Stochastic Simulation Introduction Bo Friis Nielsen

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1 Stochastic Simulation Introduction Bo Friis Nielsen Applied Mathematics and Computer Science Technical University of Denmark 2800 Kgs. Lyngby Denmark

2 Practicalities Notes will handed out next week, some will be available online Course evaluation is: passed/not passed - based on lab reports, report over final project, and possibly oral presentation of project. Teachers: Bo Friis Nielsen, room 205, building 321 ext , bfni@dtu.dk Jacob Emil Friis Frølich, Maksim Mazuryn, Anders Munch, (Jesper Fink Andersen) lecture 1 2

3 Significance One of the most (The most?)important Operations Research techniques Many modern statistical techniques rely on simulation lecture 1 3

4 What is simulation? (From Concise Oxford Dictionary): To simulate: To pretend, to act like, to mimic, to imitate. Here: Computer experiments with mathematical model Stochastic simulation To (have a computer) simulate a system which is affected by randomness. Narrow sense: To generate (pseudo)random numbers from a prescribed distribution (e.g. Gaussian) Computer experiments with mathematical model General engineering technique Analytical/numerical solutions lecture 1 4

5 Why simulate? Real system expensive Mathematical model to complex Get idea of dynamic behaviour lecture 1 5

6 Related areas Statistics Computer science Operations research lecture 1 6

7 Target group Methodology course of general interest Of special importance for students specialising in Computer science Statistics Operations Research Planning and management lecture 1 7

8 Course goal Topics related to scientific computer experimentation Specialised techniques Variance reduction methods Random number generation Random variable generation The event-by-event principle Simulation based statistical techniques Markov chain Monte Carlo Bootstrap Validition and verification of models Model building lecture 1 8

9 Recommended reading Sheldon M. Ross: Simulation, fifth edition, Academic Press 2013 Averill M. Law: Simulation Modeling and Analysis, McGraw-Hill 2015, ISBN Jerry Banks, John S. Carson II, Barry L. Nelson, David M. Nicol: Discrete-Event System Simulation, Prentice and Hall 1999, ISBN Brian Ripley: Stochastic Simulation, John Wiley & Sons 1987, ISBN Reuven Y. Rubinstein and Benjamin Melamed: Modern Simulation and Modelling, John Wiley & Sons 1998, ISBN Jack P. C. Kleijnen: Statistical Tools for Simulation Practitioneers, Marcel Dekker 1987, ISBN lecture 1 9

10 Knowledge/science in simulation Modelling skill Statistical methods - it is necessary to understand statistical methodology OR - Stochastic Processes Technical skills Random number generations Sampling from distributions Variance reduction techniques Statistical techniques bootstrap/mcmc General purpose/and specialised simulation software lecture 1 10

11 Discrete versus continuous Discrete event simulation as opposed to continuous simulation mixed models lecture 1 11

12 Probability basics 0 P(A) 1 P(Ω) = 1 P( ) = 0 A B = P(A B) = P(A)+P(B) Complement rule P(A c ) = 1 P(A) Difference rule for A B: P(B A c ) = P(B) P(A) Inclusion, exclusion for 2 events P(A B) = P(A)+P(B) P(A B) Conditional probability: for A given B (partial information): P(A B) = P(A B) P(B) Multiplication rule: P(A B) = P(B)P(A B) Law of total probability: (B i is a partitioning), P(A) = i P(B i)p(a B i ) Bayes theorem: (B i is a partitioning): P(B i A) = P(A B i)p(b i ) independence: P(A B) = P(A B c ) j P(A B j)p(b j ) (P(A B) = P(A)P(B))

13 Random variables Mapping from sample space to the real line Probabilities defined in terms of the preimage Most probabilitistic calculations are performed with only a slight reference to the underlying sample space lecture 1 13

14 Random variables Random variables: maps outcomes to real values Distribution P(X = x) P(X = x) = 1 Joint distribution P(X = x,y = y) x,y x P(X = x,y = y) = 1 Marginal distribution P X (X = x) = y P(X = x,y = y) Conditional distribution P(Y = y X = x) = P(X=x,Y=y) P X (X=x) independence P(Y = y,x = x) = P X (X = x)p Y (Y = y) Mean value E(X) = x P(X = x) General expectation E(g(X)) = g(x) P(X = x) Linearity E(aX +by +c) = ae(x)+be(y)+c

15 Continuous random variables Uniform distribution of two variables: P ((x,y) C) = A(C) A(D) Continuous random variables Density: f(x) 0, f(x)dx = 1, P(X dx) = f(x)dx Mean, variance (moments): E(X) = xf(x)dx E(g(X)) = g(x)f(x)dx, E ( X k) = x k f(x)dx Normal distribuion: f(x) = 1 2πσ e 1 2( x µ σ ) 2 Z = X µ σ Joint densities f(x,y)dxdy = P(x X x+dx,y Y y+dy),f(x,y) 0 Joint distribution F(x,y) = P(X x,y y) = y x f(u,v)dudv

16 Continuous random variables continued Conditional continous distributions f(y X = x) = f(x,y) f X (x) Integral version of law of total probability P(A) = P(A X = x)f X (x)dx Conditional expectation E(Y) = E(E(Y X)) Covariance/corellation Cov(X,Y) = E[(X E(X))(Y E(Y))] = E(XY) E(X)E(Y) Corr(X,Y) = Cov(X,Y) SD(X)SD(Y) (X,Y) independent Corr(X,Y) = 0 Variance ( of sum of variables N ) Var k=1 X k = n k=1 Var(X k)+2 1 j<k n Cov(X j,x k ) Bilinearity ( of covariance n Cov i=1 a ix i, ) m j=1 b jy j = n i=1 m j=1 a ib j Cov(X i,y j )

Random Variables. Cumulative Distribution Function (CDF) Amappingthattransformstheeventstotherealline.

Random Variables. Cumulative Distribution Function (CDF) Amappingthattransformstheeventstotherealline. Random Variables Amappingthattransformstheeventstotherealline. Example 1. Toss a fair coin. Define a random variable X where X is 1 if head appears and X is if tail appears. P (X =)=1/2 P (X =1)=1/2 Example