Delivering Integrated, Sustainable, Water Resources Solutions Monte Carlo Simulation

Transcription

1 Monte Carlo Simulation Robert C. Patev North Atlantic Division Regional Technical Specialist (978)

2 Topics Basics Reliability Reporting Demonstration

3 Monte Carlo Simulation Types of simulation methods Direct brute force method Stratified effort in regions Latin Hypercube form of stratified sampling Importance selected shift in distributions Adaptive form of importance sampling

4 Introduction Monte Carlo Simulation (MCS) Spreadsheet add-in Excel Macros User friendly interface Easy input Many probability distribution functions Graphical output

5 CAVEAT Let the engineer beware Not just a black box that gives the correct answer or decision Tool to assist in making decisions and arriving at a solution Understand the inputs to your model Understand limitations in your spreadsheets Cautiously scrutinize and review output (Does it make sense?)

6 @Risk Use within the Corps of Engineers Reliability Analysis Structural Geotechnical Economic Analysis Major Rehabilitation Projects System Studies ORMSS, GLSLS

7 @Risk Capabilities Easily adds MCS to existing spreadsheet model Fast execution time Save MCS results quickly User-defined macros Complete statistical analysis Input Output Sensitivity

8 @Risk Basics Iterations vs. simulations Iteration - an iteration is a single sampling of random variables Simulation - x number of iterations Monte Carlo Simulation methods Direct sampling Latin hypercube sampling

9 1.0 Monte Carlo Simulation 1.0 Cumulative Probability Cumulative Probability 0 0 Direct Sampling Latin Hypercube

10 @Risk Basics Random number seed generator -1 to (default = 0) Convergence Input random variables Selected output cells User-defined macros

11 @Risk Basics Random Variables Numerous discrete/continuous distributions Correlation Positive/negative Examine outputs Truncation Physical limitations to data Examine results

12 @Risk Basics Negative Positive Random Variable B Random Variable B Random Variable A Random Variable A

13 @Risk Basics Truncation 0.4 pdf Area under curve = 1 0 XL XU

14 Reliability Delivering Integrated, Sustainable, Reliability R = 1 - P(u) where, P(u) = N pu / N N pu = Number of unsatisfactory performances at limit state < 1.0 N = number of iterations

15 Random Variables Distributions Statistical parameters (min/max, mean, std. dev., ) Distribution types Questions - Why use, Where come from, How applied in model, What other distributions can be used Correlation/truncation Justification Plots of simulated distributions for random variables and selected output cells from simulation

16 Sensitivity/Convergence Sensitivity Identifies the most critical variables to the output Range: +1 to -1 (closest to (+/-)1, model most sensitive) R-squared method/rank correlation coefficient Convergence Limit state functions Probability of unsatisfactory performance