Chapter 4: Monte Carlo Methods -...

Chapter 4: Monte Carlo Methods

Paisan Nakmahachalasint

Introduction

Monte Carlo Methods are a class of computational algorithms that rely on repeated random sampling to compute their results.Monte Carlo simulation methods are especially useful in studying systems with a large number of coupled degrees of freedom.

Introduction

Monte Carlo methods describe a large and widely-used class of approaches which tend to follow a particular pattern:

Define a domain of possible inputs. Generate inputs randomly from the domain using a certain specified probability distribution.

Perform a deterministic computation using the inputs. Aggregate the results of the individual computations into the final result.

Uniform Random Number Generators

Monte Carlo simulations are inherently probabilistic, and thus make frequent use of programs to generate random numbers.

On a computer these numbers are not random at all -- they are strictly deterministic and reproducible, but they look like a stream of random numbers. For this reason such programs are more correctly called pseudo-random number generators.


Essential Properties

Repeatability -- the same sequence should be produced with the same seeds.Randomness -- should produce independent uniformly distributed random variables that pass all statistical tests for randomness. Long period – The period should be much longer than the amount of random numbers needed.Insensitive to seeds -- period and randomness properties should not depend on the initial seeds.


Additional PropertiesPortability -- should give the same results on different computers. Efficiency -- should be fast (small number of floating point operations) and not use much memory. Disjoint subsequences -- different seeds should produce long independent (disjoint) subsequences so that there are no correlations between simulations with different initial seeds. Homogeneity -- sequences of all bits should be random.


Middle-Square Method

Start with a four-digit number x0, called the seed.

Square it to obtain an eight-digit number(add a leading zero if necessary).Take the middle four digits as the next random number.

64801004104901801009742316562041xn

876543210n


Linear Congruence Generator (LCG)

Introduced by D. H. Lehmer in 1951. A great majority of pseudo-random numbers used today are based on this method.

xn+1 = (axn + b) (mod m)

Example: if x0 = 7, a = 1, b = 7, m = 10, we obtain the following sequence:

7, 4, 1, 8, 5, 2, 9, 6, 3, 0, 7, 4, …


Choice of the modulus, m.m should be as large as possible since the period can never be larger than m.One usually chooses m to be near the largest integer that can be represented.

Choice of the multiplier, aIt was proven by M.Greenberger in 1961 that the sequence will have period m if and only if

b is a relative prime to m.a – 1 is a multiple of p for all p | m.4|m ⇒ 4|a – 1.


With b = 0, one cannot get the full period, but in order to get the maximum possible, the followings should be satisfied:

x0 is a relative prime to m.a is a primitive root modulo m.

It is possible to obtain a period of length m – 1, but usually the period is around m/4.


A Disadvantage of LCGRANDU is a popular random number generator distributed by IBM in 1960’s with the algorithm

xn+1 = 65539xn (mod 231).This generator was later found to have a serious problem – all random numbers fall into hyperplanes.


5000 random points in 1 and 2 dimensions generated from RANDU.

0.2 0.4 0.6 0.8 1.

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Histogram of 5000random numbers.

Distribution of 5000 points in 2 dimensions


5000 random points in 3 dimensions generated from RANDU viewed at two different view points.


LCGs in rand() in various compilers.

In all above compilers, m = 232

16 - 30123451103515245ANSI C

32 - 631134775813Borland Delphi

16 - 302531011214013Microsoft C/C++

0 - 30123451103515245Glibc (GCC)

16 - 30122695477Borland C/C++

Output bits in rand()baSource


Lagged-Fibonacci Generator (LFG)

Lagged Fibonacci pseudo-random number generators have become increasingly popular in recent years. These generators are so named because of their similarity to the familiar Fibonacci sequence. LFG uses the algorithm

xn = xn – l + xn – k (mod m) where l > k > 0

with m initial values {x0, x1, …, xl – 1}.


For most applications of interest, m is a power of two; i.e., m = 2M.

With a proper choice of l, k and the initial values, the period P of LFG is (2l – 1)×2M – 1.

Two commonly used LFGs.

xn = xn – 17 + xn – 5 (mod 231) period ≈ 247

xn = xn – 55 + xn – 24 (mod 231) period ≈ 285


Unlike LCG, the value of the modulus, m, does not by itself limit the period of the generator.LFG is computationally simple: an integer add, a logical AND (to accomplish the mod 2M operation), and the advancing of two array pointers are the only operations required to produce a new random number.

Random Variables and Distribution

A random variable can be thought of as an unknown value that may change every time it is inspected. Thus, a random variable can be thought of as a function mapping the sample space of a random process to the real numbers.

A random variable can be discrete or continuous.


Example (Discrete RV)The sample space of tossing 2 fair coins is

S = {HH, HT, TH, TT}.

