Transcript Modeling and Inference with Relational Dynamic Bayesian Networks

Modeling and Inference with
Relational Dynamic Bayesian
Networks
Cristina Manfredotti
[email protected]
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Tracking
From: Prof. D. Hogg (University of Leeds) web site.
Estimate current position and trajectories
given uncertain sensors
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Multi Target Tracking
Priority Role
Sailing together
Thanks to Davide Piazza for the videos.
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Activity Recognition
Rendezvous
Priority Role
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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 A
Id
color
position(t)
velocity(t)
direction(t)
DecreasingVelocity(t)
Boat B
Id
color
position(t)
velocity(t)
direction(t)
DecreasingVelocity(t)
SameDirection(t)
distance(t)
SameDirection(t)
distance(t)
1Attributes
and relations are predicate in FOL.
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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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A large BN
Thanks to Mark Chavira
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RBN
• Syntax RBN:
predicate
– a set of nodes, one for each variable
– a directed, acyclic
graph graph
– a conditional distribution for each node
given its parents
parents,
To guarantee acyclicity predicates must
be ordered.
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Closing the parenthesis: Alarm RBN
Earthquacke
Neigh.DegOfDef
Alarm.Volume
Neigh.NoiseAround
NeighborCalls
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Relational State
The State of a Relational Domain is the
set of the predicates that are true in
the Domain.
State of attributes
Relational state
s 
s   r 
s
 
a
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
Id
color
position(t-1)
velocity(t-1)
…
Sensor Model
Boat
Boat
Id
color
position(t)
velocity(t)
…
SameDirection(t)
..
SameDirection(t-1)
..
Transition model
Zt-1
Zt
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Inference
Under Markov assumption
Bayesian Filter algorithm:
Belief: bel(st) = p(st|z1:t)
Transition Model
= kp(zt|st)s p(st|st-1)bel(st-1)dst-1
Sensor Model
Relations in the State result in correlating the
State of different instantiations between them
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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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Transition Model (2nd assumpt.)
p(st|st-1) = p(sat,srt|sat-1, srt-1)
Sat-1
Sat
Srt-1
Srt
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Relational
Transition
Model
Relational
Inference
p(sat,srt|sat-1,srt-1) =p(sat|sat-1,srt-1) p(srt|sat-1,srt-1, sat)
But srt independent by sat-1 given srt-1 and sat
p(sat,srt|sat-1,srt-1) = p(sat|sat-1,srt-1) p(srt|srt-1, sat)
p(zt|sat,srt) = p(zt|sat)
bel(st) = p(st|z1:t) = p(sat,srt|z1:t)
bel(st)=kp(zt|sat,srt)s p(sat,srt|sat-1,srt-1)bel(st-1)dst-1
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Particle Filtering* (general case)
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
* 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..
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Relational Particle Filter (RPF)
Fix the number of particles: M
1. Particle generation:
• sta[m] ~ p(sat|sat-1,srt-1)
• st r[m] ~ p(srt|srt-1, sat= sa[m]t)
Sense the measure at time t: zt
2a. Weight computation wt*[m]= p(zt|sat)
2b. Weight normalization wt[m]=wt*[m]/(wt*[m])
3. Resampling
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RPF (1)
Sa[m]t
Sr[m]t
Sa[m]t

p(sat|sat-1,srt-1)
Sa[m]t
sr[m]t

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p(sr[m]t|srt-1, sat=sa[m]t)
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RPF (2)
Weight (
Sa[m]t
Sr[m]t
)

p(zt|sat)
The weighting step is done according to the
instantiation part of each particle only, the
relational part follows.
The consistency of the probability function ensures the
convergence of the algorithm.
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Tracking AND Activity Recognition
1° step of sampling:
prediction of the state of attributes
Sa[m]t
Sa[m]t
Sa[m]t
Sr[m]t
Sa[m]t+1
Xao{t,(m)}
X {t,(m)}
Sr[m]t
2° step of sampling:
prediction of the state of relations
Or activity prediction
Sa[m]t
Sa[m]t+1
Xao{t,(m)}
X {t,(m)}
Xao{t,(m)}
X {t,(m)}
Sr[m]t
Sr[m]t+1
Sr[m]t
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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 ...
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