Random Walk Models

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

1-D Random Walk

Unbounded S0S0 S1S1 S2S2 S -1 S -2 p 0,-1 p -1,-2 p 1,0 p 2,1 p -2, -1 p -1, 0 p 0,1 p 1, 2 p 2, 3 p -3, -2 p -2,-3 p 3, 2 ……

1-D Random Walk 1 side bounded, 1 unbounded S0S0 S1S1 S2S2 S -1 S -2 p 0,-1 p -1,-2 p 1,0 p 2,1 p -2, -1 p -1, 0 p 0,1 p 1, 2 p 2, 2 p -3, -2 p -2,-3 …

1-D Random Walk Bounded S0S0 S1S1 S2S2 S -1 S -2 p 0,-1 p -1,-2 p 1,0 p 2,1 p -2, -1 p -1, 0 p 0,1 p 1, 2 p 2, 2 p -2,-2

1-D Random Walk 1 absorbing state S0S0 S1S1 S2S2 S -1 S -2 p 0,-1 p -1,-2 p 1,0 p 2,1 0p -1, 0 p 0,1 p 1, 2 p 2, 2 1

1-D Random Walk 2 absorbing states S0S0 S1S1 S2S2 S -1 S -2 p 0,-1 p -1,-2 p 1,0 0 0p -1, 0 p 0,1 p 1, 2 11

2-D Random Walk …… ……

1-D Random Walk Definition A 1-D random walk is a –Markov chain –where the states are ordered …, S -2, S -1, S 0, S 1, S 2, … The transition probability between states S i and S j are 0 unless S i = S j  1.

1-D Random Walk Unbounded S0S0 S1S1 S2S2 S -1 S -2 p 0,-1 p -1,-2 p 1,0 p 2,1 p -2, -1 p -1, 0 p 0,1 p 1, 2 p 2, 3 p -3, -2 p -2,-3 p 3, 2 ……

More on Random Walks Note that the states usually have real interpretations, but can be abstract placeholders.

Real Interpretations NeutralAgitatedAngryUpsetSad Loc 0 Loc 1 Loc 2 Loc -1 Loc -2

Placeholders S0S0 S1S1 S2S2 S -1 S -2

More on Random Walks Note that the time it takes to go from one state to another is often important NeutralAgitatedAngryUpsetSad The subject was “angry” for 5 mins before returning to an “agitated” state… The subject fluctuated rapidly between “neutral” and “upset”.

Probability of Absorption at S 2 S0S0 S1S1 S2S2 S -1 S -2 p 0,-1 p -1,-2 p 1,0 0 0p -1, 0 p 0,1 p 1, 2 11

Probability of Absorption at S 2

Transition “up” =.25, “down” =.75. Start in S 0. StepsS -2 S -1 S0S0 S1S1 S2S

Ad for Matrix Algebra 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 Analysis y=ax+by=ax 2 +bx+c SSE=16.10 SSE= 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 Validation y=ax+by=ax 2 +bx+c SSE when fit to 1/2 of data SSE when tested on other 1/2 of data

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

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