Section 8.2: Monte Carlo Estimation

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1 Section 8.2: Monte Carlo Estimation Discrete-Event Simulation: A First Course c 2006 Pearson Ed., Inc Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 1/ 19

2 Section 8.2: Monte Carlo Estimation Example Program craps was run once with n = 100 replications Got 56 wins and 44 losses (initial seed 12345) The true probability of winning Craps is Our point estimate of 0.56 is not accurate Using interval estimation, 56 wins in 100 tries corresponds to a 95% confidence interval estimate of 0.56 ± 0.10 Two important points: The 0.56 ± 0.10 interval estimate is correct in the sense that it contains the true probability of winning The interval is too wide indicating that more than 100 replications should have been used Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 2/ 19

3 Interval Estimation for Probability In Monte Carlo Simulation, x 1, x 2,..., x n is a random sample of size n Each x i is either 0 ( loss ) or 1 ( win ) The sample mean (probability point estimate) is x = 1 n n x i = i=1 the number of 1 s n The sample standard deviation is ( ) s = 1 n x 2 n i x 2 = x x 2 = x(1 x) i=1 Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 3/ 19

4 Interval Estimation for Probability (2) From Alg , for large n, the interval estimate for µ is x ± t x(1 x) n 1 where t = idfstudent(n 1,1 α/2) When Monte Carlo simulation is used to estimate a probability, the sample size is always large (in practice) Consistent with Section 8.1, if n is sufficiently large and if α = 0.05 t = t = idfnormal(0.0,1.0,0.975) = With negligible error the 95% confidence interval estimate can be written as x ± 2 x(1 x)/n Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 4/ 19

5 Algorithm Conventionally, ˆp (instead of x) denotes the point estimate of an unknown probability p Algorithm To estimate the unknown probability p of an event A by Monte Carlo simulation, replicate n times to obtain ˆp = the number of occurrences of event A n as the point estimate of p = Pr(A). We are 95% confident that p lies somewhere in the interval ˆp(1 ˆp) ˆp ± 2 n Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 5/ 19

6 Example Program craps was run 200 times with n = 100 For each run, a 95% confidence interval estimate was computed Each estimate is illustrated above as vertical lines (the left-most interval is 0.56 ± 0.10 from Example 8.2.1) The horizontal line corresponds to p % (191 of 200) of the intervals include p Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 6/ 19

7 Approximate Interval Estimation If ˆp is close to 0.5, ˆp(1 ˆp) is also close to 0.5. So, the 95% confidence interval estimate reduces to approximately ˆp ± 1 n The approximate interval estimate is easy to remember, conservative and illustrates the curse of n 100 replications are needed to estimate p to within ± replications are needed to estimate p to within ± replications are needed to estimate p to within ±0.001 To achieve 1 more decimal digit of precision, be prepared to do 100 times more work! Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 7/ 19

8 Example For a random sample of size 100, n = 10 If ˆp = 0.56, the 95% confidence interval estimate is approximately 0.56 ± 0.10 Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 8/ 19

9 Example 8.2.4: Specified Precision For Monte Carlo simulation, want to know a priori how many replications n are needed to achieve a specified precision ±w Using 95% confidence, solve w = 2 ˆp(1 ˆp)/n for n: 4ˆp(1 ˆp) n = A small run (100 replications) can be used to get a preliminary value for ˆp, use above equation to determine n w 2 For example, ˆp = 0.2 and w = 0.01, 4(0.2)(0.8) n = (0.01) 2 = 6400 instead of replications suggested previously Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 9/ 19

10 Algorithm Algorithm Given a specified interval half-width w and level of confidence 1 α, the algorithm computes the probability interval estimate ˆp ± w t = idfnormal(0.0, 1.0, 1 - α / 2); n = 1; x = Generate(); /* returns 0 or 1*/ ˆp = x; while ((n < 40) or (t * sqrt(ˆp * (1-ˆp)) > w*sqrt(n)){ n++; x = Generate(); /* return 0 or 1*/ ˆp = ˆp + (x - ˆp)/n } return n, ˆp Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 10/ 19

11 Monte Carlo Integration Use Monte Carlo simulation to estimate the value of I = b a g(x)dx Limited practical value: use trapezoid rule or Simpson s rule instead Let X be Uniform(a,b); pdf of X is f (x) = 1/(b a) Use function g( ) to define random variable Y = (b a)g(x) The expected value of Y is E[Y ] = b a (b a)g(x)f (x)dx = b a g(x)dx = I Modify Alg to determine interval estimate Î ± w for I Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 11/ 19

