Download CZ2105 Lecture 2

MA3264 Mathematical Modelling
Lecture 6
Monte Carlo Simulation
Motivation
Elevator Service : Should we
assign elevators to even / odd floors?
use express elevators?
Traffic Control : Where should we locate
one way streets?
traffic lights?
Experimental Method : May be disruptive since
upset customers may take stairs!
frustrated drivers may take sidewalks!
Therefore decision makers require simulation.
Random Number Generation
>> m=4; n = 2;
>> rand(m,n)
ans =
0.9501
0.2311
0.6068
0.4860
0.8913
0.7621
0.4565
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')
gives m x n matrix whose entries are
independent random numbers uniformly
distributed over the interval [0,1]
Random Number Generation
>> 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*')
Random Number Generation
Random Number Generation
Area Under a Curve
>> s = 0:.001:1;
>> fs = s.^2;
>> plot(s,fs,'ko')
>> title('f(s) = s^2')
>> count = 0;
>> for j = 1:10000
if y(j) < x(j)^2
count = count + 1;
end
end
>> 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/1000000
area_estimate = 0.3329
>> for m = 1:100
z = rand(1000000,2);
count = sum(z(:,2) < z(:,1).^2);
area_estimate(m) =
count/1000000;
end
>> z = rand(1000000,2);
>> count = sum(z(:,2) < z(:,1).^2);
>> area_estimate = count/1000000
area_estimate = 0.3330
>> z = rand(1000000,2);
>> count = sum(z(:,2) < z(:,1).^2);
>> area_estimate = count/1000000
area_estimate = 0.3328
Question How does accuracy depend on # points used ?
Volume Inside of a Solid Object
Problem 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/10000000
fraction = 0.9212
>> volume_estimate = (27/8)*fraction
volume_estimate = 3.1092
Generating Random Numbers
Middle Square Method, invented by Ulam, Metropolis +
http://en.wikipedia.org/wiki/John_von_Neumann
1. Start with a four digit number
x 0 , called the seed.
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.
3084
5129
>> x(1) = 3084;
5110
3066
Question Whats wrong?
>> for k = 1:17
s = x(k)^2;
s6 = mod(s,1000000);
r6 = s6 - mod(s6,100);
x(k+1) = r6/100;
end
1121
2566
5843
1406
9768
4138
1230
4003
240
576
3317
24
5
0
Question Where to go?
Generating Random Numbers
>> x(1) = 962947;
>> for k = 1:100000
s = x(k)^2;
s9 = mod(s,1000000000);
r9 = s9 - mod(s9,1000);
x(k+1) = r9/1000;
end
hist(x/1000000)
Question What did we do?
>> x = rand(100000,1);
>> hist(x)
Question What method is better?
Generating Random Numbers
Linear Congruence Method, invented by
http://en.wikipedia.org/wiki/Derrick_Henry_Lehmer
1. Start with a four digit number
2. Apply the transformation
x 0 , called the seed.
xn1  (axn  b) mod( c)
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 = 1:100000
x(k+1)=mod(a*x(k)+b,c);
end
x = x/(c-1);
hist(x)
Generating Random Numbers
It 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
>> numbers
numbers = 17 18 19 18 17
11
Generating Random Numbers
Queueing theory models elevator (lift) and bus service.
http://en.wikipedia.org/wiki/Queueing_theory
The time between people consecutively arriving
at an elevator or a bus stop is a random variable x
that is exponentially distributed over the interval [0,  ).
This means that
prob ( x  [a, b])  
b
1
a c
e t / c dt  e a / c  e b / c
where c is the expected value of x.
Question How can we generate this random variable
using (only) a uniform random number generator?
Generating Random Numbers
Theorem If c > 0 and y is uniformly distributed over [0,1]
then x = - c ln(y) is exponentially distributed over [0,  ).
Proof prob ( x  [a, b])  prob (a  c ln y  b) 
prob (b / c  ln y  a / c)  prob (e b / c  y  e  a / c )  e  a / c  e b / c
The MATLAB code generates 100000 random samples
>> c = 1;
>> for k = 1:100000
y(k) = rand;
x(k) = -c*log(y(k));
end
hist(x,100)
histogram of y
histogram of
x
Generating Random Numbers
Theorem If u is uniformly distributed over [0,1] and r is
independent of u , and r is exponentially distributed with
expected value 2 then and x  r cos( 2 u ) and y  r sin( 2 u )
independent and normally distributed with mean 0 and
variance 1
Proof The conditions on x and y are satisfied iff
prob (a  r  b)  
{a  r  b}
1

2
e
a2 / 2

{a  r  b}
e
r 2 / 2
b
dx dy   e
Pnormal ( x) Pnormal ( y )dxdy
r 2 / 2
a
e
b 2 / 2
b2 / 2
r dr  
t
e
 dt
a 2 / 2
 prob(a 2  r  b 2 ) when r is exponentia lly
distribute d with expected value 2
Generating Random Numbers
The MATLAB commands generate 100000 samples
>> for k = 1:50000
u(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));
end
hist(x,100)
Simulating Choosing m from n People
The MATLAB commands below do this
% people are called 1, 2, …,n
% c = vector containing chosen people
% r = vector of people not yet chosen
r = 1:n;
for k = 1:m
% compute random integer between 1 and n-k+1
j = 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
5.2 Generating Random Numbers, p 177-181, Prob 5.2
5.3 Simulating Prob. Behavior, p 186-190, Prob 5.3
Recommended Websites
http://en.wikipedia.org/wiki/Monte_Carlo_method
http://en.wikipedia.org/wiki/Random_number_generator
http://en.wikipedia.org/wiki/Queueing_theory
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 October
1. 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.