MA3264 Mathematical Modelling Lecture 6 Monte Carlo Simulation.

MA3264 Mathematical ModellingLecture 6

Monte Carlo Simulation

Motivation

assign elevators to even / odd floors? Elevator Service : Should we

Experimental Method : May be disruptive since

one way streets?

use express elevators?

Traffic Control : Where should we locate

upset customers may take stairs!

traffic lights?

frustrated drivers may take sidewalks!

Therefore decision makers require simulation.

Random Number Generation

gives m x n matrix whose entries are independent random numbers uniformly distributed over the interval [0,1]

>> m=4; n = 2;>> rand(m,n)

ans =

0.9501 0.8913 0.2311 0.7621 0.6068 0.4565 0.4860 0.0185

>> x = rand(10,1);>> x = rand(10,1);>> y = zeros(10,1);>> plot(x,y,'r*')>> hold on>> x = rand(10,1);>> plot(x,y,'bo')


>> x = rand(10,1);>> y = rand(10,1);>> plot(x,y,'r*')>> hold on>> x = rand(10,1);>> y = rand(10,1);>> plot(x,y,'bo')

>> x = rand(100,1);>> y = rand(100,1);>> plot(x,y,'r*')

Area Under a Curve

>> s = 0:.001:1;>> fs = s.^2;>> plot(s,fs,'ko')>> title('f(s) = s^2')>> count = 0;>> for j = 1:10000if y(j) < x(j)^2count = count + 1;endend>> area_estimate = count/10000

area_estimate =

0.3387

Question What is the area under the curve f(s) = s^2 ?

Area Under a Curve

>> z = rand(1000000,2);>> count = sum(z(:,2) < z(:,1).^2);>> area_estimate = count/1000000area_estimate = 0.3329



Question How does accuracy depend on # points used ?

>> for m = 1:100z = rand(1000000,2);count = sum(z(:,2) < z(:,1).^2);area_estimate(m) = count/1000000;end

Volume Inside of a Solid ObjectProblem Compute the volume of a solid that is formed by intersecting a ball of radius 1 and a cube of diameter 3/2 that has the same center as the ball?

Solution Clearly the volume of the cube = 27/8 = 3.750 Therefore is suffices to compute the fraction of random points chosen inside the cube that are inside the sphere.The following MATLAB commands do the computation>> N = 10000000;>> count = 0;>> xyz = 1.5*rand(N,3)-0.75;>> radii = sqrt(xyz(:,1).^2 + xyz(:,2).^2 + xyz(:,3).^2);>> count = sum(radii < 1);>> fraction = count/10000000fraction = 0.9212>> volume_estimate = (27/8)*fractionvolume_estimate = 3.1092

Generating Random NumbersMiddle Square Method, invented by Ulam, Metropolis +

0x1. Start with a four digit number , called the seed.

http://en.wikipedia.org/wiki/John_von_Neumann

2. Square it to obtain an eight digit number (add leading zeros if necessary).3. Take the middle four digits as the next random number.>> x(1) = 3084;>> for k = 1:17s = x(k)^2;s6 = mod(s,1000000);r6 = s6 - mod(s6,100);x(k+1) = r6/100;end

308451101121256658431406976841381230

51293066400324057633172450

Question Whats wrong?

Question Where to go?



Generating Random Numbers

>> x(1) = 962947;>> for k = 1:100000s = x(k)^2;s9 = mod(s,1000000000);r9 = s9 - mod(s9,1000);x(k+1) = r9/1000;endhist(x/1000000)

>> x = rand(100000,1);>> hist(x)

Question What did we do?

Question What method is better?

Generating Random NumbersLinear Congruence Method, invented by

0x1. Start with a four digit number , called the seed.

http://en.wikipedia.org/wiki/Derrick_Henry_Lehmer

2. Apply the transformation

Here a, b, c are three required positive integers. The random integers are between 0 and c-1.

>> a = 15625;>> b = 22221;>> c = 2^31;>> x(1) = 389274;>> for k = 1:100000x(k+1)=mod(a*x(k)+b,c);endx = x/(c-1);hist(x)

)mod()(1 cbaxx nn



Generating Random NumbersIt is often required to compute random numbers that are not uniformly distributed over an interval.Example When a dice is rolled it produces a random number in the set {1,2,3,4,5,6}, it is uniformly distributed over that (discrete) set of points. The following MATLAB commands simulates 100 throws of a dice and computes how many times each value occured.

