Formal Modeling in Cognitive Science Uniform Distribution Lecture - - PowerPoint PPT Presentation

Special Distributions Application: Eye-movement Data

Formal Modeling in Cognitive Science

Lecture 27: Special Distributions; Applications Frank Keller

School of Informatics University of Edinburgh keller@inf.ed.ac.uk

March 11, 2005

Frank Keller Formal Modeling in Cognitive Science 1 Special Distributions Application: Eye-movement Data

1 Special Distributions

Uniform Distribution Binominal Distribution Normal Distribution

2 Application: Eye-movement Data

Eye-movements and Cognition Eye-movements and Reading Probability Distributions

Frank Keller Formal Modeling in Cognitive Science 2 Special Distributions Application: Eye-movement Data Uniform Distribution Binominal Distribution Normal Distribution

Uniform Distribution

Definition: Uniform Distribution A random variable X has a discrete uniform distribution iff its probability distribution is given by: f (x) = 1 k for x = x1, x2, . . . , xk where xi = xj when i = j. The mean and variance of a uniform distribution are: µ =

k

• i=1

xi · 1 k σ2 =

k

• i=1

(xi − µ)2 1 k

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Binominal Distribution

Often we are interested in experiments with repeated trials: assume there is a fixed number of trials; each of the trial can have two outcomes (e.g., success and failure, head and tail); the probability of success and failure is the same for each trial: θ and 1 − θ; the trials are all independent. Then the probability of getting x successes in n trials in a given

• rder is θx(1 − θ)n−x. And there are

n

x

• different orders, so the
• verall probability is

n

x

• θx(1 − θ)n−x.

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Special Distributions Application: Eye-movement Data Uniform Distribution Binominal Distribution Normal Distribution

Binominal Distribution

Definition: Binomial Distribution A random variable X has a binominal distribution iff its probability distribution is given by: b(x; n, θ) = n x

• θx(1 − θ)n−xfor x = 0, 1, 2, . . . , n

Example The probability of getting five heads and seven tail in 12 flips of a balanced coin is: b(5; 12, 1 2) = 12 5

• (1

2)5(1 − 1 2)12−5

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Binominal Distribution

1 1 2 3 4 5 6 7 8 9 10 11 12 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 b(x; 12, 0.5)

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Binominal Distribution

If we invert successes and failures (or heads and tails), then the probability stays the same. Therefore: Theorem: Binomial Distribution b(x; n, θ) = b(n − x; n, 1 − θ) Two other important properties of the binomial distribution are: Theorem: Binomial Distribution The mean and the variance of the binomial distribution are: µ = nθ and σ2 = nθ(1 − θ)

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Normal Distribution

Definition: Normal Distribution A random variable X has a normal distribution iff its probability density is given by: n(x; µ, σ) = 1 σ √ 2π e− 1

2( x−µ σ )2 for − ∞ < x < ∞

Normally distributed random variables are ubiquitous in probability theory; many measurements of physical, biological, or cognitive processes yield normally distributed data; such data can be modeled using a normal distributions (sometimes using mixtures of several normal distributions).

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Special Distributions Application: Eye-movement Data Uniform Distribution Binominal Distribution Normal Distribution

Standard Normal Distribution

0.4 x 0.1 0.3 4 0.2 2

• 2
• 4

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Normal Distribution

Definition: Standard Normal Distribution The normal distribution with µ = 0 and σ = 1 is referred to as the standard normal distribution: n(x; 0, 1) = 1 √ 2π e− 1

2x2

Theorem: Standard Normal Distribution If a random variable X has a normal distribution, then: P(|x − µ| < σ) = 0.6826 P(|x − µ| < 2σ) = 0.9544 This follows from Chebyshev’s Theorem (see previous lecture).

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Normal Distribution

Theorem: Z-Scores If a random variable X has a normal distribution with the mean µ and the standard deviation σ then: Z = X − µ σ has the standard normal distribution. This conversion is often used to make results obtained by different experiments comparable: convert the distributions to Z-scores.

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Eye-movements and Cognition

Let’s apply what we’ve learned to some real data. An eye-tracker makes it possible to record the eye-movements of subjects while their are performing a cognitive task: reading a text; looking at a picture; using a computer screen and keyboard; driving a vehicle. Mind’s Eye Hypothesis: where subjects are looking indicates what they are processing. How long they are looking at it indicates how much processing effort is needed.

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Special Distributions Application: Eye-movement Data Eye-movements and Cognition Eye-movements and Reading Probability Distributions

Eye-movements and Cognition

Let’s look at subjects’ eye-movements while they read text.

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Eye-movements and Reading

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Eye-movements and Reading

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Eye-movements and Reading

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Special Distributions Application: Eye-movement Data Eye-movements and Cognition Eye-movements and Reading Probability Distributions

Eye-movements and Reading

Eye-movements are recorded while subjects read texts; very high spatial and temporal accuracy; eye movements in reading are saccadic: a series of relatively stationary periods (fixations) between very fast movements (saccades); average fixation time is about 250 ms; can be longer or shorter, depending on ease or difficulty of processing; typically test a number of subjects, with a number of test sentences, and statistical analysis done on results.

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Probability Distributions

Data: 23 subjects read a 2000 word text while their eye-movements were being recorded. To analyze the data, define two random variables: X: time taken to read a word; Y : number of regressions made while reading a word; Note that X is a continuous random variable, while Y is a discrete random variable. Plot the distributions; compute their means and standard deviations.

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Distribution: Reading Time

100 200 300 400 500 600 700 Fixation time [ms] 50 100 150 200 250 300 350 400 450 500 Frequency

µ = 269.58, σ = 132.88. Almost normal distribution.

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Distribution: Number of Regressions

1 2 3 4 5 6 7 8 9 Number of regressions 250 500 750 1000 1250 1500 1750 2000 Frequency

µ = 1.31, σ = 1.81. Almost a binominal distribution.

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Special Distributions Application: Eye-movement Data Eye-movements and Cognition Eye-movements and Reading Probability Distributions

Probability Distributions

We can therefore approximate the empirical distributions as follows: f (x) = n(x; 269.58, 132.88) = 1 132.88 √ 2π e− 1

2 ( x−269.58 132.88 )2

= 0.003e− 1

2(0.008x−2.03)2

f (y) = b(y; 20, 0.01) − 10 = 20 y

• 0.01y0.0920−y

These distributions fit the empirical distributions reasonably

• closely. They give us an idea about the process that generated

these distributions.

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Summary

The uniform distribution assigns each value the same probability; The binomial distributions models an experiment with a fixed number of independent binary trials, each with the same probability; The normal distribution models the data generated by measurements of physical, biological, or cognitive processes; Z-scores can be used to convert a normal distribution into the standard normal distribution; eye-movement data: reading times are approx. normally distributed; regressions are approx. binomially distributed.

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