General Principle of Monte Carlo

Fall 2012

By Yaohang Li, Ph.D.

Review• Last Class

– Short Resume of Statistics• Probability• Random Variables• Important Distributions

– Uniform Distribution– Exponential Distribution– Binomial Distribution– Poison Distribution– Normal Distribution

• Estimation– Sample– Estimand– Parent Distribution– Sampling Distribution– Estimator

» Important estimators• This Class

– Nature of Monte Carlo– Direct Simulation– Monte Carlo Integration

• Next Class– Analysis of Monte Carlo Integration

Mathematical Theory and Experiment

• Theoretical Mathematics– Deduce conclusions from postulates

• Experimental Mathematics– Infer conclusions from observations

– Complicated Mathematical Experiments require computers

• Difference– Between deduction and induction

Problems Handled by Monte Carlo• Probabilistic Problems

– Directly concern with the behavior and outcome of random processes

– Approach (Direct Simulation)• Choose random numbers in a certain way• Directly simulate the physical random process of the

original problem• Infer the desired solution from the behavior of these

random numbers– Usually possess little theoretical interest

• Deterministic– Not directly concern with the behavior of random processes– Usually called Sophisticated Monte Carlo

Strength and Weakness of Theoretical Mathematics

• Theoretical Mathematics– Strength

• Abstraction and Generality

• Abstract the essence of a problem and reveal its underlying structure

– Inherent weakness

• The more general and formal the theory

• The harder to provide a numerical solution in a particular application

• Main idea behind Monte Carlo approach to deterministic problems– Exploit the strength of theoretical mathematics

– Avoid its associated weakness by replacing theory by experiment whenever the theory falters

Example of Sophisticated Monte Carlo

• Electromagnetic Theory– Requires solution of Laplace’s equation subject to certain boundary

conditions

– The boundary conditions may be complicated and no analytical techniques are available

• Monte Carlo Method– Study of particles

• Diffuse randomly in a region bounded by absorbing barriers

• Guide the particles by means of random numbers

• Until they are absorbed on barriers– Barriers are chosen to represent the prescribed boundary conditions

– The distribution of particles absorbed on barriers gives the solution

Monte Carlo Solution• Monte Carlo Solution

– Uncertain

• Arise from raw observational data consisting of random numbers

• If we can manage to make the uncertainty fairly negligible, Monte Carlo can serve a useful purpose

– Good experiments

• Ensure the sample should be more rather than less representative in the Monte Carlo computation

– Good representation of conclusions

• Indicates how likely the solution can be wrong

Reducing Uncertainty

• Reducing Uncertainty– Collect more observations

• Not very economic

– Variance Reduction

• Manipulate the mathematical problem

• Reduce the inherent variance

Direct Simulation• Probabilistic Problem

– Direct simulate the physical random processes of the original problem– Direct observation of the random numbers

• Examples– Control of the floodwater and the construction of dams on the Niles

• The quantity of water in the river varies randomly from season to season

• Use data records of weather, rainfall, and water levels extending over 48 years

• To see what will happen to the water if certain dams are built and certain possible policies of water control exercised

– Telenetworking– Growth of an insect population on the basis of certain assumed vital

statistics of survival and reproduction– Service rates– Operational research– Computer games

Direct Simulation of the Lifetime of Comets

• A Long-period Comet– Described as a sequence of elliptic orbits

• Sun at one focus

• Energy of a comet is inversely proportional to the length of the semimajor axis of the ellipse

Direct Simulation of the Lifetime of Comets

• Behavior of the Comet– Most of the time

• Moves at a great distance from the sun

– A relatively short time (instantaneous)

• Passes through the immediate vicinity of the sun and the planets

• At this instant, the the gravitational field of the planet perturbs the cometary energy by a random component

• Successive energy perturbations (in suitable units of energy) may be taken as independent normally distributed random variables 1, 2, 3,…, n

1, 2, 3,…, n can be normalized to a standard normal distribution

Direct Simulation of the Lifetime of Comets

• Comets under perturbation– A comet, starting with an energy –z0, has accordingly energies

-z0, -z1=-z0+1, -z2=-z1+2, …

– The process continues until the first occasion on which z changes sign

• Once z changes sign, the comet departs on a hyperbolic orbit and is lost from the solar system

• Kepler’s Third Law– the time taken to describe an orbit with energy –z is z-3/2

