1. 1 Simulation: Modeling Uncertainty with Monte Carlo 12-706 / 19-702

2. 2 Notes on Uncertainty zUncertainty is inherent in everything we do yThere are no right answers zOur goal: yUnderstand and model it --> make better decisions zWe ‘internalize’ uncertainty using ranges or distributions of our inputs yThis is a computationally intensive idea

3. 3 First: Reading PDFs & CDFs zWhat information can we get from simply looking at a PDF or CDF?

4. 4 Some types of distributions zDiscrete zUniform zNormal zTriangular zExponential zLognormal z… zAll defined/expressed in the text

5. Some Reminders zFor 2 variables X and Y: yE(X+Y) = E(X) + E(Y) yE[f(x)] does not necessarily equal f(E[x]) zFor 2 independent X and Y (not typical) yE(X*Y) = E(X)*E(Y) yVar(X+Y) = Var(X-Y) = Var(X) + Var(Y) yStd dev = sqrt(Var) 5

6. Adding Distribution Functions zTriangular distribution with min -10, mid 0, max 10.. (mean 0, st dev 4) yIf X and Y both modeled like this.. yThen E(X+Y)=0 ; st dev ~ 5.6 (as above) 6

7. Simulation alternatives zPhysical – hydraulic tanks used for blood flow models in Porter Hall

8. 3D CAD Simulations

9. Deterministic (system) dynamics

10. Monte Carlo (Stochastic) Simulation

11. 11 Monte Carlo History zNot a new idea: “Statistical sampling” zFermi, Ulam, working on Manhattan Project in 1940’s yDeveloped method for doing many iterations of picking inputs to generate a distribution of “answers” yFinally have “fast” computers zLater, named after Monte Carlo (famous for gambling)

12. Monte Carlo History - Hendrickson zI used Monte Carlo Simulation for my doctoral dissertation research in 1977. zSimulated dial-a-ride tours in a service area to validate an analytical model. (used to model service tours performance). zGenerated service call times and locations, then constructed vehicle tours and calculated performance. zSeveral hundred computer punch cards for simulation of tours in areas. zNow could be done rapidly on a spreadsheet.

13. Computer Punch Cards

14. 14 Monte Carlo Method zMonte Carlo analysis: 3 steps 1.Specify probability distributions in place of constants/variables in a model 2.Trial by random draws 3.Repeat for many* trials zMonte Carlo does not yield “the right answer”! xProduces a distribution of results xMany random draws simulated -> convergence * hundreds or thousands

15. Random Draws zHistorically, developing a random number generator was difficult. zNow, straightforward using numerical methods and modern computers.

16. Validating Simulation Models zValidation is difficult! zCheck internal consistency. zCheck input distribution assumptions. zCheck against known data. zDocument assumptions.

17. 17 Monte Carlo Simulations zWe’ll use @RISK (part of DecisionTools Suite) zAdds special probability functions to Excel yExcel has some, but these are better yBunch of distributions: Binomial, Discrete, Exponential, Normal, Poisson, Triangular, Uniform (more on these in a moment) zHas a lot of nice post-simulation analysis yStatistics, graphs, reports yAgain, these are not “answers”

18. 18 Let’s Test It! zLook at test-montecarlo-07.xls spreadsheet yNumber of trials sheet first yCan random draws from a normal distribution give us the parameters of that distribution? xi.e. given a distribution w/mean, st dev “can we sample it”? xSee Excel formula to do so (and link on web page for more) yWhat difference does 10, 100, 1000 or 10000 trials make? yLook at histogram of trials worksheet to visualize yDon’t worry about MC mechanics: tutorial next week

19. 19 Let’s talk about distributions zYou (should) have seen these before: yUniform, Normal, Triangular, Binomial, Discrete, Poisson, possibly Exponential, Lognormal, Weibull yImportant part of MC is picking “correct” distributions & parameters yBe careful of over-thinking this choice! yFirst – lets talk about normal (bell curve)

20. 20 Distribution Examples?

21. 21 Homework Grades – Normal?

22. 22 Distributions – Thought Examples zWhat processes / items could be modeled with: zUniform zNormal zTriangular zBinomial zDiscrete zPoisson zExponential zLognormal zWeibull

23. Bootstrap Approach zApplicable when you have a sample of observations. zSample repeatedly from the observations with equal probability of each being used. zDoesn’t require underlying distribution assumption. zCan compare bootstrap and classic approaches.

24. Results zWhen you add distributions yYou get expected results as in E(X+Y) above zWhat happens when you multiply them? 24

25. 25 zUsing examples familiar to us zInstead of point estimates, use probabilistic functions yPick up penny example? Examples Using Monte Carlo

26. Penny Example zWhy do we get big weights at bottom/top of output distributions? zHow can we fix? 26

27. 27 Wrap Up zWe have much better models - and knowledge of our results now zImportant take away messages: yDon’t introduce unnecessary uncertainty with your input choices yMonte Carlo doesn’t give you the answer yInterpreting output (PDFs/CDFs) gives you an answer