Prediction in Interacting Systems: Applications & Simulations Jarett Hailes November 1, 2002 dX t =...

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

Page 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 + …

Prediction in Interacting Systems: Prediction in Interacting Systems: Applications & SimulationsApplications & Simulations

Jarett HailesNovember 1, 2002

dXt = μ(Xt)dt + σ(Xt)dBt

dx = this->mu()*dt + …

Page 2: 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 + …

OutlineOutline

• Refining Grid Stochastic Filter

– Description

– Characteristics

• Performer Tracking Problem

– Model

– Simulation

Page 3: 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 + …

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 N1 x N2 x … x Nd cells

N1

N2

Page 4: 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 + …

Each cell contains:

42- Particle count

- Associated Rate

Page 5: 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 + …

Refining Grid Stochastic Filter (REST Filter)

Particles used to approximate unnormalized conditional distribution

Page 6: 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 + …

1

-12-1 -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

Page 7: 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 + …

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

Net Birth Rates

O

B

S

E

R

V

A

T

I

O

N

Page 8: 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 + …

Tree Node Cell Node

4

2

1

6

1 7

3

Observation: -2 Particles

14

Page 9: 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 + …

N2

N1

Dynamic Cell Sizing

Zoom in: N1Zoom out: N1

Page 10: 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 + …

Dynamic Cell Sizing Example

Page 11: 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 + …

REST Advantages

- Less simulation noise than particle filters

- Dynamic cell sizing, inherent parameter estimation

- Dynamic domain problems

Page 12: 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 + …

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

Page 13: 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 + …

Audience

Page 14: 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 + …

Performer Model

t

tt

tt

tt

ttttt

ftttftavgft

ztttzzt

ttt

ttt

dB

dtyx

y

yx

xXd

dBffffdtffdf

dBzzzzdtzz

dz

dtfdy

dtfdx

2

22

2

222

minmax

minmaxminmax

))(sin())(cos()(

))(()(

))((3

2

)sin(

)cos(

1

θAudience

Page 15: 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 + …

Audience

Observation Model

otherwise 1

)1,0( if ),(

)()+()+(),( 222

pUWSXhY

zSySxSSXh

mtl

tt

lz

ly

lx

lt

mm

m

m

S1(x,y,z)

S3(x,y,z) S4(x,y,z)

S2(x,y,z)

Page 16: 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 + …