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Analysis of RT distributions with 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 3 I INTRODUCTION TO PROBABILITY THEORY 4 I Probability space Ω Probability space P 1 Probability measure A Subsets of interest 0 5 Probability Space (Ω, A, P) I A Ω 1 { } 3 2 { 1; 2; 3 } {1} {2} {3} { 1; 2 } { 1; 3 } { 2; 3 } Sample space: set of all possible outcomes Set of events : collection of subsets (σ-Algebra) P 0 1/4 1/2 3/4 1 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) 7 Example: Rolling a die I • 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 8 Example: RT Distribution I Ex-Gaussian distribution y 2 f ( y | , , ) exp 2 2 y 1 9 Modelling behavioural experiments I „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 II 10 Important Laws in Probability theory I • Law of large numbers • Central limit theorem 11 Law of large numbers I • „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 13 Importance of Law of large numbers I • It justifies aggregation of data to its mean • (will be important again in III ) 14 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 15 • Binomial distributions B(n, p), e.g. Tossing a coin n-times with prob(head) = p • increasing n Normal distribution 16 Importance of Central limit theorem I • 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 Excursion MATRIX CALCULUS 18 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 a1,1 a1, 2 a1,n a a 2 , 1 2 , 2 A am,1 am,n n 19 Matrix operations (I) • Addition of two 2-by-2 matrices A, B performed component-wise: 1 4 2 1 3 3 0 2 1 1 1 1 A B A+B • Note that „+“ is commutative, i.e. A+B = B+A 20 Matrix operations (II) • Scalar Multiplication of a 2-by-2 matrix A with a scalar c 1 4 2 8 2 0 2 0 4 c A cA • Again commutativity, i.e. c*A = A*c 21 Matrix operations (III) • Transposition of a 2-by-3 matrix A AT 1 0 1 2 4 2 6 0 6 9 4 9 A T AT • It holds, that ATT= A. 22 Matrix operations (IV) • Matrix multiplication of matrices C (2-by-3) and D (3-by-2) to E (2-by-2): 3 1 1 0 2 5 1 2 1 1 3 1 4 2 1 0 C D E 23 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 25 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... 26 AND NOW TO 27