>> for k = 1:100>> y = rand;>> x(k) = round(6*y+1/2);>> end>> for val = 1:6>> numbers(val) = sum(x == val);>> end>> numbersnumbers = 17 18 19 18 17 11

Generating Random NumbersQueueing theory models elevator (lift) and bus service.

http://en.wikipedia.org/wiki/Queueing_theory

Question How can we generate this random variable using (only) a uniform random number generator?

The time between people consecutively arriving at an elevator or a bus stop is a random variable that is exponentially distributed over the interval ).,0[

This means thatcbcab

a

ctc eedtebax ///1]),[(prob

where c is the expected value of

x

.x



Generating Random NumbersTheorem If c > 0 and y is uniformly distributed over [0,1] then x = - c ln(y) is exponentially distributed over Proof

).,0[

The MATLAB code generates 100000 random samples

)ln(prob]),[(prob bycabax

>> c = 1;>> for k = 1:100000y(k) = rand;x(k) = -c*log(y(k));endhist(x,100)

cbcacacb eeeyecaycb //// )(prob)/ln/(prob

x

histogram of

histogram of

y

Generating Random NumbersTheorem If is uniformly distributed over [0,1] and isindependent of and is exponentially distributed with expected value 2 then and and independent and normally distributed with mean 0 and variance 1

Proof The conditions on x and y are satisfied iff

b}r{a)()()(prob dxdyyPxPbra normalnormal

,u

dtedrredydxeb

a

tb

a

rr

2/

2/

2/

b}r{a

2/

2

2

22

2

1

u

)2cos( urx )2sin( ury r

r

llyexponentia iswhen )( 222/2/ 22

rbraprobee ba

2 valueexpected with ddistribute

Generating Random NumbersThe MATLAB commands generate 100000 samples >> for k = 1:50000u(k) = rand;r(k) = -2*log(rand);x(2*k-1) = sqrt(r(k))*cos(2*pi*u(k));x(2*k) = sqrt(r(k))*sin(2*pi*u(k));endhist(x,100)

Simulating Choosing m from n PeopleThe MATLAB commands below do this

% people are called 1, 2, …,n % c = vector containing chosen people% r = vector of people not yet chosenr = 1:n;for k = 1:m

% compute random integer between 1 and n-k+1j = round((n-k+1)*rand+0.5);c (k) = r(j);r(j) = r(n-k+1);r = r(1:n-k);

end

Suggested Reading&Problems in Textbook

5.1 Area Under a Curve, p 177-181, Prob 5.1

Recommended Websites

5.2 Generating Random Numbers, p 177-181, Prob 5.2

5.3 Simulating Prob. Behavior, p 186-190, Prob 5.3

http://en.wikipedia.org/wiki/Monte_Carlo_method

http://en.wikipedia.org/wiki/Random_number_generator








Tutorial 6 Due Week 6-10 October

Page 181. Problem 1. Imagine that every week you purchase lottery tickets until you win a prize. Design and run a simulation program to compute the average number of lottery tickets that you purchase per week.

Page 181. Problem 3. Choose a number of random points so that your estimated value of pi is accurate to about 4 significant digits

Page 190. Problem 1 (not project 1)

Homework 2 Due Friday 10 October1. Write a computer program to simulate waiting times at a bus station during rush hours using the assumptions and steps below. Run the program 200 times to generate a histogram of the waiting times that people experience and a histogram of the total number of bus arrivals. Explain in detail your logic and algorithms.

a. People start to arrive at the bus station at 5:00 at the average rate of 8 per minute. They stop arriving at 7:00. The times between consecutive arrivals is exponentially distributed. For each simulation, first compute an array containing random samples of the times (in order) that people arrive between 5-7PM.

b. Buses arrive promptly every 10 minutes starting at 5:10 and continue until the last passenger is picked up. Each bus arrives empty and picks up exactly 60 people.

c. Each time a bus arrives the people waiting scramble to board the bus. Simulate this by choosing 60 people randomly, from among those waiting, during each bus arrival.

MA3264 Mathematical Modelling Lecture 6 Monte Carlo Simulation.

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