Change Detection in Dynamic Environments Mark Steyvers Scott Brown UC Irvine This work is supported...
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Transcript of Change Detection in Dynamic Environments Mark Steyvers Scott Brown UC Irvine This work is supported...
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
50
60
70
80
90
100
Changes (%)
Acc
ura
cy
0 20 40 6030
40
50
60
70
80
90
100
Changes (%)
Acc
ura
cy
= Subjectideal
ideal
INFERENCE PREDICTION
0 20 40 6030
40
50
60
70
80
90
100
Changes (%)
Acc
ura
cy
ideal
0 20 40 6030
40
50
60
70
80
90
100
Changes (%)
Acc
ura
cy
ideal
INFERENCE PREDICTION
= Process model
= Subject