1. Monte Carlo Process Risk Analysis for Water Resources Planning and Management Institute for Water Resources 2008

2. Learning Objectives At the end of this session participants will understand: The difference between an analytical solution and a simulation. The two steps of the Monte Carlo process. A simulation comprises many iterations. The purpose of more iterations in a simulation

3. Analytical Solution Solution meets all criteria of problem 2 + x = 4 x = 2 is a solution that works Some problems have more than one analytical solution x^2 = 9 Some problems have no analytical solution

4. Simulation Numerical technique used to estimate analytical solutions to a problem Not an optimization technique, answers what-if questions Results are not analytical solutions Analytical solutions are preferred

5. Monte Carlo Process Code name for simulations relating to development of atomic bomb 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

6. Time to Understand We’ve used Monte Carlo process Two-step process in each cell in spreadsheet

7. Monte Carlo Process Two steps Generate a simple random number. Transform it into a useful value using a specific probability distribution.

8. Random Number Generation Pseudorandom Numbers [0,1] Seed = 4745 (any number) Mid-square Method (John von Neumann) (4745)^2 = 22515025; r1=.5150 (5150)^2 = 26522500; r2=.5225 (5225)^2 = 27300625; r3=.3006 etc. More sophisticated method

9. Transformation Assume Uniform Distribution, U(a,b) where a = 10 and b = 50. To obtain a value, x, we use x = a + (b - a)u. In this case, x = 10 + 40u. Generate U~U(0,1), say u =.5150 then x = 10 +(50 - 10).5150 = 30.6 x = 10 +(50 - 10).5225 = 30.9 x = 10 +(50 - 10).3006 = 22.0, etc. Other distributions are similar but more complex transformations.

10. Some Language Simulation--technique for calculating a model output value many times with different input values. Purpose is to get complete range of all possible scenarios. Iteration--one recalculation of the model during a simulation. Uncertain variables are sampled once during each iteration according to their probability distributions.

11. Monte Carlo Simulation Simulation model that uses the Monte Carlo process Deterministic values replaced by distributions Values randomly generated for each probabilistic variable & calculations completed Process repeated desired # times

12. Monte Carlo Simulation X = 20 10

13. How Many Iterations? Means often stabilize quickly (10 2 ) Estimating probabilities of outcomes (10 3 ) Defining tails of output distribution (10 4 ) If extreme events are important (10 5 )

14. Reading and Reporting Results Means Minimums and maximums (extremes?) Percentiles 95% confidence interval Graphs and tables

15. Take Away Points Don’t simulate when analytical solutions exist Monte Carlo process is not a simulation Generate random number Transform it into useful value Desired iterations depends on purpose