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Modeling and Inference with Relational Dynamic Bayesian Networks Cristina Manfredotti

Modeling and Inference with Relational Dynamic Bayesian Networks Cristina Manfredotti cristina.manfredotti@gmail.com. Tracking. Estimate current position and trajectories given uncertain sensors. From: Prof. D. Hogg (University of Leeds) web site. Multi Target Tracking. Priority Role.

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Modeling and Inference with Relational Dynamic Bayesian Networks Cristina Manfredotti

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  1. Modeling and Inference with Relational Dynamic Bayesian Networks Cristina Manfredotti cristina.manfredotti@gmail.com Cristina Manfredotti

  2. Cristina Manfredotti Tracking Estimate current position and trajectories given uncertain sensors From: Prof. D. Hogg (University of Leeds) web site.

  3. Cristina Manfredotti Multi Target Tracking Priority Role Sailing together Thanks to Davide Piazza for the videos.

  4. Cristina Manfredotti Activity Recognition Rendezvous Priority Role

  5. Cristina Manfredotti Desiderata • Model relations and • 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

  6. Cristina Manfredotti Boat Boat B Id color position(t)‏ velocity(t)‏ direction(t)‏ DecreasingVelocity(t) ‏ SameDirection(t)‏ distance(t)‏‏ Relational Domain Relational Domain: set of objects characterized by attributes1 and with relations1between them A Id color position(t)‏ velocity(t)‏ direction(t)‏ DecreasingVelocity(t)‏ SameDirection(t)‏ distance(t)‏‏ 1Attributes and relations are predicate in FOL.

  7. Cristina Manfredotti A Parenthesis: To model uncertainty in a Relational Domain we will use Relational (Dynamic) Bayesian Networks

  8. Cristina Manfredotti BN: the Alarm example

  9. 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. Cristina Manfredotti

  10. Cristina Manfredotti A large BN Thanks to Mark Chavira

  11. Cristina Manfredotti RBN • Syntax RBN: • a set of nodes, one for each predicate • a directed, graph • a conditional distribution for each node given its parents, • Syntax RBN: • a set of nodes, one for each variable • a directed, acyclic graph • a conditional distribution for each node given its parents To guarantee acyclicity predicates must be ordered.

  12. Cristina Manfredotti Closing the parenthesis: Alarm RBN Earthquacke Neigh.DegOfDef Alarm.Volume Neigh.NoiseAround NeighborCalls

  13. Cristina Manfredotti Relational State The State of a Relational Domain is the set of thepredicates that are true in the Domain. State of attributes Relational state State of relations

  14. Cristina Manfredotti State evolves with time Dynamics The State of a Relational Domain is the set of the predicates that are true in the Domain. We extend a RBN to a RDBN as we are used to extend a BN to a DBN.

  15. Cristina Manfredotti Relational Dynamic Bayesian Nets Sensor Model Boat Boat Id color position(t-1)‏ velocity(t-1)‏ … Id color position(t)‏ velocity(t)‏ … SameDirection(t)‏ ..‏‏ SameDirection(t-1)‏ ..‏‏ Transition model Zt-1 Zt

  16. Cristina Manfredotti Transition Model Sensor Model Under Markov assumption Bayesian Filter algorithm: Inference Belief: bel(st) = p(st|z1:t) = kp(zt|st)sp(st|st-1)bel(st-1)dst-1 Relations in the State result in correlating the State of different instantiations between them

  17. Cristina Manfredotti 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.

  18. Cristina Manfredotti Sat-1 Sat Srt Srt-1 Transition Model (2nd assumpt.) p(st|st-1) = p(sat,srt|sat-1, srt-1)

  19. Cristina Manfredotti 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 srtindependent by sat-1 given srt-1and 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)sp(sat,srt|sat-1,srt-1)bel(st-1)dst-1

  20. Particle Filtering* (general case) Fix the number of particles: M 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.. Cristina Manfredotti

  21. Cristina Manfredotti Relational Particle Filter (RPF) Fix the number of particles: M 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

  22. Cristina Manfredotti Sa[m]t Sr[m]t RPF (1) Sa[m]t p(sat|sat-1,srt-1) Sa[m]t p(sr[m]t|srt-1, sat=sa[m]t) sr[m]t

  23. Cristina Manfredotti RPF (2) Sa[m]t Weight ( p(zt|sat) ) Sr[m]t 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.

  24. Cristina Manfredotti Sa[m]t Sa[m]t+1 Xa{t,(m)} Xa{t,(m)} 1° step of sampling: prediction of the state of attributes Xo{t,(m)} Xo{t,(m)} Sa[m]t Sa[m]t Sr[m]t Sr[m]t+1 Sa[m]t Sa[m]t+1 Sr[m]t Xa{t,(m)} Sr[m]t Xo{t,(m)} 2° step of sampling: prediction of the state of relations Or activity prediction Sr[m]t Tracking AND Activity Recognition

  25. Cristina Manfredotti Exp: Canadian Harbor Constant speed

  26. Cristina Manfredotti Exp: Canadian Harbor Same speed

  27. Cristina Manfredotti FOPT for sat

  28. Cristina Manfredotti FOPT for srt

  29. Cristina Manfredotti Results

  30. Cristina Manfredotti 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

  31. Cristina Manfredotti Adding decisions ...

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