Let X = number of heads, then

X = {(HH,2), (HT,1), (TH,1), (TT,0)}.


Example (Cont’d)The probability function

of X is given by

The mean of X is

The variance of X is

141214

if 0

if 1( )

if 2

0 otherwise

x

xP X x

x

⎧⎪ =⎪⎪⎪ =⎪⎪= = ⎨⎪ =⎪⎪⎪⎪⎪⎩

1 1 14 2 4

E[ ] ( ) 0 1 2 1X i i

i

X x P xμ = = = ⋅ + ⋅ + ⋅ =∑

2 2 2 12

Var[ ] E[( ) ] E[ ]X X

X X Xμ μ= − = − =


Example: Binomial Distribution X ~ B(n,p)

( )( ) 1 0,1, ,n xxn

P X x p p x nx−⎛ ⎞⎟⎜ ⎟= = − =⎜ ⎟⎜ ⎟⎜⎝ ⎠

…

( ) ( )0

22

0

E[ ] ( )

Var[ ] ( ) 1

n

xn

x

X xP x np

X x P x np np p

=

=

= =

⎛ ⎞⎟⎜ ⎟= − = −⎜ ⎟⎜ ⎟⎜⎝ ⎠

∑

∑


Example: Continuous Uniform DistributionX ~ U(a,b)

1 for( )

0 otherwiseb a

X

a x bf x −

⎧⎪ ≤ ≤⎪= ⎨⎪⎪⎩

( )

( )2

22

12

112

E[ ] ( )

Var[ ] ( )2

X

X

X xf x dx a b

a bX x f x dx b a

∞

−∞

∞

−∞

= = +

⎛ ⎞⎛ ⎞ + ⎟⎜⎟⎜ ⎟= − = −⎜⎟⎜ ⎟⎟ ⎜⎜ ⎟⎜⎝ ⎠ ⎝ ⎠

∫

∫


Cumulative Distribution Function

Example( ) ( ) ( )

x

X XF x P X x f t dt

−∞= ≤ = ∫

0

( ) for

1

x aX b a

x a

F x a x b

x b

−−

⎧⎪ <⎪⎪⎪= ≤ ≤⎨⎪⎪ >⎪⎪⎩

1 for( )

0 otherwiseb a

X

a x bf x −

⎧⎪ ≤ ≤⎪= ⎨⎪⎪⎩

-1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Example: X ~ U(0,1) and Y = X2.

( )2

0 if 0

( ) if 0 1

1 if 1Y

y

F y P X y y y

y

⎧⎪ <⎪⎪⎪⎪= ≤ = ≤ ≤⎨⎪⎪ >⎪⎪⎪⎩

1

2if 0 1( )

( )0 otherwise

Y yY

ydF yf y

y

⎧⎪ ≤ ≤⎪⎪= = ⎨⎪⎪⎪⎩-0.5 0.0 0.5 1.0 1.5

1

2

3

4


Example: X ~ U(0,1) and

( )212

14

0 if 0

( ) ( ) 2 if 0 1

1 ifY

y

F y P X y y y

y

⎧⎪ <⎪⎪⎪⎪= − ≤ = ≤ ≤⎨⎪⎪ >⎪⎪⎪⎩

1 14

if 0( )( )

0 otherwiseY y

Y

ydF yf y

y

⎧⎪ ≤ ≤⎪⎪= = ⎨⎪⎪⎪⎩

( )212

Y X= −

0.0 0.2 0.4 0.6 0.8 1.0

2

4

6

8


Example: X ~ U(0,1), Y ~ U(0,1), Z = X+Y.

( )2

22

(2 )

2

0 if 0

if 0 1( )

1 if 1 2

1 if 2

z

Z z

z

zF z P X Y z

z

z

−

⎧⎪ <⎪⎪⎪ ≤ ≤⎪⎪= + ≤ = ⎨⎪ − < ≤⎪⎪⎪ >⎪⎪⎩

1 1 if 0 2( )( )

0 otherwiseZ

Z

z zdF zf z

z

⎧⎪ − − ≤ ≤⎪⎪= = ⎨⎪⎪⎪⎩

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

0.2

0.4

0.6

0.8

1.0

Non-uniform Random Numbers

Inverse CDF MethodIf X is a random variable having the probability density function f X(x) and the cumulative distribution function (CDF)

then the random variable U = FX(X)is uniformly distributed on the interval [0,1].

Thus has the desired distribution.

