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

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
In a Poisson distribution mean variance ?
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
(No Transcript)
10
One Week of Freeway Data
11
Detecting Unusual Events Baseline Method
Baseline model
car count
Unsupervised learning faces a chicken and egg
dilemma
12
Persistent Events
Notion of Persistence missing from Baseline model
13
Quantifying Event Popularity
Ideal model
Baseline model
14
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.

15
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
16
Directed Graphical Models

• Nodes ? variables

hidden
Observed Count
observed
Event
Rate Parameter
17
Directed Graphical Models

• Nodes ? variables
• Edges ? direct dependencies

18
Graphical Models Modularity
Observed Countt
Observed Countt-2
Observed Countt-1
Observed Countt2
Observed Countt1
19
Graphical Models Modularity
Poisson Rate ?(t)
Day, Timet-1
Day, Timet
Day, Timet1
Normal Countt-1
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
Observed Countt1
20
Graphical Models Modularity
21
Graphical Models Modularity
Poisson Rate ?(t)
Day, Timet-1
Day, Timet
Day, Timet1
Normal Countt-1
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
Observed Countt1
Eventt
Eventt-1
Eventt1
22
Graphical Models Modularity
Eventt
Eventt-1
Eventt1
Event State Transition Matrix
23
Poisson Rate ?(t)
Day, Timet-1
Day, Timet
Day, Timet1
Normal Countt-1
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
Observed Countt1
Event Countt
Event Countt-1
Event Countt1
Eventt
Eventt-1
Eventt1
Event State Transition Matrix
24
a
Poisson Rate ?(t)
Day, Timet-1
Day, Timet
Day, Timet1
Normal Countt-1
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
Observed Countt1
Event Countt
Event Countt-1
Event Countt1
Eventt
Eventt-1
Eventt1
?
?
?
Event State Transition Matrix
ß
25
Poisson Rate ?(t)
Day, Timet-1
Day, Timet
Day, Timet1
Normal Countt-1
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
Observed Countt1
Event Countt
Event Countt-1
Event Countt1
Eventt
Eventt-1
Eventt1
Event State Transition Matrix
Markov Modulated Poisson Process (MMPP) model
e.g., see Heffes and Lucantoni (1994), Scott
(1998)
26
Approximate Inference
27
Gibbs Sampling


















28
Gibbs Sampling





y



x
29
Block Sampling
30
Gibbs Sampling
Poisson Rate ?(t)
Day, Timet-1
Day, Timet
Day, Timet1
Normal Countt-1
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
Observed Countt1
Event Countt
Event Countt-1
Event Countt1
Eventt
Eventt-1
Eventt1
Event State Transition Matrix
31
Gibbs Sampling
Poisson Rate ?(t)
Poisson Rate ?(t)
Poisson Rate ?(t)
For the ternary valued event variable with
chain length of 64,000 Brute
force complexity
Day, Timet-1
Day, Timet
Day, Timet1
Normal Countt-1
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
Observed Countt1
Event Countt
Event Countt-1
Event Countt1
Eventt
Eventt-1
Eventt1
Event State Transition Matrix
Event State Transition Matrix
Event State Transition Matrix
32
Gibbs Sampling
Poisson Rate ?(t)
Poisson Rate ?(t)
Day, Timet-1
Day, Timet-1
Observed Countt-1
Observed Countt-1
Event Countt-1
Event Countt-1
Normal Countt-1
Normal Countt-1
Eventt
Eventt-1
Eventt1
A
A
A
33
(No Transcript)
34
Chicken/Egg Delima
car count
35
Event Popularity
car count
car count
36
Persistent Event
Notion of Persistence missing from Baseline model
37
Persistent Event
38
Detecting Real Events Baseball Games
Remember the model training is completely
unsupervised, no ground truth is given to the
model
39
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
40
Where are the People?
Building Level
City Level
41
Sensor Measurements Reflect Dynamic Human Activity
Optical People Counter at a Building Entrance
Loop Sensors on Southern California Freeways
42
Application Multi-sensor Occupancy Model
CalIt2 Building, UC Irvine campus
43
Building Occupancy, Raw Measurements
Occt Occt-1 inCountst-1,t outCountst-1,t
44
Building Occupancy Raw Measurements
Under-counting
Noisy sensors make raw measurements of little
value
45
Adding Noise Model
Poisson Rate ?(t)
Day, Timet-1
Day, Timet
Normal Countt-1
Normal Countt-1
Observed Countt
Observed Countt-1
True Countt-1
True Countt
Event Countt
Event Countt-1
Eventt
Eventt-1
Event State Transition Matrix
46
Probabilistic Occupancy Model
Time t
Time t1
Constraint
Occupancy
Occupancy
Out(Exit) Sensors
Out(Exit) Sensors
In(Entrance) Sensors
In(Entrance) Sensors
47
24 hour constraint
Geometric Distribution, p0.5
Constraint
? ? ?
Occupancy
? ? ?
? ? ?
Building Occupancy
47
48
Learning and Inference

• Gibbs Sampling Forward-Backward Complexity

Occupancy
Occupancy
Out(Exit) Sensors
Out(Exit) Sensors
In(Entrance) Sensors
In(Entrance) Sensors
49
Typical Days
Building Occupancy
Thursday Friday
Saturday
50
Missing Data
Building Occupancy
time
51
Corrupted Data
Building Occupancy
Thursday
Friday
52
Future Work

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

53
Sensor Failure Extension
54
Spatial Correlation
55
Four Off-Ramps
56
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 2007.
• 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 2006.
• 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