Sequential Inference for Evolving Groups of Objects

2012-07-19

이범진Biointelligence Lab

Seoul National University

What are we going to do?

Think about dynamically evolving groups of objects¨ Ex)

flocks of birds Schools of fish Group of aircraft

However...

Difficulties on this research¨ 1. recognizing groups are hard¨ 2. incorporating new members into the groups,

Ex) splitting and merging of groups

How many groups?

Merging

Spliting

Proposed solution

Implementation rule¨ 1. Targets themselves are dynamic¨ 2. Targets’ grouping can change overtime¨ 3. Assignment of a target to a group affects the probabilistic

properties of the target dynamics¨ 4. Group statistics belong to a second hidden layer, target

statistics belong to the first hidden layer and the observation process usually depends only on the targets

¨ 5. Number of targets is typically unknown

Framework (1)

Dynamic group tracking model

𝒑 (𝑿𝟏: 𝒕 ,𝑮𝟏: 𝒕 ,𝒁𝟏 :𝒕 )=𝒑 (𝑿𝟏|𝑮𝟏 )𝒑 (𝑮𝟏)𝒑 (𝒁𝟏|𝑿𝟏)×∏𝒕 ′=𝟐

𝒕

𝒑 (𝑿 𝒕|𝑿 𝒕 −𝟏 ,𝑮 𝒕 ,𝑮𝒕 −𝟏 )𝒑 (𝑮𝒕|𝑮𝒕 −𝟏 , 𝑿 𝒕 −𝟏 )𝒑 (𝒁 𝒕∨𝑿 𝒕 )

G1

X1

Z1

G2

X2

Z2

Gt

Xt

Zt

Gt+1

Xt+1

Zt+1

Framework (2)

Main components of the group tracking model¨ 1. group dynamical model :

Describes motion of members in a group

¨ 2. group structure transition model Describes the way the group membership or group matic

states Xt

Markovian assumption

𝑝 (𝑋 𝑡∨𝑋 𝑡− 1 ,𝐺𝑡 ,𝐺𝑡− 1)

𝑝 (𝐺𝑡∨𝐺𝑡 −1 ,𝑋 𝑡− 1)

How do we inference?

Proposed MCMC-particle algorithm

Why is it better!?

No resampling is required ¨ Particle filters use MCMC to rejuvenate degenerate samples

after resampling

Less computationally intensive than the MCMC-based particle filter¨ Because avoids numerical integration of the predictive den-

sity at every MCMC iteration

Consider the general joint distribution of St and St-1

¨

,

How good is it?

How good is it?

Framework (2)

Main components of the group tracking model¨ 1. group dynamical model :

Describes motion of members in a group

¨ 2. group structure transition model Describes the way the group membership or group matic

states Xt

Markovian assumption

𝑝 (𝑋 𝑡∨𝑋 𝑡− 1 ,𝐺𝑡 ,𝐺𝑡− 1)

𝑝 (𝐺𝑡∨𝐺𝑡 −1 ,𝑋 𝑡− 1)

Experiments(1)

Ground target tracking¨ For group dynamical model(with repulsive force, virtual

leader) Use stochastic differential equations (SDEs) and Itô stochastic

calculus– Using velocity, position, acceleration, restoring force, etc.

¨ For state-dependent group structure transition model

¨ For observation model Using single discretized sensor model which scans a fixed rec-

tangular region, and track-before-detect approach(TBD)

𝑝 (𝐺𝑡|𝑋 𝑡− 1,𝑒𝑡−1 ,𝐺𝑡 −1 )={ 𝑃𝑁𝐶

(1−𝑃𝑁𝐶)�̂� (𝐺𝑡∨𝐺𝑡 −1 ,𝑋 𝑡 −1 ,𝑒𝑡− 1)If ) otherwise

Experiments(1)

Experiments(1) result

MCMC-particles algorithm is used to detect and track the group targets

Nburn = 1000 iteration for burn-in

Experiments(1) result cont.

Experiments(2)

Group stock selection¨ For group stock mean reversion model (dynamical model)

Use stochastic differential equations (SDEs) Formulation with ‘force’ which changes stock prices that

brings the stocks back into equilibrium¨ For state-independent group structure transition model

K possible groups G = group assignment πt = models the underlying proportion of targets in various

groups at time t |

Experiments(2) cont.

Experiments(2) cont.

Dynamic Dirichlet distribution ¨ Assumption

All the stocks are independent Stock prices starts at Z1,i = 0

¨ Transition is obtained from the log-distribution of the group stock mean

reversion model

Experiments(2) result

MCMC-particles algorithm is used to inference {Gt, πt}

These models can identify groupings of stock based only on their stock price behaviour

Thank you