( ) ( )x

X XF x f t dt

−∞= ∫

1( )X

X F U−=


Example: An exponential distribution

for 0( )

0 for 0

x

X

e xf x

x

λλ −⎧⎪ >⎪⎪= ⎨⎪ ≤⎪⎪⎩

0 if 0

( ) 1 if 0 1

1 if 1

xX

x

F x e x

x

λ−

⎧⎪ <⎪⎪⎪= − ≤ ≤⎨⎪⎪ >⎪⎪⎩

lnwhere U(0,1)

UX U

λ= − ∼

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.20.40.60.81.0


Example: A triangular distribution

2 2 4 for 0 1( )

0 otherwiseX

x xf x

⎧⎪ − − ≤ ≤⎪⎪= ⎨⎪⎪⎪⎩

2 12

2 12

0 if 0

2 if 0( )

1 2(1 ) if 1

1 if 1

X

x

x xF x

x x

x

⎧⎪ <⎪⎪⎪ ≤ ≤⎪⎪= ⎨⎪ − − < ≤⎪⎪⎪ >⎪⎪⎩

12 2

1 12 2

ifwhere U(0,1)

1 if

U

U

UX U

U−

⎧⎪ ≤⎪⎪= ⎨⎪ − >⎪⎪⎩

∼

-0.5 0.5 1.0 1.5

0.5

1.0

1.5

2.0

-0.5 0.5 1.0 1.5

0.20.40.60.81.0


Simulated Result: A triangular distribution3N = 10

4N = 10

5N = 10 6N = 10


Acceptance/Rejection MethodX is a random variable with pdf fX(x). We want to generate a random variable Ywith pdf fY(x) and there is a constant c for which ( ) ( ).

Y Xf x cf x≤

( )Yf x

( )X

cf x

x


Acceptance/Rejection AlgorithmGenerate a random value x from X. Generate a random value u from U(0,1).

If , then x is accepted.

Otherwise, x is rejected (and repeat with the step of the generation of x).

( )

( )Y

X

f xu

cf x≤


Box-Muller Method for Normal DistributionLet U ~ U(0,1) andgenerate u1 and u2 from U. Then

are independently distributed as the standard normal distribution.

1 1 2

2 1 2

cos(2 ) 2 ln

sin(2 ) 2 ln

x u u

x u u

π

π

= −

= −

Central Limit Theorem

Central Limit Theorem (CLT)

The sum of a sufficiently large number of independent and identically distributed (i.i.d.) random variables, each with finite mean and variance, will be approximately normally distributed.


Let X1, X2, ..., Xn be a sequence of n i.i.d. random variables each having finite values of expectation μ and variance σ2 > 0.

Let Sn = X1 + X2 + … + Xn. If we define

then Zn → N(0,1).

nn

X nZ

nμ

σ−

=


Example: If a fair coin is tossed 106 times, and we let X = number of heads, then

E[ ] 500, 000

Var[ ] (1 ) 250, 000

X np

X np p

= =

= − =500, 000

(0,1)500

XN

−→

499 000 500 000 501000 502000

0.0002

0.0004

0.0006

0.0008

Monte Carlo Simulation

Monte Carlo simulation was named for Monte Carlo, Monaco, where the primary attractions are casinos containing games of chance. Games of chance such as roulette wheels, dice, and slot machines, exhibit random behavior. Monte Carlo simulation randomlygenerates values for uncertain variables over and over to simulate a model.


Why Simulation?In some circumstances, it may not be feasible to observe the behavior directly or to conduct experiment.

A system of elevators during rush hour.

Automobile traffic controlling.

In other situations, the system for which alternative procedures need to be tested may not even exist yet.

Communication networks in an office buildingLocations of machines in a new industrial plant.


In such instances, the modeler might simulate the behavior and test the various alternatives being considered to estimate how each affects the behavior.

Build a scaled model.Drag force on a proposed submarine.Designs of a jet airplane.

Monte Carlo simulation which is typically accomplished with the aid of a computer.


Processes with an element of chance involved are called probabilistic, as opposed to deterministic.

Monte Carlo simulation is a probabilistic model. It can also be used to approximate a deterministic behavior.

Behavior

Probabilistic

Deterministic

Model

Probabilistic

Deterministic


Probabilistic Behavior: an Unfair DieConsider a probability model where each event is not equally likely. Assume a die is biased according to the following distribution.

0.16

0.25

0.34

0.23

0.12

0.11

P(roll)Roll value

SIX(0.9, 1.0]

FIVE(0.7, 0.9]

FOUR(0.4, 0.7]

THREE(0.2, 0.4]

TWO(0.1, 0.2]

ONE[0, 0.1]

AssignmentValue of xi


0.10.10450.10440.1040.1200.1506

0.20.20110.20120.2010.1840.2805

0.30.30810.30820.3080.3200.3504

0.20.19620.19620.1920.1990.2103

0.10.09920.09920.0990.0990.0802

0.10.09480.09480.0940.0980.1301

Expected Results400001000050001000100Die

Value

Monte Carlo Integration

Let p be the probability of a die having a desired value.Let N be the number of total tosses,and n be the number of tosses with the desired value.

