Modeling and Inference with Relational Dynamic Bayesian Networks PhD Dissertation Cristina E....
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Modeling and Inference with Relational Dynamic Bayesian Networks PhD Dissertation Cristina E. Manfredotti XXII Ciclo (DISCo) Università degli Studi Milano-Bicocca
Modeling and Inference with Relational Dynamic Bayesian
Networks PhD Dissertation Cristina E. Manfredotti XXII Ciclo
(DISCo) Universit degli Studi Milano-Bicocca
[email protected] Enza Messina, Advisor David Fleet,
Co-advisor Domenico G. Sorrenti, Tutor
Cristina Manfredotti 3 Tracking Estimate current position and
trajectories given uncertain sensors From: Prof. D. Hogg
(University of Leeds) web site.
Slide 4
Cristina Manfredotti 4 Multi Target Tracking Thanks to Davide
Piazza for the videos. Sailing together Priority Role
Slide 5
Cristina Manfredotti 5 Activity Recognition Priority Role
Rendezvous
Slide 6
Cristina Manfredotti 6 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
Slide 7
Cristina Manfredotti 7 Relational Domain Relational Domain: set
of objects characterized by attributes 1 and with relations 1
between them Boat 1 Attributes and relations are predicate in FOL.
Id color position(t) velocity(t) direction(t) DecreasingVelocity(t)
SameDirection(t) distance(t) A Boat B Id color position(t)
velocity(t) direction(t) DecreasingVelocity(t) SameDirection(t)
distance(t)
Slide 8
Cristina Manfredotti 8 Relational Bayesian Networks To model
uncertainty in a Relational Domain we will use Relational (Dynamic)
Bayesian Networks
Slide 9
Cristina Manfredotti, DISCo, September 8 th 2009 9 RBNs Syntax
RBN: a set of nodes, one for each variable a directed, acyclic
graph a conditional distribution for each node given its parents
This distribution must take into account the actual complexity of
the nodes! 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.
Slide 10
Cristina Manfredotti 10 Relational State The State of a
Relational Domain is the set of the predicates that are true in the
Domain. Relational state State of attributes State of
relations
Slide 11
Cristina Manfredotti 11 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.
Slide 12
Cristina Manfredotti 12 Relational Dynamic Bayesian Nets Boat
Id color position(t-1) velocity(t-1) SameDirection(t-1).. Boat Id
color position(t) velocity(t) SameDirection(t).. Z t-1 ZtZt
Transition model Sensor Model
Slide 13
Cristina Manfredotti 13 Inference Under Markov assumption
Bayesian Filter algorithm: Belief: bel(s t ) = p(s t |z 1:t )
Relations in the State result in correlating the State of different
instantiations between them = kp(z t |s t ) s p(s t |s t-1 )bel(s
t-1 )ds t-1 Sensor Model Transition Model
Slide 14
Cristina Manfredotti 14 Measurement model (1 st assumpt.) part
of the state relative to relations, s r, not directly observable
p(z t |s t ) = p(z t |s a t ) observation z t independent by the
relations between objects. This measurement model only depends on
the part of the state of instances.
Slide 15
Cristina Manfredotti 15 p(s t |s t-1 ) = p(s a t,s r t |s a
t-1, s r t-1 ) S a t-1 S r t-1 SatSat SrtSrt Transition Model (2nd
assumpt.)
Slide 16
Cristina Manfredotti 16 Relational Transition Model p(s a t,s r
t |s a t-1,s r t-1 ) = But s r t independent by s a t-1 given s r
t-1 and s a t p(s a t,s r t |s a t-1,s r t-1 ) = p(s a t |s a t-1,s
r t-1 ) p(s r t |s r t-1, s a t ) bel(s t ) = p(s t |z 1:t ) = p(s
a t,s r t |z 1:t ) p(z t |s a t,s r t ) = p(z t |s a t ) Relational
Inference p(s a t |s a t-1,s r t-1 ) p(s r t |s a t-1,s r t-1, s a
t ) bel(s t )=kp(z t |s a t,s r t ) p(s a t,s r t |s a t-1,s r t-1
)bel(s t-1 )ds t-1
Slide 17
Cristina Manfredotti 17 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 s t [m] ~ p(s t |s t-1
) Sense the measure at time t: z t 2a. Weight computation w t *[m]
=p(z t |s t [m] ) 2b. Weight normalization w t [m] =w t *[m] / w t
*[m] ) 3. Resampling
Slide 18
Cristina Manfredotti 18 Relational Particle Filter (RPF) Fix
the number of particles: M 1.Particle generation: s t r [m] ~ p(s r
t |s r t-1, s a t = s a[m] t ) Sense the measure at time t: z t 2a.
Weight computation w t *[m] = p(z t |s a t ) 2b. Weight
normalization w t [m] =w t *[m] / w t *[m] ) 3. Resampling s t a
[m] ~ p(s a t |s a t-1,s r t-1 )
Slide 19
Cristina Manfredotti 19 RPF (1) S a[m] t S r[m] t S a[m] t p(s
a t |s a t-1,s r t-1 ) S a[m] t p(s r[m] t |s r t-1, s a t= s a[m]
t ) s r[m] t
Slide 20
Cristina Manfredotti 20 RPF (2) The consistency of the
probability function ensures the convergence of the algorithm. S
a[m] t S r[m] t Weight ( ) p(z t |s a t ) The weighting step is
done according to the instantiation part of each particle only, the
relational part follows.
