Monte Carlo Simulation

310 Views

Download Presentation
## Monte Carlo Simulation

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Monte Carlo Simulation**Robert C. Patev North Atlantic Division – Regional Technical Specialist (978) 318-8394**“Monte Carlo” is the method (code name) for simulations**relating to development of atomic bomb during WWII • Traditional – static not dynamic (not involve time), U(0,1) • Non-Traditional – multi-integral problems, dynamic (time) • Applied to wide variety of complex problems involving random behavior • Procedure that generates values of a random variable based on one or more probability distributions • Not simulation method per se – just a name!**Analytical solution**• For many problems analytical solution difficult or may not exist • An approximate solution • Monte Carlo Simulation • Monte Carlo process • Frequently used in risk analysis to generate sample of realizations of the model • Not always necessary • Can enumerate all the end-points • Model collapsed into single period • Full enumeration may not be feasible • Simulation produces no new information about contributing variables • The analyst must provide the distributions for the contributing variables • Monte Carlo simulation is a numerical technique for estimating a distribution • Resulting statistics are reflection of sample**Simple Example:**Determine the expected value and distribution of the sum of two die Each face of each die has equal probability = 1/6 Multiple ways of getting the same sum Analytical Solution: Enumerate all possible combinations EX. Pr(3) = Pr(1 and 2) + Pr(2 and 1) = 1/36+1/36 = 0.05556**Requirements for Monte Carlo Simulation**(1)Model describing in quantitative terms the variable(s) of interest and the relationship among them (2)Estimate distributions of the contributing variables (3)Generate random numbers (4)Transform random numbers into useful values using a specific probability distribution (5)Criteria for determining sample size (6)Perform a statistical analysis on the random sample to determine**Results from Monte Carlo Simulation**• Means often stabilize quickly--few hundred • Estimating probabilities of outcomes take MANY more • Defining tails of output distribution takes MANY MANY more iterations • If extreme events are important it make take MANY MANY MANY more • Convergence is key to a believable result from a simulation. Documentation is critical. • Depends on the degree of accuracy desired**Criteria for Determining Sample Size**Number of iterations required increases with • increases in variance and skew • reductions in probability • the number of variables simulated • Many rules of thumb in literature….. • Really need to set and examine convergence criteria • Convergence • Sample mean • Sample variance • Extremes--maximum and minimum • Percentiles • Confidence intervals**Stopping Rules**• Some commercial Monte Carlo simulation software rely on convergence • User specified percentage change in sample statistics • Careful…..THIS MAY NOT BE TRUE CONVERGENCE OF THE LIMIT STATE! • Careful….CONVERGENCE ON TIME-DEPENDENT PROBLEMS NEEDS TO BE EXAMINED ON EACH TIME INCREMENT!