1. Novel approach to nonlinear/non- Gaussian Bayesian state estimation N.J Gordon, D.J. Salmond and A.F.M. Smith Presenter: Tri Tran 7-2005

2. Outline Motivations Recursive Bayesian estimations Bayesian Bootstrap filter algorithms Evaluations of the algorithms Summary

3. Motivations In many applications of positioning, navigation and tracking, we want to estimate the position of a moving object at discrete times. Positioning: position of an object is to be estimated when an navigation system is used to provide measurements of movement. Navigation: besides the position also velocity, attitude and heading, accelerating and angular rates are included in the problem. Target tracking: another object’s position is to be estimated based on measurement of relative range and angles to one’s own position.

4. Motivations (cont.) The problems can be described by state space models where the state vector contains the position and derivatives of the position. Bayesian methods is to construct a probability density function (PDF) of the state based on all the available information. For linear/Gaussian models, the required PDF remains Gaussian at every iteration of the filter, Kalman filter can be used to propagate and update the mean and covariance of the distribution. For nonlinear/non-Gaussian models, there is no general analytic expression for the required PDF  a new way of representing and recursively generating an approximation to the state PDF. Central idea: represent the required PDF as a set of random samples, rather than as a function over state space.

5. Particle Filters Recursive methods to estimate the dynamic state (a set of random samples) of a moving object in a discrete time nonlinear model based on posterior distributions.

6. Recursive Bayesian estimation System model x k+1 = f k (x k, w k ) where f k : R n xR m  R n the system transition function w k is white noise sequence of known PDF, independent of past and current states. Measurement model (observation equation) y k = h k (x k, v k ) where h k : R n xR r  R p the system transition function v k is another white noise sequence of known PDF, independent of past and current states. Assume that the initial PDF p(x 1 |D 0 ) of the state vector, the functional forms f i and h i for i=1,…, k are known. At time step k, available information is the set of measurements D k = {y k :i=1,…,k} Need to construct the PDF of the current state x k, given all the available information: p(x k |D k )

7. Bayesian bootstrap filter A set of random samples {x k-1 (i): i=1,…,N} from the PDF p(x k-1 |D k-1 ). Propagate and update the set of random samples to obtain {x k (i): i=1,…,N} which approximates as p(x k |D k ) Prediction: each sample is passed through the system model to obtain samples from the prior at time step k: x * k (i) = f k-1 (x k-1 (i), w k-1 (i)) Update: with the measurement y k, evaluate a normalized weight for each sample Thus define a discrete distribution over {x* k (i): i=1,…,N} with probability q i associated with element i Resample N times from the discrete distribution to generate samples {x k (i): i=1,…,N}, so that for any j, Pr{x k (j)=x* k (i)} =q i

8. Bootstrap filter Advantages: no restrictions on the functions f k and h k or on the distributions of the system or measurement noise. Requirements: (a) p(x 1 ) is available for sampling (b) the likelihood p(y k |x k ) is a known functional form (c) p(w k ) is available for sampling The number of random samples N depends on: (a) the dimension of the state space (b) the typical ‘overlap’ between the prior and the likelihood (c) the required number of time steps

9. Evaluations of the bootstrap filter algorithm One-dimensional nonlinear example Consider the following nonlinear model X k = 0.5x k-1 +25x k-1 /(1+x 2 k- 1 )+8cos(1.2(k-1))+w k Y k =x 2 k /20+v k W k and v k are zero-mean Gaussian white noise with variances 10.0 and 1.0, respectively. K=50, x 0 = 0.1, N=500

10. Evaluations of the bootstrap filter algorithm Bearings-only tracking example Target moves within the x-y plan: A fixed observer at the origin of the plane takes noisy measurements z k of the target bearing k=24, N=4000, initial state x 1 =(-0.05, 0.001, 0.7, -0.055)

11. Evaluations of the bootstrap filter algorithm Bearings-only tracking example (cont.) The actual co-ordinate value is always within the 95% probability region.

12. Summary A bootstrap filter for implementing recursive Bayesian filters The method is applied for nonlinear models and approximates the posterior distribution as a set of random samples.

13. References N.J Gordon, D.J. Salmond and A.F.M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation”, in IEE Processinds-F, Vol. 140, 1993. Fredrik Gustafsson, Fredrik Gunnarsson, Niclas Bergman, Urban Forssell, Jonas Jansson, Rickard karlsson, Per-Johan Nordlund, “Prticle Filters for Positioning, Navigation and Tracking”, IEEE Transactions on Signal Processing, vol. 50, pp. 425- 435, Feb. 2002