change detection in dynamic environments mark steyvers scott brown uc irvine this work is supported...
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Change Detection in Dynamic Environments
Mark Steyvers
Scott Brown
UC Irvine
This work is supported by a grant from the US Air Force Office of Scientific Research (AFOSR grant number FA9550-04-1-0317)
Overview
• Experiments with dynamically changing environments
• Task: Given a sequence of random numbers, predict the next one
• Questions:
– How do observers detect changes?
– What are the individual differences? “Jumpiness”
• Bayesian models + simple process models
= observed data
= prediction
Two-dimensional prediction task
• 11 x 11 button grid• Touch screen monitor• 1500 trials • Self-paced • Same sequence for all
subjects
0
5
10
0
5
10
Subject 4
40 50 60 70 80 90 100 110 120 1300
5
10
Time
Subject 12
Sequence Generation
• (x,y) locations are drawn from two binomial distributions of size 10, and parameters θ
• At every time step, probability 0.1 of changing θ to a new random value in [0,1]
• Example sequence:
Time
θ=.12 θ=.95 θ=.46 θ=.42 θ=.92 θ=.36
Example Sequence
0
5
10
Optimal Bayesian Solution
0
5
10
Subject 4
40 50 60 70 80 90 100 110 120 1300
5
10
Time
Subject 12
= observed sequence
Bayesian Solution= prediction
0
5
10
Optimal Bayesian Solution
0
5
10
Subject 4
40 50 60 70 80 90 100 110 120 1300
5
10
Time
Subject 12
Subject 4 – change detection too slow0
5
10
Optimal Bayesian Solution
0
5
10
Subject 4
40 50 60 70 80 90 100 110 120 1300
5
10
Time
Subject 12
Subject 12 – change detection too fast
(sequence from block 5)
Tradeoffs
• Detecting the change too slowly will result in lower accuracy and less variability in predictions than an optimal observer.
• Detecting the change too quickly will result in false detections, leading to lower accuracy and higher variability in predictions.
0.5 1 1.5 2 2.5 3 3.52.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
Mean Absolute Movement
Me
an
Abs
olu
te T
ask
Err
or
12
34
5
6
7
8 9
OPTIMAL SOLUTION
Average Error vs. Movement
= subject
“Ideal” observer: inferring the HMM that generated the data
2x
2
2y
1x
1
1y
tx
t
ty
...
Time 1 2 t t+1
Measurements
states
changepoints
Change probability
1tx
1t
?
Gibbs sampling
Model predictionInferred changepoint
• Sample from distribution over change points. • Prediction is based on average measurement after last inferred
changepoint
0.5 1 1.5 2 2.5 3 3.52.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
Mean Absolute Movement
Me
an
Abs
olu
te T
ask
Err
or
12
34
5
6
7
8 9
OPTIMAL SOLUTION
Average Error vs. Movement
= subject
A simple process model
1. Make new prediction some fraction α of the way between recent outcome and old prediction α = change proportion
2. Fraction α is a linear function of the error made on last trial
3. Two free parameters: A, B
A<B bigger jumps with higher error
A=B constant smoothing
1 (1 )t t tp y p
t t tError y p
α
0
1
A
B
BA
Average Error vs. Movement
0.5 1 1.5 2 2.5 3 3.52.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
Mean Absolute Movement
Me
an
Abs
olu
te T
ask
Err
or
12
34
5
6
7
8 9
OPTIMAL SOLUTION
= subject
= model
One-dimensional Prediction Task
1
2
3
4
5
6
7
8
9
10
11
12
12 PossibleLocations
• Where will next blue square arrive on right side?
Average Error vs. Movement
0 0.5 1 1.5 2
1.2
1.4
1.6
1.8
2
2.2
2.4
Mean Absolute Movement
Me
an
Abs
olu
te T
ask
Err
or
12
3
4
56
7
8
910
11
12
13
14
15
16
17
1819
2021
OPTIMAL SOLUTION
= subject
= model
New Experiments
• Prediction judgments might not be best measurement for assessing psychological change
• New experiments:
– Inference judgment: what currently is the state of the system?
Inference Task(aka filtering)
2x
2
2y
1x
1
1y
tx
t
ty
...
Time 1 2 t t+1
1tx
?
1ty
What is the cause of yt+1?
Tomato Cans Experiment
• Cans roll out of pipes A, B, C, or D
• Machine perturbs position of cans (normal noise)
• At every trial, with probability 0.1, change to a new pipe (uniformly chosen)
(real experiment has response buttons and is subject paced)
A B C D
Tomato Cans Experiment
(real experiment has response buttons and is subject paced)
A B C D • Cans roll out of pipes A, B, C, or D
• Machine perturbs position of cans (normal noise)
• At every trial, with probability 0.1, change to a new pipe (uniformly chosen)
• Curtain obscures sequence of pipes
Tasks
A B C D• Inference:
what pipe produced the last can?
A, B, C, or D?
Cans Experiment
• 136 subjects
• 16 blocks of 50 trials
• Vary change probability across blocks
– 0.08
– 0.16
– 0.32
• Question: are subjects sensitive to the number of changes?
50
60
70
80
90
100low alpha
50
60
70
80
90
100
% A
ccu
racy
(ag
ain
st T
rue
)med alpha
0 10 20 30 40 50 6050
60
70
80
90
100
% Changes
high alpha
ideal
ideal
ideal
Prob. = .08
Prob. = .16
Prob. = .32
Plinko/ Quincunx Experiment
Physical version Web version
Conclusion
• Adaptation in non-stationary decisionenvironments
• Individual differences
– Over-reaction: perceiving too much change
– Under-reaction: perceiving too little change
Number of Perceived Changes per Subject
Low medium high
Change Probability
(Red line shows ideal number of changes)
Subject #1
Number of Perceived Changes per Subject
55% of subjects show increasing pattern
45% of subjects show non-increasing pattern
Low, medium, high change probability Red line shows ideal number of changes
Tasks
A B C D• Inference:
what pipe produced the last can?
A, B, C, or D?
• Prediction: in what region will the next can arrive?
1, 2, 3, or 4?
1 2 3 4
Cans Experiment 2
• 63 subjects
• 12 blocks
– 6 blocks of 50 trials for inference task
– 6 blocks of 50 trials for prediction task
– Identical trials for inference and prediction
INFERENCE PREDICTION
Sequence
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1
2
3
4
5
6
1 2 3 4 5 6A B C D
Trial
INFERENCE PREDICTION
Sequence
Ideal Observer
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1
2
3
4
5
6
1 2 3 4 5 6A B C D
Trial
INFERENCE PREDICTION
Sequence
Ideal Observer
Individualsubjects
Trial0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1
2
3
4
5
6
1 2 3 4 5 6A B C D
INFERENCE PREDICTION
Sequence
Ideal Observer
Trial0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1
2
3
4
5
6
1 2 3 4 5 6A B C D
Individualsubjects
INFERENCE PREDICTION
Sequence
Ideal Observer
Trial0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1
2
3
4
5
6
1 2 3 4 5 6A B C D
Individualsubjects
0 20 40 6030
40
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90
100
Changes (%)
Acc
ura
cy
0 20 40 6030
40
50
60
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80
90
100
Changes (%)
Acc
ura
cy
= Subjectideal
ideal
INFERENCE PREDICTION
0 20 40 6030
40
50
60
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80
90
100
Changes (%)
Acc
ura
cy
ideal
0 20 40 6030
40
50
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90
100
Changes (%)
Acc
ura
cy
ideal
INFERENCE PREDICTION
= Process model
= Subject