNC), Bayesian Methods in Positioning Applications p.21/21
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