Random Walk Models

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Random Walk Models
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Agenda

• Final project presentation times?
• Random walk overview
• Local vs. Global model analysis
• Nosofsky Palmeri, 1997

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1-D Random Walk
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1-D Random Walk
Unbounded
p0,-1
p-1,-2
p1,0
p2,1
p-2,-3
p3, 2
S0
S1
S2
S-1
S-2


p-2, -1
p-1, 0
p0,1
p1, 2
p2, 3
p-3, -2
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1-D Random Walk
1 side bounded, 1 unbounded
p0,-1
p-1,-2
p1,0
p2,1
p-2,-3
S0
S1
S2
S-1
S-2
p2, 2

p-2, -1
p-1, 0
p0,1
p1, 2
p-3, -2
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1-D Random Walk
Bounded
p0,-1
p-1,-2
p1,0
p2,1
S0
S1
S2
S-1
S-2
p2, 2
p-2,-2
p-2, -1
p-1, 0
p0,1
p1, 2
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1-D Random Walk
1 absorbing state
p0,-1
p-1,-2
p1,0
p2,1
S0
S1
S2
S-1
S-2
p2, 2
1
0
p-1, 0
p0,1
p1, 2
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1-D Random Walk
2 absorbing states
p0,-1
p-1,-2
p1,0
0
S0
S1
S2
S-1
S-2
1
1
0
p-1, 0
p0,1
p1, 2
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2-D Random Walk
. . .
. . .
. . .
. . .
. . .




. . .
. . .
. . .
. . .
. . .
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1-D Random Walk Definition

• A 1-D random walk is a
• Markov chain
• where the states are ordered , S-2, S-1, S0, S1,
  S2,
• The transition probability between states Si and
  Sj are 0 unless Si Sj ? 1.

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1-D Random Walk
Unbounded
p0,-1
p-1,-2
p1,0
p2,1
p-2,-3
p3, 2
S0
S1
S2
S-1
S-2


p-2, -1
p-1, 0
p0,1
p1, 2
p2, 3
p-3, -2
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More on Random Walks

• Note that the states usually have real
  interpretations, but can be abstract placeholders.

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Real Interpretations
Neutral
Agitated
Angry
Upset
Sad
Loc0
Loc1
Loc2
Loc-1
Loc-2
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Placeholders
S0
S1
S2
S-1
S-2
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More on Random Walks

• Note that the time it takes to go from one state
  to another is often important

Neutral
Agitated
Angry
Upset
Sad
The subject was angry for 5 mins before
returning to an agitated state The subject
fluctuated rapidly between neutral and upset.
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Probability of Absorption at S2
p0,-1
p-1,-2
p1,0
0
S0
S1
S2
S-1
S-2
1
1
0
p-1, 0
p0,1
p1, 2
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Probability of Absorption at S2
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Probability of Absorption at S2
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Probability of Absorption at S2

• Transition up .25, down .75.
• Start in S0.

20
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• For many predictions, all this ugly algebra
  pretty much goes away if you use matrix algebra.

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Other Possible Calculations

• What is the probability that a particular state
  will be visited.
• How many times will a state be visited before
  absorption.
• What is the likelihood of a sequence of states
  being visited.
• How long will it take before absorption.

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Diffusion Process

• A diffusion process is a random walk in which
• The distance between states is very small
  (infinitesimal).
• The time it takes to transition between states is
  very small (infinitesimal).
• The process appears/is continuous.

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Local Fit Measures

• Local measure are based solely on the best
  fitting parameters
• How close can the model come to the data?
• Some measures are
• SSE
• ML
• PVAF
• A good fit is necessary for a model to be taken
  seriously.

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Sensitivity Analysis

• Sensitivity analyses
• Vary the parameters to see how robust the model
  fits are.
• If a good fit reflects a fundamental property of
  the model, then its behavior should be stable
  across parameter variation.
• Human data is noisy. A robust model will not be
  perturbed by small parameter changes.

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Sensitivity Analysis
yaxb
yax2bxc
SSE16.10
SSE11.45
SSE when Perturb params by Gau(0, .5)
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Cross Validation

• Cross validation
• Is a related to sensitivity analyses.
• Is a method by which a model if fit to half the
  data and tested on the other half.

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Cross Validation
yaxb
yax2bxc
SSE when fit to 1/2 of data
43.05
40.27
SSE when tested on other 1/2 of data
23.16
48.86
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Global Fit Measures

• Global measures try to incorporate information
  about the full range of behaviors that the model
  exhibits.
• Global measures tend to focus on how well a model
  can fit future, unseen data.
• Bayesian methods
• MDL
• Landscaping

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Global Fit Measures
Quadratic
Goodness of Fit (Bigger is better)
Linear
X
Data Space