Modeling Count Data over Time Using Dynamic Bayesian Networks Jonathan Hutchins Advisors: Professor Ihler and Professor Smyth

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

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

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

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

mean people count variance Building Data

mean people count variance Freeway Data

One Week of Freeway Observations

One Week of Freeway Data

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

Persistent Events Notion of Persistence missing from Baseline model

Quantifying Event Popularity Ideal model Baseline model

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 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.

"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 Graphical Models

Nodes  variables Directed Graphical Models observed Observed Count hidden EventRate Parameter

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

Graphical Models: Modularity Observed Count t Observed Count t-2 Observed Count t-1 Observed Count t+2 Observed Count t+1

Graphical Models: Modularity hidden observed Poisson Rate λ(t) Normal Count t-1 Observed Count t Observed Count t-1 Observed Count t+1 Day, Time t-1 Normal Count t-1 Day, Time t Normal Count t-1 Day, Time t+1

Graphical Models: Modularity hidden observed Poisson Rate λ(t) Normal Count t-1 Observed Count t Observed Count t-1 Observed Count t+1 Day, Time t-1 Normal Count t-1 Day, Time t Normal Count t-1 Day, Time t+1

Graphical Models: Modularity Event t Event t-1 Event t+1 hidden observed Poisson Rate λ(t) Normal Count t-1 Observed Count t Observed Count t-1 Observed Count t+1 Day, Time t-1 Normal Count t-1 Day, Time t Normal Count t-1 Day, Time t+1

Graphical Models: Modularity Event t Event t-1 Event t+1 hidden observed Poisson Rate λ(t) Normal Count t-1 Observed Count t Observed Count t-1 Observed Count t+1 Day, Time t-1 Normal Count t-1 Day, Time t Normal Count t-1 Day, Time t+1 Event State Transition Matrix

Event t Event t-1 Event t+1 Event State Transition Matrix Observed Count t Observed Count t-1 Observed Count t+1 Event Count t Event Count t-1 Event Count t+1 hidden observed Poisson Rate λ(t) Normal Count t-1 Day, Time t-1 Normal Count t-1 Day, Time t Normal Count t-1 Day, Time t+1

Event t Event t-1 Event t+1 Event State Transition Matrix Observed Count t Observed Count t-1 Observed Count t+1 Event Count t Event Count t-1 Event Count t+1 hidden observed Poisson Rate λ(t) Normal Count t-1 Day, Time t-1 Normal Count t-1 Day, Time t Normal Count t-1 Day, Time t+1 β α η ηη

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

Approximate Inference

Gibbs Sampling * ** * * ** * ** * * * * * * * *

* x y ** * * ** *

Block Sampling

Gibbs Sampling Event t Event t-1 Event t+1 Event State Transition Matrix Observed Count t Observed Count t-1 Observed Count t+1 Event Count t Event Count t-1 Event Count t+1 Poisson Rate λ(t) Normal Count t-1 Day, Time t-1 Normal Count t-1 Day, Time t Normal Count t-1 Day, Time t+1

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

Gibbs Sampling Event t Event t-1 Event t+1 A AA Poisson Rate λ(t) Day, Time t-1 Observed Count t-1 Normal Count t-1 Event Count t-1 Poisson Rate λ(t) Day, Time t-1 Observed Count t-1 Normal Count t-1 Event Count t-1 Poisson Rate λ(t) Day, Time t-1 Observed Count t-1 Normal Count t-1 Event Count t-1

Chicken/Egg Delima car count

Event Popularity car count

Notion of Persistence missing from Baseline model Persistent Event

Detecting Real Events: Baseball Games Total Number Of Predicted Events Graphical Model Detection of the 76 known events Baseline Model Detection of the 76 known events %86.8% %81.6% %72.4% %60.5% Remember: the model training is completely unsupervised, no ground truth is given to the model

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

Where are the People? Building LevelCity Level

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

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

Building Occupancy, Raw Measurements Occ t = Occ t-1 + inCounts t-1,t – outCounts t-1,t

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

Adding Noise Model Event t Event t-1 Event State Transition Matrix Event Count t Event Count t-1 Poisson Rate λ(t) Normal Count t-1 Day, Time t-1 Normal Count t-1 Day, Time t Observed Count t Observed Count t-1 True Count t-1 True Count t

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

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

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

Typical Days Thursday Friday Saturday Building Occupancy

Missing Data Building Occupancy time

Corrupted Data Building Occupancy Thursday Friday

Future Work Freeway Traffic On and Off ramps 2300 sensors 6 months of measurements

Sensor Failure Extension

Spatial Correlation

Four Off-Ramps

Publications 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 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 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 Prediction and ranking algorithms for event-based network data J. O Madadhain, J. Hutchins, P. Smyth ACM SIGKDD Explorations: Special Issue on Link Mining, 7(2), 23-30, December 2005