12 Algorithm Algorithm Given w, 1 α, compute interval estimate Î ± w for I = R b a g(x)dx t = idfnormal(0.0, 1.0, 1 - α/2); n = 1; x = Uniform(a, b); /*modify as appropriate*/ y = (b - a) * g(x); /*modify as appropriate*/ ȳ = y; v = 0.0; while((t*sqrt(v/n) > w*sqrt(n-1))or(n <40)){ n++; x = Uniform(a, b); /*modify as appropriate*/ y = (b - a)* g(x); /*modify as appropriate*/ d = y - ȳ; v = v + d * d * (n - 1) / n; ȳ = ȳ + d / n; } return n, ȳ; /* ȳ is Î */ Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 12/ 19

13 Example Algorithm was used to evaluate I = 1 2π 4 3 exp( x 2 /2)dx with w = 0.01 and α = 0.05, using g(x) = exp( x2 /2) 2π n = replications produced Î ± w = ± 0.01 We are 0.95% confident that < I < and it is consistent with the correct value of I = Φ(4) Φ( 3) function evaluations were required to achieve precision ±0.01 Trapezoid rule: 30 function evaluations achieve precision ±0.0001; Monte Carlo integration would require about function evaluations to do the same Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 13/ 19

14 Importance Sampling In Example 8.2.5, the slow convergence is due to the use of Uniform(a,b) random variates That is, it is wasteful to repeatedly sample and sum g(x) for values of x where g(x) = 0 The integration process should concentrate on sampling and summing g(x) for values of x where g(x) is large Generalization of Algorithm 8.2.3: use general random variable X with pdf f (x) > 0 defined for all possible values x X Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 14/ 19

15 Importance Sampling (2) Define new random variable The expected value of Y is E[Y ] = b a Y = g(x)/f (X) g(x) b f (x) f (x)dx = g(x)dx = I a An interval estimate Î ± w for I = E[Y ] can be calculated by using a modified version of Algorithm to determine an interval estimate for E[Y ] The key feature is a clever choice of X or the pdf f ( ) Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 15/ 19

16 Example Evaluate the integral I = 4 0 exp( x 2 /2)dx Example uses Uniform(0,4) for X, converges slowly If X is an Exponential(1) random variate truncated to X = (0, 4), better convergence should be achieved The pdf of X is f (x) = exp( x) F(4) where F(4) = 1 exp( 4) is the cdf of an un-truncated Exponential(1) random variable Algorithm 8.2.3, with indicated assignments modified accordingly, can be used to compute an interval estimate for I Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 16/ 19

17 Time-Averaged Statistics The time-averaged number in a single-server service node is l = 1 τ τ 0 l(t)dt where l(t) is the number in the service node at time t and the average is over the interval 0 < t < τ Section 2.1 claimed: if we were to observe(sample) the number in the service node at many different times chosen at random between 0 and τ and then calculate the arithmetic average of all these observations, the result should be close to l The next example clarifies this statement Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 17/ 19

18 Example Let T be Uniform(0,τ) and define new random variable L = l(t) By definition, the pdf of T is f (t) = 1/τ for 0 < t < τ The expected value of L is E[L] = τ 0 l(t)f (t)dt = 1 τ τ 0 l(t)dt = l The observation times t 1,t 2,,t n be generated as an n-point random variate sample of T l(t 1 ),l(t 2 ),,l(t n ) is a random sample of L and the mean is an (unbiased) point estimate of E[L] = l ; If the sample size is large, the sample mean should be close to l Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 18/ 19

19 Random Sampling The random times t i at which the l(t i ) samples are taken can be generated by the same algorithm used to generate a stationary Poisson arrival process Algorithm to generate n random sample times (in sorted order) with an average inter-sample time of δ Random Sampling t 0 = 0; for (i = 0; i < n; i++) t i+1 = t i + Exponential(δ); return t 1, t 2,..., t n ; Discrete-Event Simulation: A First Course Section 8.2: Monte Carlo Estimation 19/ 19

Hand and Spreadsheet Simulations

1 / 34 Hand and Spreadsheet Simulations Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 9/8/16 2 / 34 Outline 1 Stepping Through a Differential Equation 2 Monte