1. Lecture 10 Outline Monte Carlo methods History of methods
Sequential random number generators
Parallel random number generators
Generating non-uniform random numbers
Monte Carlo case studies

2. Monte Carlo Methods
Monte Carlo is another name for statistical sampling methods of great importance to physics and computer science
Applications of Monte Carlo Method
Evaluating integrals of arbitrary functions of 6+ dimensions
Predicting future values of stocks
Solving partial differential equations
Sharpening satellite images
Modeling cell populations
Finding approximate solutions to NP-hard problems

3. An Interesting History of Statistical Physics
In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases: great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.
In 1859, Scottish physicist James Clerk Maxwell formulated the distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell used a simple thought experiment: particles must move independent of any chosen coordinates, hence the only possible distribution of velocities must be normal in each coordinate.
In 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell’s paper and was so inspired by it that he spent much of his long, distinguished, and tortured life developing the subject further.

4. History of Monte Carlo Method
Credit for inventing the Monte Carlo method is shared by Stanislaw Ulam, John von Neuman and Nicholas Metropolis.
Ulam, a Polish born mathematician, worked for John von Neumann on the Manhattan Project. Ulam is known for designing the hydrogen bomb with Edward Teller in In a thought experiment he conceived of the MC method in 1946 while pondering the probabilities of winning a card game of solitaire.
Ulam, von Neuman, and Metropolis developed algorithms for computer implementations, as well as exploring means of transforming non-random problems into random forms that would facilitate their solution via statistical sampling. This work transformed statistical sampling from a mathematical curiosity to a formal methodology applicable to a wide variety of problems. It was Metropolis who named the new methodology after the casinos of Monte Carlo. Ulam and Metropolis published a paper called “The Monte Carlo Method” in Journal of the American Statistical Association, 44 (247), , in 1949.

5. Errors in Estimation and Two Important Questions for Monte Carlo
Errors arise from two sources
Statistical: using finite number of samples to estimate an inifinte sum.
Computational: using finite state machines to produce statistically independent random numbers.
Questions
To ensure a level of statistical accuracy, i.e., m significant digits with probability 1-1/n, how many samples are needed?
Given n samples, how statistically accurate is the estimate?
“the best methods rarely offers more than 2-3 digit accuracy”-- G. Fishman, Monte Carlo Methods, Springer

6. Controlling Error Two tenets from statistics:
Weak Law of Large Numbers
value of a sum of i.i.d random variables converges to me n u ( u is mean).
So as n grows approx error vanishes
Central Limit Theorem
Sum converges to a normal distribution with mean nu and variance n var sqrt(n)
So as n grows error can be estimated using normal tables and theorems about tails of distribution.
Caveats
Convergence rates differ and can be complex
Results in additional error estimates or need larger sample size than normal distribution would indicate

7. Sample Size and Chebyshev’s Theorem
Theorem: If X is a discrete random variable with mean  and standard deviation , then for every positive real number
or equivalently
That is, the probability that a random variable will assume a value more than h standard deviations away from its mean is never greater than the square of the reciprocal of h.

8. An Example:
200 years ago, Comte de Buffon, proposed the following problem. If a needle of length l is dropped at random on the middle of a horizontal surface ruled with parallel lines a distance d > l apart, what is the probability that the needle will cross one of the lines? 
The positioning of the needle relative to nearby lines can be described with a random vector (A, theta) is uniformly distributed on the region [0,d) [0,pi). Accordingly, it has probability density function above. The probability that the needle will cross one of the lines is given by the integral

9. Estimation and variance reduction
This integral is easily solved to obtain the probability
Suppose Buffon’s experiment is performed with the needle being dropped n times. Let M be the random variable for the number of times the needle crosses a line,
Certain techniques can be used to reduce variance, e.g., an experimenter Fox used five inch needle and made the lines on the surface just two inches apart. Now, the needle could cross as many as three lines in a single toss. In 590 tosses, Fox obtained 939 line crossings. This was a precursor of today’s variance reduction techniques.

10. Variance Reduction There are several ways to reduce the variance
Importance Sampling
Stratified Sampling
Quasi-random Sampling
Metropolis Random Mutations

11. Simulation Often rely on Other Distributions
Analytical transformations
Box-Muller Transformation
Rejection method

12. Analytical Transformation -probability density function f(x) -cumulative distribution F(x) In theory of probability, a quantile function of a distribution is the inverse of its cumulative distribution function.

13. Exponential Distribution: An exponential distribution arises naturally when modeling the time between independent events that happen at a constant average rate and are memoryless. One of the few cases where the quartile function is known analytically.
1.0

14. Example 1:
Produce four samples from an exponential distribution with mean 3
Uniform sample: 0.540, 0.619, 0.452, 0.095
Take natural log of each value and multiply by -3
Exponential sample: 1.850, 1.440, 2.317, 7.072

15. Example 2: Simulation advances in time steps of 1 second
Probability of an event happening is from an exponential distribution with mean 5 seconds
What is probability that event will happen in next second?
F(x=1/5) =1 - exp(-1/5)) =
Use uniform random number to test for occurrence of event (if u < then ‘event’ else ‘no event’)

16. Normal Distributions: Box-Muller Transformation
Cannot invert cumulative distribution function to produce formula yielding random numbers from normal (gaussian) distribution
Box-Muller transformation produces a pair of standard normal deviates g1 and g2 from a pair of normal deviates u1 and u2

17. Box-Muller Transformation
repeat
v1  2u1 - 1
v2  2u2 - 1
r  v12 + v22
until r > 0 and r < 1
f  sqrt (-2 ln r /r )
g1  f v1
g2  f v2
This is a consequence of the fact that the chi-square distribution with two degrees of freedom is an easily-generated exponential random variable.

18. Example
Produce four samples from a normal distribution with mean 0 and standard deviation 1 u1 u2 v1 v2 r f g1 g2
0.234
0.784
-0.532
0.568
0.605
1.290
-0.686
0.732
0.824
0.039
0.648
-0.921
1.269
0.430
0.176
-0.140
-0.648
0.439
1.935
-0.271
-1.254

19. Von Neumann’s Rejection Method

20. Case Studies (Topics Introduced)
Temperature inside a 2-D plate (Random walk)
Two-dimensional Ising model (Metropolis algorithm)
Room assignment problem (Simulated annealing)
Parking garage (Monte Carlo time)
Traffic circle (Simulating queues)

21. Temperature Inside a 2-D Plate
Random walk

22. Example of Random Walk 3 2 2 1 2 1 3 1

23. Parking Garage Parking garage has S stalls
Car arrivals fit Poisson distribution with mean A: Exponentially distributed inter-arrival times
Stay in garage fits a normal distribution with mean M and standard deviation M/S

24. Implementation Idea 64.2 Times Spaces Are Available 101.2 142.1 70.3
91.7
223.1
Current Time
Car Count
Cars Rejected
64.2 15 2

25. Summary Concepts revealed in case studies Monte Carlo time Random walk
Metropolis algorithm
Simulated annealing
Modeling queues