1. Prediction in Interacting Systems: Applications & Simulations Jarett Hailes November 1, 2002 dX t = μ(X t )dt + σ(X t )dB t dx = this- >mu()*dt + …

2. Outline Refining Grid Stochastic Filter –Description –Characteristics Performer Tracking Problem –Model –Simulation

3. Refining Grid Stochastic Filter (REST) Filter - Given a signal that evolves on regular Euclidean subset - Divide signal state space into a finite number of cells In general N 1 x N 2 x … x N d cells N1N1 N2N2

4. Each cell contains: 42 - Particle count - Associated Rate

5. Refining Grid Stochastic Filter (REST Filter) Particles used to approximate unnormalized conditional distribution

6. 1 2 -2 1 Cell Rates Cell rates are used to calculate net birth (death rate) in a cell Rates are determined by cell’s particle count and immediate neighbour’s rates = Net Birth Rate +1

7. Net birth rates are used to mimic particle movement in observation- dependent manner. Net Birth Rates OBSERVATIONOBSERVATION

8. Tree Node Cell Node 4 2 1 6 17 3 Observation: -2 Particles 14

9. N2N2 N1N1 Dynamic Cell Sizing Zoom in: N 1 Zoom out: N 1

10. Dynamic Cell Sizing Example

11. REST Advantages - Less simulation noise than particle filters - Dynamic cell sizing, inherent parameter estimation - Dynamic domain problems

12. Performer Problem - Acoustic tracking system designed to have lighting equipment follow performer on large stage - Due to mechanical lags, system must be able to predict performer’s future position based on current state

13. Audience

14. Performer Model θ Audience

15. Observation Model          otherwise 1 )1,0( if ),( )()+( (),( 222 pUWSXh Y zSySxSSXh mt l t t l z l y l x l t mm m m  S 1 (x,y,z) S 3 (x,y,z) S 4 (x,y,z) S 2 (x,y,z)