Download 2-by-3

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
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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(!)
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Excursion
MATRIX CALCULUS
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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)
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!
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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...
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AND NOW TO
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