– the total lifetime of the comet is

• zT is the first negative quantity in the sequence of z0, z1, …

1

0

2/3T

iizG

Direct Simulation of the Lifetime of Comets

• Solution– Very difficult problem in theoretical mathematics

– Easy to simulate

• f(g)=P(Gg) is the required distribution solution

Direct Simulation Examples

• Computational Financing– http://www.mathfinance.wagner.com/PRODUCTS/RETIREM

ENT/RSP/Applets/BernApplet.html

Numerical Integration• Methods for approximating

definite integrals– Rectangle Rule

– Trapezoidal Rule

• Divide the curve into N strips of thickness h=(b-a)/N

• Sum the area of each trip– Approximate to that of a

trapezium

– Simpson’s Rule

• Calculate quadratic instead

Monte Carlo Method• General Principles

– Every Monte Carlo computation that leads to quantitative results may be regarded as estimating the value of a multiple integral

• Efficiency– Definition

• Suppose there are two Monte Carlo methods

• Method 1: n1 units of computing time, 12

• Method 2: n2 units of computing time, 22

• Methods comparison

222

211

n

n

Monte Carlo Integration• Consider a simple integral

• Definition of expectation of a function on random variable

• If is uniformly distributed, then

1

0

)( dxxf

1

0

)()())(( dxxpxffE

1

0

)())(( dxxffE

Crude Monte Carlo

• Crude Monte Carlo– If 1, …, n are independent random numbers

• Uniformly distributed

– then fi=f(i) are random variates with expectation

– is an unbiased estimator of – The variance is

– The standard error is /n1/2

n

iifn

f1

1

ndxxfn

fE /))((1

))(( 21

0

22

Hit-or-Miss Monte Carlo Revisit

• Suppose 0 f(x) 1 when 0 x 1• Main idea

– draw a curve y=f(x) in the unit square 0x,y1

– is the proportion of the are of the square beneath the curve

– or we can write

y f(x) if 1

yf(x) if

0),(

),()(1

0

yxg

dyyxgxf

Analysis of Hit-or-Miss Monte Carlo

can be estimated as the a double integral

• The estimator of hit-or-miss Monte Carlo

1

0

1

0

),( dxdyyxg

n

iii n

ng

ng

1

*

212 ),(1

Hit-or-Miss Monte Carlo

• Hit-or-Miss Monte Carlo– We take n points at random in the unit square, and count the

proportion of them which lie below the curve y=f(x)

– The points are either in or out of the area below the curve

• The probability that a point lies under the curve is

• The Hit-or-Miss Monte Carlo is a Bernoulli trial– the estimator of Hit-or-Miss Monte Carlo is binomial

distributed

Binomial Distribution Revisit• Binomial Distribution

– Discrete probability distribution Pp(n|N) of obtaining exactly n successes out of N Bernoulli trials

– Each Bernoulli trial is true with probability p and false with probability q=1-p

• variance N(1-p)p= =

Comparison of Hit-or-Miss Monte Carlo and Crude Monte Carlo

• Standard error of Hit-or-Miss Monte Carlo

• Standard error of Crude Monte Carlo

• Hit-or-Miss Monte Carlo is always worse than Crude Monte Carlo– Why?

n

)1(

n

dxf 1

0

2)(

Why Hit-or-Miss Monte Carlo is worse?

• Fact– The hit-or-miss to crude sampling is equivalent to replacing

g(x, ) by its expectation f(x)

– The y variable in g(x,y) is a random variable

• Leads uncertainty

• Leads extra uncertainty in the final results

• Can be replace by exact value

General Principle of Monte Carlo

If, at any point of a Monte Carlo calculation, we can replace an estimate by an exact value, we shall replace an estimate by an exact value, we shall reduce the sampling error in the final result

Summary• Nature of Monte Carlo

– Experimental Mathematics and Theoretical Mathematics– Problems that Monte Carlo can handle

• Probabilistic– Direct Simulation

• Deterministic– Sophisticated Monte Carlo

– Monte Carlo Solution• Good Experiment• Good Conclusion

• Direct Simulation– Comet Lifetime Problem

• Monte Carlo Integration– Hit or Miss Monte Carlo– Crude Monte Carlo

What I want you to do?

• Review Slides• Review basic probability/statistics concepts• Work on your Assignment 1