1. Analysis of RT distributions with R Emil Ratko-Dehnert WS 2010/ 2011 Session 02 – 16.11.2010

2. Course roadmap Introduction to probability theory Random variables and their characterization Estimation Theory Model testing 2 I II III IV

3. 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

4. INTRODUCTION TO PROBABILITY THEORY 4 I

5. Probability space 5 0 1 Ω P A I Subsets of interest Probability measure Probability space

6. Probability Space (Ω, A, P) { 1 } { } { 2 } { 3 } { 1; 2 } { 1; 3 } { 2; 3 } { 1; 2; 3 } 1 2 3 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

7. Probability measure P Is governed by „Kolmogorov-Axioms“  P(A) ≥ 0; A event(non-negativity)  P({}) = 0 and P(Ω) = 1(normality)  P(Σ A i ) = Σ P(A i ); for A i disjoint(σ-additivity) 7 I

8. 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 8 I

9. Example: RT Distribution 9 I Ex-Gaussian distribution

10. Modelling behavioural experiments „Response times to a pop-out experiment?“ What is the probability space (Ω, A, P)? Ω RT = („all times between 0 and +∞ ms“) A = B (R) = ( [x, y); x, y є R ) P([x, y)) = ?  this will be addressed in 10 I II

11. Important Laws in Probability theory Law of large numbers Central limit theorem 11 I

12. Law of large numbers „The sample average X n (of a random variable X n ) 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 I

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14. Importance of Law of large numbers It justifies aggregation of data to its mean (will be important again in ) 14 I III

15. 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. 15 N n  ∞ I

16. Binomial distributions B(n, p), e.g. Tossing a coin n-times with prob(head) = p increasing n  Normal distribution 16

17. 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 I

18. MATRIX CALCULUS Excursion 18

19. Excursion: Matrix Calculus Def: A matrix A = (a i,j ) is an array of numbers It has m rows and n columns (dim = m*n) 19 m n

20. 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 ABA+B

21. Matrix operations (II) Scalar Multiplication of a 2-by-2 matrix A with a scalar c Again commutativity, i.e. c*A = A*c 21 c AcA

22. Matrix operations (III) Transposition of a 2-by-3 matrix A  A T It holds, that A TT = A. 22 A ATAT

23. Matrix operations (IV) Matrix multiplication of matrices C (2-by-3) and D (3-by-2) to E (2-by-2): 23 C D E

24. 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

25. Geometric interpretation Matrices can be interpreted as linear transformations in a vector space 25

26. 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

27. AND NOW TO 27