Slide 21
Cristina Manfredotti 21 Exp: Canadian Harbor Constant
velocity
Slide 22
Cristina Manfredotti 22 Exp: Canadian Harbor Same velocity
Cristina Manfredotti 26 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
Slide 27
Cristina Manfredotti 27 Adding decisions...
Slide 28
Cristina Manfredotti 28 Challenges
Slide 29
Cristina Manfredotti 29 Particle filtering operations
Represents the posterior pdf with a set of random samples paired
with weights. Computes the filtering based on these weights: Sample
space Posterior pdf
Slide 30
Cristina Manfredotti 30 Relational Inference bel(s t ) = p(s t
|z 1:t ) = p(s i t,s r t |z 1:t ) bel(s t )=kp(z t |s i t,s r t )
p(s i t,s r t |s i t-1,s r t-1 )bel(s t-1 )ds t-1 p(z t |s i t,s r
t ) = p(z t |s i t ) p(s i t,s r t |s i t-1,s r t-1 ) = p(s i t |s
i t-1,s r t-1 ) p(s r t |s i t-1,s r t-1, s i t ) But s r t
independent by s i t-1 given s r t-1 and s i t p(s i t,s r t |s i
t-1,s r t-1 ) = p(s i t |s i t-1,s r t-1 ) p(s r t |s r t-1, s i t
)
Slide 31
Cristina Manfredotti 31 The alarm (famous) example I'm at work,
neighbor John calls to say my alarm is ringing, but neighbor Mary
doesn't call. Sometimes it's set off by minor earthquakes. Is there
a burglary? Variables: BurglarEnter, EarthquakeAppens, AlarmRings,
JohnCalls, MaryCalls Network topology reflects "causal" knowledge:
A burglar can set the alarm off An earthquake can set the alarm off
The alarm can cause Mary to call The alarm can cause John to
call
Slide 32
Cristina Manfredotti 32 Alarm Volume Sensibility ToRing...
Person DegOfDef NoiseAround Teleph DegOfBelieve
Being_honer/Being_neigh... Listening Calling Burglary Red words:
predicates, that concern only the object itself Dashed arrows:
relation between an object and an attribute of the object (or a
predicate) Green arrows: dependence between two attributes
Continouse arrows: relations between two objects Bold black words:
objects names Black words: objects attributes (caracteristic of the
variables, they make an instanciation of each object different by
each other). The Alarm Relational Domain (2)
Slide 33
Cristina Manfredotti 33 The overall system Belief about
correlated behavior Better prediction for the next state Better
track positions Activity recognition
Slide 34
Cristina Manfredotti 34 The importance of the context
Cristina Manfredotti 36 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.
Slide 37
Cristina Manfredotti 37 Conditional Probability Distribution
FOPT: a Probabilistic Tree whose nodes are FOL formulas CPD pos t
(x): relation t-1 (x,y) G(pos t-1 (x)) F(pos t-1 (x), pos t (y)) T
F CPD relation t (x,y): relation t-1 (x,y) F(pos t (x), pos t (y))
T F G(pos t (x), pos t (y))
Slide 38
Cristina Manfredotti 38 Related works Complex tracking tasks:
Heuristics, Mixed-States models Complex activity recognition:
Stochastic grammar Free, First Order Logic
Slide 39
Cristina Manfredotti 39 Activity Recognition: Stochastic
Parsing Y.A.Ivanov and A.F.Bobick Recognition of Visual Activities
and Interactions by Stochastic Parsing
Slide 40
Cristina Manfredotti 40 Activity recognition: First Order Logic
S. Tran and L. Davis, Visual Event Modeling and Recognition using
Markov Logic Networks
Slide 41
Cristina Manfredotti 41 Multi Target Tracking
Slide 42
Cristina Manfredotti 42 Activity Recognition
Slide 43
Cristina Manfredotti 43 The Alarm Relational Domain (1)
Relational Domain contains a set of objects with relations and/or
predicates between them Object-types e.g.: Relation neighbor alarm
burglar toCall (the honer of the house) toHear (the alarm)
neighbors attributes: his capacity of hearing, his attention,...
alarms attributes: its volume, its sensibility,... e.g.: Predicate
toRing
Slide 44
Cristina Manfredotti 44 BN: the Alarm example
Slide 45
Cristina Manfredotti 45 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.
Slide 46
Cristina Manfredotti 46 Thanks to Mark Chavira A large BN
Slide 47
Cristina Manfredotti 47 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
Cristina Manfredotti 49 Tracking AND Activity Recognition S
a[m] t S r[m] t S a[m] t S r[m] t S a[m] t X a {t,(m)} X o {t,(m)}
S r[m] t S a[m] t+1 1 step of sampling: prediction of the state of
attributes S a[m] t X a {t,(m)} X o {t,(m)} S r[m] t S a[m] t+1 X a
{t,(m)} X o {t,(m)} S r[m] t+1 2 step of sampling: prediction of
the state of relations Or activity prediction
Slide 50
System for activity recognition Cristina Manfredotti 50 From:
Prof. D. Hogg (University of Leeds) web site.
Slide 51
Cristina Manfredotti 51 Multi Target Tracking Thanks to Davide
Piazza for the videos. Sailing together Priority Role
Slide 52
Cristina Manfredotti 52 Activity Recognition Priority Role
Rendezvous
