1. Tracking Unknown Dynamics - Combined State and Parameter Estimation Tracking Unknown Dynamics - Combined State and Parameter Estimation Presenters: Hongwei Long and Nikki Hu Joint work with M.A. Kouritzin University of Alberta Supported by NSERC, MITACS, PIMS, Lockheed Martin Naval Electronics and Surveillance System, Lockheed Martin Canada, APR. Inc

2. Outline 1.Introduction 2.Algorithm for Combined State and Parameter Estimation 3.Simulation Results 4. Convergence of the Algorithm

3. 1. Introduction 1. Introduction Signal contains unknown parameters Parameter estimation - difficult for nonlinear partially-observed stochastic systems Some typical methods: –Least squares methods –Methods of moments –Maximum likelihood methods –Filtering methods

4. Develop novel algorithm –Recursive particle prediction error identification method Branching particle filter method: –For state estimation Wide applications: –Financial markets –Signal processing –Communication networks –Target detection and tracking, pollution tracking –Search and rescue

5. 2. Algorithms for Combined State and Parameter Estimation Particle Prediction Error Identification Method –Signal model : (n-dimensional) (1) –Observation Model : sequence of k-dimensional random vectors

6. : unknown parameter vector, D is a compact subset of, :state noise, : observation noise, Gaussian random variable with mean zero and variance :actual observation data, : “true” value of the parameter vector Define Find best estimator of in least squares sense:

7. Algorithm: (i) Initialization: N particles, : initial guess (ii) Evolution: each evolves according to the law of the signal process; also

8. (iii) Parameter Estimation: where which is approximated by

9. And which is approximated by Let,

10. (iv) Selection: calculate and compare it with uniform random variable to determine if the particle should: (1) be branched into two or more particles, or (2) stay in the current state, or (3) be removed The steps (ii)-(iv) are repeated at each observation time

11. We consider where is unknown parameter vector. 3. Simulation Results 3. Simulation Results

12. Observation has the form where is the total number of pixels on the domain, and In our simulation, n=1, =0.450758, =0.792296 and =0.650651, =0.2, =0.4, =0.1 and =10.

13. Simulation

14. 4. Convergence of the Algorithm Under certain regularity and stability conditions, almost surely. : the global minimum of an ordinary differential equation The convergence is associated to the asymptotic stability of the ordinary differential equation

15. Proof ideas: –Establishing some uniform moment estimates for the signal process as well as its gradient with respect to –Using mixing conditions, ergodic theorem and Gronwall lemma (discrete version) to prove the almost sure convergence