E[ ]

Var[ ] (1 )

n pN

n p p N

== −


If then

Example: If p = 0.1, then

( )2

1ˆE[ ] E[ ]

11ˆVar[ ] Var[ ]

p n pN

p pp n

NN

= =

−= =

( )ˆ

1 0.3ˆVar[ ]

p

p pp

N Nσ

−= = =

ˆ ,n

pN

=


Area under a curve

area under curve number of points counted below curvearea of rectangle total number of random points

≈


The area under the curve y = cos x over the interval -π/2 ≤ x ≤ π/2.

2.01186300001.96641000

2.01093200001.99666900

2.00978150001.99491800

2.00873100002.02857700

2.0066980002.09440600

2.0231960002.04832500

2.0143950002.12058400

1.9996240002.01064300

1.9771130002.13628200

1.9446520002.07345100

Approximation to Area

Number of points

Approximation to area

Number of points


Let the area under the curve be I, while the area of the rectangle is R. Suppose that n out of N points fall under the curve, then we estimate the integral to be

There is a probability I/R that each random point will fall under the curve.

ˆ nI R

N= ⋅


Thus, the number of random points fall under the curve will be binomially distributed.

E[ ]

Var[ ] 1

In N

RI I

n NR R

= ⋅

⎛ ⎞⎟⎜ ⎟= ⋅ −⎜ ⎟⎜ ⎟⎜⎝ ⎠


Example: For

( )2

ˆE[ ] E[ ]

ˆVar[ ] Var[ ]

RI n I

NI R IR

I nN N

= =

⎛ ⎞ −⎟⎜ ⎟= =⎜ ⎟⎜ ⎟⎜⎝ ⎠

/2

/2cos , 2, .x dx I R

π

ππ

−= =∫

( )ˆ

2 2 1.5ˆVar[ ]I

IN N

πσ

−= = ≈

Monte Carlo Optimization

Powerful applications of Monte Carlo simulation lie in numerical optimization.There are many Monte Carlo optimization methods including Genetic Algorithm and Simulated Annealing.

The method is relatively simple, but can effectively solve many famous complex problems.


Simulated AnnealingSimulated annealing is an analogy with thermodynamics when liquids freeze and crystallize, or metals cool and anneal.At high temperatures, the molecules of liquids move freely. If the liquid is cooled slowly, thermal mobility is lost. Atoms line up to form a pure crystal, which is the state of minimum energy.


If the liquid is cooled quickly or quenched, it will not reach the ground state but ends up in some other states having somewhat higher energy.The essence of the process is slowcooling, allowing ample time for redistribution of the atoms as they lose mobility. This is the technical definition of annealing.


A system in thermal equilibrium at temperature T has its energy probabilistically distributed among all different energy states E according to the Boltzman distribution:

where k is the Boltzman’s constant.

/( ) E kTP E e−∼


At low temperatures, the system can still be in a high energy state with a small probability. This allow the system to get out of a local minimum in favor of finding a better, more global, one.

As the temperature is lowered, the system will be less likely to be in a higher energy state.


In 1953, Metropolis et al propose that a system will change its configuration from energy E1 to energy E2 with probability

This general scheme is known as the Metropolis algorithm.

2 1( )/2 1

1 21 2

if ( )

1 if

E E kTe E EP E E

E E

− −⎧⎪ ≥⎪⎪→ = ⎨⎪ <⎪⎪⎩


To make use of the Metropolis algorithm in general systems, we must provide:

Possible system configurations.

Random changes in the configuration

An objective function E to be minimized.

A control parameter T and an annealing schedule.

Could be accomplished with a physical insight or an trial-and-error experiments.


The Traveling Salesman ProblemA salesman visits N cities with given positions (xi,yi), finally returning to the city of origin. Each city is to be visited only once, and the route is to be made as short as possible.

The cities are number i = 1, 2, …, N.


Configuration: A configuration is the order in which the cities are visited; thus, it is a permutation of 1, 2, …, N.Rearrangments: An efficient suggested moves are done by

Reversal: A section of path is removed and replaced with the same cities in opposite order.

Transport: A section of path is removed and replaced in between two cities on another randomly chosen part of the path.


Objective Function: E is just the total length of journey

where the city N+1 is just the city 1.

( ) ( )2 2

1 11

N

i i i ii

E x x y y+ +=

= − + −∑


Annealing Schedule: Require some experiments. First generate some random rearrangements to determine the range of ΔE, then proceed in multiplicative steps each amounting to a 10% decrease in T. Hold a new T for 100Nreconfiguration or 10N successful reconfiguration, whichever comes first. Until efforts to reduce E further becomes discouraging.

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