Monte Carlo Methods

Versatile methods for analyzing the behavior of

some activity, plan or process that involves uncertainty

What are Monte Carlo Methods?

Monte Carlo (MC) methods are stochastic techniques--meaning they are based on the use of random numbers and probability statistics to investigate problems

A technique that provides approximate solutions to problems expressed mathematically. Using random numbers and trial and error, it repeatedly calculates the equations to arrive at a solution.

MC methods are used in everything from economics to nuclear physics to regulating the flow of traffic

http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html

Monte Carlo Methods - cont

Solving equations which describe the interactions between two atoms is fairly simple; solving the same equations for hundreds or thousands of atoms is impossible

With MC methods, a large system can be sampled in a number of random configurations, and that data can be used to describe the system as a whole.

To call something a "Monte Carlo" experiment, all you need to do is use random numbers to examine some problem.

http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html

Monte Carlo Applications

Design and visuals - Monte Carlo methods have also proven efficient in solving coupled integral differential equations that helps in computer generated films, special effects in cinema etc

Finance and Business - Often used to calculate the value of companies, to evaluate investments in projects at a business unit or corporate level, or to evaluate financial derivatives.

Games - Monte Carlo methods have recently been applied in game playing related artificial intelligence theory. Most notably the game of Go has seen remarkably successful Monte Carlo algorithm based computer players

Monte Carlo 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.

Monte Carlo Calculation of Pi

The value of π can be approximated using a Monte Carlo method:

Draw a square on the ground, then inscribe a circle within it. From plane geometry, the ratio of the area of an inscribed circle to that of the surrounding square is π/4.

Uniformly scatter some objects of uniform size throughout the square. For example, grains of rice or sand.

Since the two areas are in the ratio π/4, the objects should fall in the areas in approximately the same ratio. Thus, counting the number of objects in the circle and dividing by the total number of objects in the square will yield an approximation for π/4. Multiplying the result by 4 will then yield an approximation for π itself.

Monte Carlo Calculation of Pi

Total number of darts that hit within the square, the number of darts that hit the shaded part (circle quadrant) is proportional to the area of that part.

How good it is, depends on how many iterations (throws) are done

π approximation follows the general pattern of Monte Carlo algorithms.

First, we define a domain of inputs: in this case, it's the square which circumscribes our circle.

Next, we generate inputs randomly (scatter individual grains within the square), then perform a computation on each input (test whether it falls within the circle).

At the end, we aggregate the results into our final result, the approximation of π.

C Function to calculate Pi

Monte Carlo Pi Simulations

http://pagesperso-orange.fr/jpq/proba/montecarlo/index.htm

Components of Monte Carlo Algorithms Random number generator

Sampling rule

Probability Distribution Functions (PDF)

Error estimation

Markov Process A Markov process – a mathematical model for the random evolution of a

memoryless system, that is, one for which the likelihood of a given future state, at any given moment, depends only on its present state, and not on any past states.

Board games played with dice Monopoly, Snakes and Ladders

Transitions of Markov Chain is determined by a stochastic matrix or a probability matrix. Each row (or column) of a stochastic matrix is a probability vector, which are sometimes called stochastic vectors. A vector whose elements consist should always sum to 1

Example of Markov Chains

Weather Prediction The weather on day 0 is known to be sunny. This is represented by a vector in which

the "sunny" entry is 100%, and the "rainy" entry is 0%:

The weather on day 1 can be predicted by:

Thus, there is an 90% chance that day 1 will also be sunny.

General Rule for day n are:

http://en.wikipedia.org/wiki/Examples_of_Markov_chains

References

1. http://en.wikipedia.org/wiki/Monte_Carlo_method

2. http://www.answers.com/topic/monte-carlo-method