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Dynamic Bayesian Networks for Modeling Count Data Over Time

Learn to model dynamic count data over time using Bayesian networks. Explore single and multiple sensor results, future work, event detection, and adaptive event detection methods. Discover the marriage of probability theory and graph theory with Graphical Models.

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Dynamic Bayesian Networks for Modeling Count Data Over Time

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  1. Modeling Count Data over Time Using Dynamic Bayesian NetworksJonathan HutchinsAdvisors: Professor Ihler and Professor Smyth

  2. Sensor Measurements Reflect Dynamic Human Activity Optical People Counter at a Building Entrance Loop Sensors on Southern California Freeways

  3. Outline • Introduction, problem description • Probabilistic model • Single sensor results • Multiple sensor modeling • Future Work

  4. Modeling Count Data p(count|λ) count In a Poisson distribution: mean = variance = λ

  5. Simulated Data variance mean people count 15 weeks, 336 time slots

  6. Building Data variance mean people count

  7. Freeway Data variance mean people count

  8. One Week of Freeway Observations

  9. One Week of Freeway Data

  10. Detecting Unusual Events: Baseline Method Ideal model car count Baseline model car count Unsupervised learning faces a “chicken and egg” dilemma

  11. Persistent Events Notion of Persistence missing from Baseline model

  12. Quantifying Event Popularity Ideal model Baseline model

  13. My contribution Adaptive event detection with time-varying Poisson processes A. Ihler, J. Hutchins, and P. Smyth Proceedings of the 12th ACM SIGKDD Conference (KDD-06), August 2006. • Baseline method, Data sets, Ran experiments • Validation Learning to detect events with Markov-modulated Poisson processes A. Ihler, J. Hutchins, and P. Smyth ACM Transactions on Knowledge Discovery from Data, Dec 2007 • Extended the model to include a second event type (low activity) • Poisson Assumption Testing Modeling Count Data From Multiple Sensors: A Building Occupancy Model J. Hutchins, A. Ihler, and P. Smyth IEEE CAMSAP 2007,Computational Advances in Multi-Sensor Adaptive Processing, December 2007.

  14. Graphical Models "Graphical models are a marriage between probability theory and graph theory. They provide a natural tool for dealing with two problems that occur throughout applied mathematics and engineering -- uncertainty and complexity” Michael Jordan 1998

  15. Directed Graphical Models • Nodes  variables hidden Observed Count observed Event Rate Parameter

  16. B A C Directed Graphical Models • Nodes  variables • Edges  direct dependencies

  17. Graphical Models: Modularity Observed Countt+1 Observed Countt+2 Observed Countt-2 Observed Countt-1 Observed Countt

  18. Graphical Models: Modularity Poisson Rate λ(t) hidden Day, Timet-1 Day, Timet Day, Timet+1 observed Normal Countt-1 Normal Countt-1 Normal Countt-1 Observed Countt+1 Observed Countt-1 Observed Countt

  19. Graphical Models: Modularity Poisson Rate λ(t) hidden Day, Timet-1 Day, Timet Day, Timet+1 observed Normal Countt-1 Normal Countt-1 Normal Countt-1 Observed Countt+1 Observed Countt-1 Observed Countt

  20. Graphical Models: Modularity Poisson Rate λ(t) hidden Day, Timet-1 Day, Timet Day, Timet+1 observed Normal Countt-1 Normal Countt-1 Normal Countt-1 Observed Countt+1 Observed Countt-1 Observed Countt Eventt-1 Eventt Eventt+1

  21. Graphical Models: Modularity Poisson Rate λ(t) hidden Day, Timet-1 Day, Timet Day, Timet+1 observed Normal Countt-1 Normal Countt-1 Normal Countt-1 Observed Countt+1 Observed Countt-1 Observed Countt Eventt-1 Eventt Eventt+1 Event State Transition Matrix

  22. Poisson Rate λ(t) Day, Timet-1 Day, Timet Day, Timet+1 hidden Normal Countt-1 Normal Countt-1 Normal Countt-1 observed Observed Countt-1 Observed Countt Observed Countt+1 Event Countt-1 Event Countt Event Countt+1 Eventt-1 Eventt Eventt+1 Event State Transition Matrix

