Random Walk Models. Agenda Final project presentation times? Random walk overview Local vs. Global...
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Transcript of Random Walk Models. Agenda Final project presentation times? Random walk overview Local vs. Global...
Agenda
• Final project presentation times?
• Random walk overview
• Local vs. Global model analysis
• Nosofsky & Palmeri, 1997
1-D Random Walk
Unbounded
S0 S1 S2S-1S-2
p0,-1p-1,-2 p1,0 p2,1
p-2, -1 p-1, 0 p0,1 p1, 2 p2, 3p-3, -2
p-2,-3 p3, 2
… …
1-D Random Walk
1 side bounded, 1 unbounded
S0 S1 S2S-1S-2
p0,-1p-1,-2 p1,0 p2,1
p-2, -1 p-1, 0 p0,1 p1, 2
p2, 2
p-3, -2
p-2,-3
…
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.
1-D Random Walk
Unbounded
S0 S1 S2S-1S-2
p0,-1p-1,-2 p1,0 p2,1
p-2, -1 p-1, 0 p0,1 p1, 2 p2, 3p-3, -2
p-2,-3 p3, 2
… …
More on Random Walks
• Note that the states usually have real interpretations, but can be abstract placeholders.
More on Random Walks
• Note that the time it takes to go from one state to another is often important
Neutral Agitated AngryUpsetSad
The subject was “angry” for 5 mins before returning toan “agitated” state…
The subject fluctuated rapidly between “neutral” and “upset”.
Probability of Absorption at S2
S0 S1 S2S-1S-2
p0,-1p-1,-2 p1,0 0
0 p-1, 0 p0,1 p1, 2
11
€
No Loops : p0,1p1,2
One Loop : p0,1p1,0p0,1p1,2 or p0,−1p−1,0p0,1p1,2
Two Loops : p0,1p1,0p0,1p1,0p0,1p1,2 or p0,−1p−1,0p0,1p1,0p0,1p1,2 or
p0,1p1,0p0,−1p−1,0p0,1p1,2 or p0,−1p−1,0p0,−1p−1,0p0,1p1,2
Probability of Absorption at S2
€
No Loops : p0,1p1,2
One Loop : p0,1p1,0p0,1p1,2 or p0,−1p−1,0p0,1p1,2
= p0,1p1,0 + p0,−1p−1,0( )p0,1p1,2
Two Loops : p0,1p1,0p0,1p1,0p0,1p1,2 or p0,−1p−1,0p0,1p1,0p0,1p1,2 or
p0,1p1,0p0,−1p−1,0p0,1p1,2 or p0,−1p−1,0p0,−1p−1,0p0,1p1,2
= p0,1p1,0( )2
+ 2 × p0,1p1,0p0,−1p−1,0( ) + p0,−1p−1,0( )2
[ ]p0,1p1,2
= p0,1p1,0 + p0,−1p−1,0( )2p0,1p1,2
Three Loops : p0,1p1,0 + p0,−1p−1,0( )3p0,1p1,2
Probability of Absorption at S2
€
P(Absorption at S2) = P(Absorption at S2 after n loops)n= 0
∞
∑
= p0,1p1,2 p0,1p1,0 + p0,−1p−1,0( )n
n= 0
∞
∑
...after some algebra...
=p0,1p1,2
p0,1p1,2 + p0,−1p−1,−2
Probability of Absorption at S2
• Transition “up” = .25, “down” = .75.
• Start in S0.
Steps S-2 S-1 S0 S1 S2
1 0 .75 0 .25 0
2 .5625 0 .3750 0 .0625
3 .5625 .2812 0 .0938 .0625
1000 .9 0 0 0 .1
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• For many predictions, all this ugly algebra pretty much goes away if you use matrix algebra.
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.• …
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.
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.
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.
Sensitivity Analysisy=ax+b y=ax2+bx+c
19 61
47 131
145 152
56 30
18 105
SSE=16.10 SSE=11.45
11033 3092
1394 2820
4786 6091
2386 12024
5322 9671
SSE when Perturb params by Gau(0, .5)
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.
Cross Validationy=ax+b y=ax2+bx+c
SSE when fit to 1/2 of data
SSE when tested on other 1/2 of data
43.05
23.16
40.27
48.86
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