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 + …

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

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

Each cell contains: 42 - Particle count - Associated Rate

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

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

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

Tree Node Cell Node Observation: -2 Particles 14

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

Dynamic Cell Sizing Example

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

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

Audience

Performer Model θ Audience

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)