Analysis of RT distributionswith R

Emil Ratko-DehnertWS 2010/ 2011

Session 02 – 16.11.2010

Last time ...

• Organisational Information ->see webpage

• Why response times? -> ratio-scaled, math. treatment

• Why use R? -> standard, free, powerful, extensible

• Sources of randomness in the brain -> neurons,

bottom-up and top-down factors, measuring procedure

• Mathematical modelling of phenomena in the world

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INTRODUCTION TOPROBABILITY THEORY

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I

Probability space

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0

1

Ω

P

A

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Subsets of interest

Probability measure

Probability space

Probability Space (Ω, A, P)

{ 1 }{ }

{ 2 } { 3 }{ 1; 2 }

{ 1; 3 }{ 2; 3 }

{ 1; 2; 3 }1

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0 1/21/4 3/4 1

Ω A

P

Sample space:

set of all possible outcomes

Set of events :

collection of subsets (σ-Algebra)

Probability measure:

Governed by Kolmogorov-Axioms 6

I

Probability measure P • Is governed by „Kolmogorov-Axioms“

P(A) ≥ 0; A event (non-negativity)

P({}) = 0 and P(Ω) = 1 (normality)

P(Σ Ai) = Σ P(Ai); for Ai disjoint (σ-additivity)

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Example: Rolling a die

• Ω = {1, 2, 3, 4, 5, 6}

• A = Powerset(A) = { {1}, {2}, ..., {6}, {1, 2},

{1,3} , ..., {5, 6}, {1,2,3}, ..., {1, 2, 3, 4, 5, 6} }

• P(ω) = 1/6, for all ω є Ω

• A = { „even pips“ } = {2, 4, 6}

• P(A) = 3/6 = 1/2

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Example: RT Distribution

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y

yyf

2

2

2exp

1),,|(

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Ex-Gaussian distribution

Modelling behavioural experiments

„Response times to a pop-out experiment?“

• What is the probability space (Ω, A, P)?

• ΩRT= („all times between 0 and +∞ ms“)

• A = B(R) = ( [x, y); x, y є R )

• P([x, y)) = ? this will be addressed in

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I

II

Important Laws in Probability theory

• Law of large numbers

• Central limit theorem

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I

Law of large numbers

• „The sample average Xn (of a random variable

Xn) converges towards the theoretical

expectation μ of X“

• Example:

– Expected value of rolling a die is 3.5

– Average value of 1000 dice should be

3500 / 1000 = 3.5 12

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Importance of Law of large numbers

• It justifies aggregation of data to its mean

• (will be important again in )

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I

III

Central limit theorem

• The average of many iid random variables

with finite variance tends towards a normal

distribution irrespective of the distribution

followed by the original random variables.

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Nn ∞

I

• Binomial distributions

B(n, p), e.g. Tossing a

coin n-times with

prob(head) = p

• increasing n Normal

distribution

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Importance of Central limit theorem

• Why is this important:

– It argues that the sum of many random processes

(whatever distribution they may follow) behaves like

a normal random process

– i.e. If you have a system, where many random

processes interact, you can just treat the overall

effect like a normal error/ noise(!)17

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MATRIX CALCULUSExcursion

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Excursion: Matrix Calculus

• Def: A matrix A = (ai,j) is an array of numbers

• It has m rows and n columns (dim = m*n)

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nmm

n

aa

aa

aaa

A

,1,

2,21,2

,12,11,1

m

n

Matrix operations (I)

• Addition of two 2-by-2 matrices A, B performed

component-wise:

• Note that „+“ is commutative, i.e. A+B = B+A 20

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33

11

12

20

41

A B A+B

Matrix operations (II)

• Scalar Multiplication of a 2-by-2 matrix A with

a scalar c

• Again commutativity, i.e. c*A = A*c21

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82

20

412

c A cA

Matrix operations (III)

• Transposition of a 2-by-3 matrix A AT

• It holds, that ATT= A.22

94

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01

960

421T

AAT

Matrix operations (IV)

• Matrix multiplication of matrices C (2-by-3)

and D (3-by-2) to E (2-by-2):

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01

12

13

131

201

CD E

Matrix operations (V)

!Warning!

One can only multiply matrices if their dimensions

correspond, i.e. (m-by-n) x (n-by-k) (m-by-k)

• And generally: if A*B exists, B*A need not

• Furthermore: if A*B, B*A exists, they need not be

equal!24

Geometric interpretation

• Matrices can be interpreted as linear

transformations in a vector space

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Significance of matrices

• Matrix calculus is relevant for

– Algebra: Solving linear equations (Ax = b)

– Statistics: LLS, covariance matrices of r. v.

– Calculus: differentiation of multidimensional functions

– Physics: mechanics, linear combinations of quantum

states and many more...

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AND NOW TO

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