Cristina Manfredotti 1

Modeling and Inference with Relational Dynamic Bayesian

Networks

Cristina Manfredotti

cristina.manfredotti@gmail.com

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Tracking

Estimate current position and trajectories given uncertain sensors

From: Prof. D. Hogg (University of Leeds) web site.

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Multi Target Tracking

Thanks to Davide Piazza for the videos.

Sailing together

Priority Role

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Activity Recognition

Priority Role

Rendezvous

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Desiderata

1. Model relations and 2. Maintain beliefs over particular

relations between objects

In order to simultaneously:

• Improve tracking with informed predictions and

• Identify complex activities based on observations and prior knowledge

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Relational Domain

Relational Domain: set of objects characterized by attributes1 and with relations1 between them

Boat

1Attributes and relations are predicate in FOL.

Idcolorposition(t)velocity(t)direction(t)DecreasingVelocity(t)

SameDirection(t)distance(t)

A

Boat BIdcolorposition(t)velocity(t)direction(t)DecreasingVelocity(t)

SameDirection(t)distance(t)

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A Parenthesis:

To model uncertainty in a Relational Domain we will use

Relational (Dynamic) Bayesian Networks

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BN: the Alarm example

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BNs: a drawback

Each node is a variable:

Two different nodes

If we would have 4 neighbors? We have to construct a graph with 2 more nodes.

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Thanks to Mark Chavira

A large BN

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• Syntax RBN:– a set of nodes, one for each variable

– a directed, acyclic graph – a conditional distribution for each node

given its parents

• Syntax RBN:– a set of nodes, one for each predicate

– a directed, graph– a conditional distribution for each node

given its parents,

To guarantee acyclicity predicates must be ordered.

RBN

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Closing the parenthesis: Alarm RBN

Alarm.Volume

NeighborCalls

Earthquacke

Neigh.DegOfDef

Neigh.NoiseAround

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Relational State

The State of a Relational Domain is the set of the predicates that are true in the Domain.

r

a

s

ss

Relational state

State of attributes

State of relations

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Dynamics

The State of a Relational Domain is the set of the predicates that are true in the Domain.

State evolves with time

We extend a RBN to a RDBN as we are used to extend a BN to a DBN.

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Relational Dynamic Bayesian Nets

Boat

Idcolorposition(t-1)velocity(t-1)…

SameDirection(t-1)..

Boat

Idcolorposition(t)velocity(t)…

SameDirection(t)..

Zt-1 Zt

Transition modelS

ensor Model

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Inference

Under Markov assumptionBayesian Filter algorithm:

Belief: bel(st) = p(st|z1:t)

Relations in the State result in correlating the State of different instantiations between them

= kp(zt|st)s p(st|st-1)bel(st-1)dst-1Sensor Model

Transition Model

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Measurement model (1st assumpt.)

part of the state relative to relations, sr, not directly observable

p(zt|st) = p(zt|sat)

observation zt independent by the relations between objects.

This measurement model only depends on the part of the state of instances.

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p(st|st-1) = p(sat,sr

t|sat-1, sr

t-1)

Sat-1

Srt-1

Sat

Srt

Transition Model (2nd assumpt.)

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Relational Transition Model

p(sat,sr

t|sat-1,sr

t-1) =

But srt independent by sa

t-1 given srt-1 and sa

t

p(sat,sr

t|sat-1,sr

t-1) = p(sat|sa

t-1,srt-1) p(sr

t|srt-1, sa

t)

bel(st) = p(st|z1:t) = p(sat,sr

t|z1:t)

bel(st)=kp(zt|sat,sr

t)s p(sat,sr

t|sat-1,sr

t-1)bel(st-1)dst-1

p(zt|sat,sr

t) = p(zt|sat)

Relational Inference

p(sat|sa

t-1,srt-1) p(sr

t|sat-1,sr

t-1, sat)

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Particle Filtering* (general case)

* It is a technique that implements a recursive Bayesian Filter through a Monte Carlo simulation. The key idea is to represent the posterior pdf as a set of samples (particles) paired with weights and to filter the mesurament based on these weights..

Fix the number of particles: M

1. Particle generation st[m] ~ p(st|st-1)

Sense the measure at time t: zt

2a. Weight computation wt*[m]=p(zt|st

[m])

2b. Weight normalization wt[m]=wt

*[m]/(wt*[m])

3. Resampling

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Relational Particle Filter (RPF)

Fix the number of particles: M

1. Particle generation:

• st r[m] ~ p(sr

t|srt-1, sa

t= sa[m]t)

Sense the measure at time t: zt

2a. Weight computation wt*[m]= p(zt|sa

t)

2b. Weight normalization wt[m]=wt

*[m]/(wt*[m])

3. Resampling

• sta[m] ~ p(sa

t|sat-1,sr

t-1)

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RPF (1)

Sa[m]t

Sr[m]t

Sa[m]t p(sa

t|sat-1,sr

t-1)

Sa[m]t p(sr[m]

t|srt-1, sa

t=sa[m]t)

sr[m]t

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RPF (2)

The consistency of the probability function ensures the convergence of the algorithm.

Sa[m]t

Sr[m]t

Weight ( ) p(zt|sat)

The weighting step is done according to the instantiation part of each particle only, the relational part follows.

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Tracking AND Activity Recognition

Sa[m]t

Sr[m]t

Sa[m]t

Sr[m]t

Sa[m]t

Xa{t,(m)}Xo{t,(m)}

Sr[m]t

Sa[m]t+1

1° step of sampling: prediction of the state of attributes

Sa[m]t

Xa{t,(m)}Xo{t,(m)}

Sr[m]t

Sa[m]t+1

Xa{t,(m)}Xo{t,(m)}

Sr[m]t+1

2° step of sampling: prediction of the state of relationsOr activity prediction

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Exp: Canadian Harbor

Constant speed

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Exp: Canadian Harbor

Same speed

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FOPT for sat

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FOPT for srt

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Results

RPF

True Positive rate 0.895

True Negative rate 0.611

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To conclude ...

• Modeling Relations “dynamically”:– To improve multi target tracking– To recognize complex activities

• Inference in Dynamic Relational Domain– In theory complex BUT

– Simplified by

• “smart decomposition” of the transition model

• “non-relational” sensor model

• Results are promising

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Adding decisions ...