  23. α Poisson Rate λ(t) Day, Timet-1 Day, Timet Day, Timet+1 hidden Normal Countt-1 Normal Countt-1 Normal Countt-1 observed Observed Countt-1 Observed Countt Observed Countt+1 Event Countt-1 Event Countt Event Countt+1 Eventt-1 Eventt Eventt+1 η η η Event State Transition Matrix β

  24. Poisson Rate λ(t) Day, Timet-1 Day, Timet Day, Timet+1 hidden Normal Countt-1 Normal Countt-1 Normal Countt-1 observed Observed Countt-1 Observed Countt Observed Countt+1 Event Countt-1 Event Countt Event Countt+1 Eventt-1 Eventt Eventt+1 Event State Transition Matrix Markov Modulated Poisson Process (MMPP) model e.g., see Heffes and Lucantoni (1994), Scott (1998)

  25. Approximate Inference

  26. Gibbs Sampling * * * * * * * * * * * * * * * * * *

  27. Gibbs Sampling * * * * * y * * * x

  28. Block Sampling

  29. Gibbs Sampling Poisson Rate λ(t) Day, Timet-1 Day, Timet Day, Timet+1 Normal Countt-1 Normal Countt-1 Normal Countt-1 Observed Countt-1 Observed Countt Observed Countt+1 Event Countt-1 Event Countt Event Countt+1 Eventt-1 Eventt Eventt+1 Event State Transition Matrix

  30. Gibbs Sampling Poisson Rate λ(t) Poisson Rate λ(t) Poisson Rate λ(t) For the ternary valued event variable with chain length of 64,000 Brute force complexity ~ Day, Timet-1 Day, Timet Day, Timet+1 Normal Countt-1 Normal Countt-1 Normal Countt-1 Observed Countt-1 Observed Countt Observed Countt+1 Event Countt-1 Event Countt Event Countt+1 Eventt-1 Eventt Eventt+1 Event State Transition Matrix Event State Transition Matrix Event State Transition Matrix

  31. Gibbs Sampling Poisson Rate λ(t) Poisson Rate λ(t) Day, Timet-1 Day, Timet-1 Observed Countt-1 Observed Countt-1 Poisson Rate λ(t) Event Countt-1 Event Countt-1 Day, Timet-1 Normal Countt-1 Normal Countt-1 Observed Countt-1 Event Countt-1 Normal Countt-1 Eventt-1 Eventt Eventt+1 A A A

  32. Chicken/Egg Delima car count car count

  33. Event Popularity car count car count

  34. Persistent Event Notion of Persistence missing from Baseline model

  35. Persistent Event

  36. Detecting Real Events: Baseball Games Remember: the model training is completely unsupervised, no ground truth is given to the model

  37. Multi-sensor Occupancy Model Modeling Count Data From Multiple Sensors: A Building Occupancy Model J. Hutchins, A. Ihler, and P. Smyth IEEE CAMSAP 2007,Computational Advances in Multi-Sensor Adaptive Processing, December 2007

  38. Where are the People? Building Level City Level

  39. Sensor Measurements Reflect Dynamic Human Activity Optical People Counter at a Building Entrance Loop Sensors on Southern California Freeways

  40. Application: Multi-sensor Occupancy Model CalIt2 Building, UC Irvine campus

  41. Building Occupancy, Raw Measurements Occt = Occt-1 + inCountst-1,t – outCountst-1,t

  42. Over-counting Building Occupancy: Raw Measurements Under-counting Noisy sensors make raw measurements of little value

  43. Adding Noise Model Poisson Rate λ(t) Day, Timet-1 Day, Timet Normal Countt-1 Normal Countt-1 True Countt-1 Observed Countt-1 True Countt Observed Countt Event Countt-1 Event Countt Eventt-1 Eventt Event State Transition Matrix

  44. Probabilistic Occupancy Model Time t Time t+1 Constraint Time Occupancy Occupancy Out(Exit) Sensors Out(Exit) Sensors In(Entrance) Sensors In(Entrance) Sensors

  45. 24 hour constraint Geometric Distribution, p=0.5 Constraint    Occupancy       Building Occupancy 47

  46. Learning and Inference Gibbs Sampling | Forward-Backward | Complexity Occupancy Occupancy Out(Exit) Sensors Out(Exit) Sensors In(Entrance) Sensors In(Entrance) Sensors

  47. Typical Days Building Occupancy Thursday Friday Saturday

  48. Missing Data Building Occupancy time

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