Modeling and Inference with Relational Dynamic Bayesian Networks Cristina Manfredotti
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Transcript of Modeling and Inference with Relational Dynamic Bayesian Networks Cristina Manfredotti
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Modeling and Inference with Relational Dynamic Bayesian
Networks
Cristina Manfredotti
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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 ...