1. General Principle of Monte Carlo Fall 2013
By Yaohang Li, Ph.D.

2. Review Last Class This Class Next 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

3. 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

4. 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

5. Strength and Weakness of Theoretical Mathematics
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

6. 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

7. 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

8. Reducing Uncertainty Reducing Uncertainty Collect more observations
Not very economic
Variance Reduction
Manipulate the mathematical problem
Reduce the inherent variance

9. Direct Simulation Probabilistic Problem Examples
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

10. 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

11. 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

12. 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, …

13. Direct Simulation of the Lifetime of Comets
Solution
Very difficult problem in theoretical mathematics
Easy to simulate
f(g)=P(Gg) is the required distribution solution

14. Direct Simulation Examples
Computational Financing

15. 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

16. Monte Carlo Method General Principles Efficiency
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

17. Monte Carlo Integration
Consider a simple integral
Definition of expectation of a function on random variable 
If  is uniformly distributed, then

18. 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

19. 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 0x,y1
is the proportion of the are of the square beneath the curve
or we can write

20. Analysis of Hit-or-Miss Monte Carlo
 can be estimated as the a double integral
The estimator of hit-or-miss Monte Carlo

21. 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

22. Binomial Distribution Revisit
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 =

23. 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?

24. 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

25. 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

26. Summary Nature of Monte Carlo Monte Carlo Integration
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
Comet Lifetime Problem
Monte Carlo Integration
Hit or Miss Monte Carlo
Crude Monte Carlo

27. What I want you to do? Review Slides
Review basic probability/statistics concepts
Work on your Assignment 1