Simulation Wrap-up, Statistics COS 323

Transcription

1 Simulation Wrap-up, Statistics COS 323

2 Today Simulation Re-cap Statistics Variance and confidence intervals for simulations Simulation wrap-up FYI: No class or office hours Thursday

3 Simulation wrap-up

4 Last time Time-driven, event-driven Simulation from differential equations Cellular automata, microsimulation, agentbased simulation see e.g. %20microsimulation.htm Example applications: SIR disease model, population genetics

5 Simulation: Pros and Cons Pros: Building model can be easy (easier) than other approaches Outcomes can be easy to understand Cheap, safe Good for comparisons Cons: Hard to debug No guarantee of optimality Hard to establish validity Can t produce absolute numbers

6 Simulation: Important Considerations Are outcomes statistically significant? (Need many simulation runs to assess this) What should initial state be? How long should the simulation run? Is the model realistic? How sensitive is the model to parameters, initial conditions?

7 Statistics Overview

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9 Random Variables A random variable is any probabilistic outcome e.g., a coin flip, height of someone randomly chosen from a population A R.V. takes on a value in a sample space space can be discrete, e.g., {H, T} or continuous, e.g. height in (0, infinity) R.V. denoted with capital letter (X), a realization with lowercase letter (x) e.g., X is a coin flip, x is the value (H or T) of that coin flip

10 Probability Mass Function Describes probability for a discrete R.V. e.g.,

11 Probability Density Function Describes probability for a continuous R.V. e.g.,

12 [Population] Mean of a Random Variable aka expected value, first moment for discrete RV: E[ X] = µ = x i p i i for continuous RV: E X [ ] = µ = x p(x) dx

13 [Population] Variance σ 2 = E [(X µ) 2 ] = E[ X 2 2Xµ + µ 2 ] = E[ X 2 ] µ 2 [ ] E X = E X 2 ( [ ]) 2 for discrete RV: i σ 2 = p i (x i µ) 2 for continuous RV: σ 2 = (x µ) 2 p(x) dx

14 Sample mean and sample variance Suppose we have N independent observations of X: x 1, x 2, x N Sample mean: 1 N N i=1 x i = x Sample variance: N 1 (x N 1 i x ) 2 = s 2 i=1 E[x ] = µ E[s 2 ] = σ 2

15 1/(N-1) and the sample variance The N differences x i x (x i x ) = 0 are not independent: If you know N-1 of these values, you can deduce the last one i.e., only N-1 degrees of freedom Could treat sample as population and compute population variance: 1 N N i=1 (x i x ) 2 BUT this underestimates true population variance (especially bad if sample is small)

16 Sample variance using 1/(N-1) is unbiased [ ] = E E s 2 1 N 1 = 1 N 1 E N i=1 N i=1 (x i x ) 2 x 2 i Nx 2 = 1 N 1 N σ 2 + µ 2 = σ 2 ( ) N σ 2 N + µ2

17 Computing sample variance Can compute as s 2 = 1 N 1 N i=1 (x i x ) 2 Prefer: s 2 = N 2 x i i=1 N(x ) 2 N 1

18 The Gaussian Distribution 1 p(x) = σ 2π e E[X] = µ Var[X] = σ 2 1 x µ 2 σ 2

19 Why so important? sum of independent observations of a random variable converges to Gaussian in nature, events having variations resulting from many small, independent effects tend to have Gaussian distributions demo: e.g., measurement error if effects are multiplicative, logarithm is often normally distributed

20 Central Limit Theorem Suppose we sample x 1, x 2, x N from a distribution with mean μ and variance σ 2 Let then x x = 1 N N x i i=1 z = x µ σ / N N(0,1) i.e., distributed normally with mean μ, variance σ 2 /N

21 Important Properties of Normal Distribution 1. Family of normal distributions closed under linear transformations: if X ~ N(μ, σ 2 ) then (ax + b) ~ N(aμ+b, a 2 σ 2 ) 2. Linear combination of normals is also normal: if X 1 ~ N(μ 1, σ 12 ) and X 2 ~ N(μ 2, σ 22 ) then ax 1 +bx 2 ~ N(aμ 1 + bμ 2, a 2 σ b 2 σ 22 )

22 Important Properties of Normal Distribution 3. Of all distributions with mean and variance, normal has maximum entropy Information theory: Entropy like uninformativeness Principle of maximum entropy: choose to represent the world with as uninformative a distribution as possible, subject to testable information If we know x is in [a, b], then uniform distribution on [a, b] has least entropy If we know distribution has mean µ, variance σ 2, normal distribution N(µ, σ 2 ) has least entropy

23 Important Properties of Normal Distribution 4. If errors are normally distributed, a least-squares fit yields the maximum likelihood estimator Finding least-squares x st Ax b finds the value of x that maximizes the likelihood of data A under some model

24 Important Properties of Normal Distribution 5. Many derived random variables have analytically-known densities e.g., sample mean, sample variance 6. Sample mean and variance of n identical independent samples are independent; sample mean is a normally-distributed random variable X n ~ N(µ,σ 2 /n)

25 Distribution of Sample Variance s 2 = 1 N 1 (For Gaussian R.V. X) N i=1 (x i x ) 2 (n 1)s2 define U = σ 2 then U has a χ 2 distribution with (n -1) d.o.f. p(x) = 2 n / 2 Γ n 2 1 ( x) n 2 1 e x / 2, x 0 E[ U] = n 1, Var[ U] = 2(n 1)

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27 The Chi-Squared Distribution

28 What if we don t know true variance? Sample mean is normally distributed R.V. X n ~ N(µ,σ 2 /n) Taking advantage of this presumes we know σ 2 x µ has a t distribution with (n-1) d.o.f. s n / n

29 [Student s] t-distribution

30 Forming a confidence interval e.g., given that I observed a sample mean of, I m 99% confident that the true mean lies between and. Know that x µ s n / n has t distribution Choose q 1, q 2 such that student t with (n-1) dof has 99% probability of lying between q 1, q 2

31 Confidence interval for the mean if P q 1 < x n µ s n / n < q 2 = 0.99 s then P x n q n 2 n < µ < x q n 1 s n n = 0.99

32 Interpreting Simulation Outcomes How long will customers have to wait, on average? e.g., for given # tellers, arrival rate, service time distribution, etc.

33 Simulate bank for N customers Let x i be the wait time of customer i Is mean(x) a good estimate for µ? How to compute a 95% confidence interval for µ? Problem: x i are not independent!

34 Replications Run simulation to get M observations Repeat simulation N times (different random numbers each time) Treat the sample mean of different runs as approximately uncorrelated s 2 = 1 n 1 i (X i X ) 2

35 Batch Means Run simulation for N (large) Divide x i into k consecutive batches of size b If b large enough, mean(batch1) approx. uncorrelated with mean(batch2), etc.

36 Other approaches Use estimation of autocorrelation between x i s to derive better estimate of variance that can be used for confidence interval Regenerative method: Take advantage of regeneration points or cycles in behavior e.g., points when bank is empty of customers

37 Simulation Wrap-up

38 Finally

39 Implications Who designed it all? How should we behave? What if we start running too many of our own simulations?

40 Software List_of_computer_simulation_software