Analysis of RT distributions with R

Analysis of RT distributionswith R Emil Ratko-Dehnert WS 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

I Introduction toProbability theory

I Probability space Ω Probability space 1 P Probability measure A 0 Subsets of interest

I Probability Space (Ω, A, P) A Ω { 1; 2; 3 } 1 { } { 1 } 3 { 1; 3 } 2 { 3 } { 2; 3 } { 2 } { 1; 2 } Sample space: set of all possible outcomes P 1/4 1/2 3/4 0 1 Set of events : collection of subsets (σ-Algebra) Probability measure: Governed by Kolmogorov-Axioms

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)

I 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 ωєΩ • A = { „even pips“ } = {2, 4, 6} • P(A) = 3/6 = 1/2

I Example: RT Distribution Ex-Gaussian distribution

I 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 єR ) • P([x, y)) = ?  this will be addressed in II

I Important Laws in Probability theory • Law of large numbers • Central limit theorem

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

I Importance of Law of large numbers • It justifies aggregation of data to its mean • (will be important again in ) III

I 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. N n  ∞

Binomial distributions B(n, p), e.g. Tossing a coin n-times with prob(head) = p • increasing n  Normal distribution

I 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(!)

Excursion Matrix Calculus

Excursion: Matrix Calculus • Def: A matrix A= (ai,j) is an array of numbers • It has m rows and n columns (dim = m*n) 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 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*c c A cA

Matrix operations (III) • Transposition of a 2-by-3 matrix A  AT • It holds, that ATT= A. A AT

Matrix operations (IV) • Matrix multiplication ofmatrices C (2-by-3) and D (3-by-2) to E (2-by-2): C E D

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!

Geometric interpretation • Matrices can be interpreted as linear transformations in a vector space

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...

And now to

Analysis of RT distributions with R