Application of Graphical Models

1
Application of Graphical Models

• Modeling Physiological Data with Conditional
  Random Fields

Chieu Hai Leong 23 March 2005
2
Outline

• The Problem
• Physiological Data for Activity Recognition
• Physiological Data Modeling Contest
• The Approach
• Conditional Random Fields (CRF)

3
Physiological Data

• Examples of Physiological signals
• Temperatures
• Galvanic skin response (electrical skin
  conductance)
  
• Electrocardiogram
• Wearable devices for signal collection
• Health/fitness tracking
• Weight management

4
Physiological Data Physiological Data Modeling
Contest

• An ICML 2004 Workshop

Gender
3004 Watching TV
5102 Sleeping

• Age
• Handedness (left or right)

5
Physiological Data

• Bodymedia armband
• measures
• Heat flux
• Galvanic skin response
• Skin temperature
• Near body temperature
• Accelerometers
• Pedometer

6
Physiological Data

• Data
• Minute-by-minute sensor readings
• Organized into sessions
• Each minute annotated with activity code
• Some statistics
• Training data 10,000 hours from 18 subjects
• Test data 12,000 hours from 30 subjects

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Physiological Data Physiological Data Modeling
Contest

• Before the workshop

Gender
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 5
Sensor 6
Sensor 7 Sensor 8 Sensor 9
Activity 3004
Activity 5102
Characteristics 1 Characteristics 2
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Physiological Data Physiological Data Modeling
Contest

• After the workshop

Gender

• GSR
• Heat flux
• Near body temperature
• Pedometer
• Skin temperature
• Longitudinal acceleration (2 values)
• Traverse acceleration (2 values)

3004 Watching TV
5102 Sleeping

• Age
• Handedness (left or right)

9
Physiological Data Physiological Data Modeling
Contest

• Activity 1
• Positive examples 3004
• Ambiguous activities 0, 3003, 5199, 5101
• Negative examples All other annotations
• Activity 2
• Positive examples 5102
• Ambiguous activities 0, 5103, 2901, 2902
• Negative examples All other annotations

10
The Approach

• Conditional Random Fields

11
Conditional Random Fields
CRFs are discriminative, undirected graphical mo
dels
12
Conditional Random Fields

• Discriminative
• Directly estimates P(YX)
• Does not model P(X)
• Undirected vs. directed

undirected
directed
13
Conditional Random Fields

• Linear Chain CRF

Definition Let G (V,E) be a graph such that
Y(Yv)v?V, so that Y is indexed by the vertices
of G. Then (X,Y) is a conditional random field in
case, when conditioned on X, the random variables
Yv obey the Markov property with respect to the
graph p(YvX,Yw, w?v) p(YvX,Yw, wv), where
wv means that w and v are neighbors in G.
For example, P(IntelligenceG,X) P(Intelligence
Grade,SAT,X)
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Conditional Random Fields

• CRF in activity recognition
• Linear chain structure

Yi Activity at minute i
One Physiological Session
Xi Sensor/Characteristics at minute i
15
Conditional Random Fields

• Why not Hidden Markov Models?
• HMM is generative
• need to model continuous sensor values
• are Gaussians (or mixtures) good models of sensor
  values and characteristics?
  

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ApproachConditional Random Fields

• Advantage over HMM
• HMM
• Label Bias Problem of HMM
• States with a single outgoing transition ignores
  their observations

P0.99
1
99x
Locally Normalized
0
2
1x
P0.01
17
Conditional Random Fields

• Exponential family
• where C is the set of cliques in the graph.

Globally Normalized
18
Conditional Random Fields

• Linear Chain CRF

Cliques
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Conditional Random Fields

• Linear Chain CRF

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Conditional Random Fields

• Linear Chain CRF
• Likelihood
• Log Likelihood
• Objective maximize log likelihood of training
  data
  
• Gradient descent for a convex objective function
• Guaranteed convergence to global minimum

21
Conditional Random Fields

• Gradient
• In the form of
• empirical feature counts expected feature
  counts
  
• Gradient requires calculation of marginal
  probabilities
  
• P(yix) and P(yi,yi1x)
• Forward backward algorithm

22
Conditional Random Fields

• Forward Backward Algorithm
• Transition Matrix

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Conditional Random Fields

• Forward Backward Algorithm
• Forward, backward vectors

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Conditional Random Fields

• Forward Backward Algorithm
• Probabilities

25
Conditional Random Fields

• Inference
• Prediction of labels given observations

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Conditional Random Fields

• Partially Labeled Chain

start
Y1
Yp-1
Yp
Yq
Yn
stop
Yq1
Yp1



CRF
X1
Xp-1
Xp
Xq
Xn
Xq1
Xp1
Labeled
Unlabeled or ambiguous
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Conditional Random Fields

• Partially Labeled Chain
• Expectation Maximization (E.M.) within the
  unlabeled chain
  
• E-step expected feature counts in unlabeled
  nodes
  
• M-step maximize log likelihood as before

start
Y1
Yp-1
Yp
Yq
Yq1
Yp1



X1
Xp-1
Xp
Xq
Xq1
Xp1
28
Physiological Data Physiological Data Modeling
Contest

• After the workshop

?
Gender

• GSR
• Heat flux
• Near body temperature
• Pedometer
• Skin temperature
• Longitudinal acceleration (2 values)
• Traverse acceleration (2 values)

3004 Watching TV
5102 Sleeping

• Age
• Handedness (left or right)

29
Conditional Random Fields

• Gender prediction

Mixture CRF
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Conditional Random Fields

• Cliques

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Conditional Random Fields

• Gender prediction

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Conditional Random Fields

• Gender prediction

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Conditional Random Fields

• Gender prediction

Y1
Yi-1
Yi
Yi1
Yn
Y1
Yi-1
Yi
Yi1
Yn
stop
start
stop
start




G0
G1
X1
Xi-1
Xi
Xi1
Xn
X1
Xi-1
Xi
Xi1
Xn
34
Conditional Random Fields

• Gender prediction

35
Conditional Random Fields

• Inference
• Prediction of labels given observations

36
Conditional Random Fields

• Inference
• Prediction of labels given observations

37
Experimental Results
38
Results
Black CRF-Mixture, Gray CRFLinear, Grady
dotted CRFLinear-EM
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Results
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Comparison of Mixture CRF with Linear Chain CRF
Sleep
TV
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Usefulness of unlabeled instances
X-axis number of sequences in training data
Y-axis score