Guaranteed Fixed-Width Confidence Intervals for Monte Carlo and Quasi-Monte Carlo Simulation

Transcription

1 Guaranteed Fixed-Width Confidence Intervals for Monte Carlo and Quasi-Monte Carlo Simulation Fred J. Hickernell Department of Applied Mathematics, Illinois Institute of Technology Joint work with Sou-Cheng Choi (NORC, U Chicago), Lan Jiang (IIT), Lluís Antoni Jiménez Rugama (IIT), and Art Owen (Stanford) Supported by NSF-DMS Thank you for your kind invitation to speak. It is a privilege to visit the country of my forefather, who came from Haguenau in Alsace June 1, 215 hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 1 / 15

2 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ^µ n := 1 n nÿ Y i, Y 1, Y 2,... Y Monte Carlo = sampling with computers originated at Los Alamos National Laboratory ca (Eckhardt, 1987) µ = E(Y) =?, where Y complicated» Y(4) %random # generator ans = hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 2 / 15

3 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ^µ n := 1 n nÿ Y i, Y 1, Y 2,... Y Monte Carlo = sampling with computers Given ε a, want n such that P[ µ ^µ n ď ε a ] ě 99% a fixed-width confidence interval µ = E(Y) =?, where Y complicated» Y(4) %random # generator ans = hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 2 / 15

4 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n nÿ Y i, Y 1, Y 2,... Y Monte Carlo = sampling with computers Given ε a, want n such that P[ µ ^µ n ď ε a ] ě 99% Central Limit Theorem: ^µ n «N(µ, σ 2 /n), suggesting S (2.58 ) W 2 ˆ 1.2 ˆ ^σ n = ^σ 2 ε a = sample variance option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray µ = E(Y) =?, where Y complicated» Y(4) %random # generator ans = hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 2 / 15

5 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n nÿ Y i, Y 1, Y 2,... Y Monte Carlo = sampling with computers Given ε a, want n such that P[ µ ^µ n ď ε a ] ě 99% Central Limit Theorem: ^µ n «N(µ, σ 2 /n), suggesting S (2.58 ) W 2 ˆ 1.2 ˆ ^σ n = ^σ 2 ε a = sample variance option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray µ = E(Y) =?, where Y complicated» Y(4) %random # generator ans = » meanmc_clt(y,.2) ans = 1.1 +/-.2» meanmc_clt(y,.2) ans = /-.2» meanmc_clt(y,.2) ans =.99 +/-.2 What is wrong? hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 2 / 15

6 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n nÿ Y i, Y 1, Y 2,... Y Monte Carlo = sampling with computers Given ε a, want n such that P[ µ ^µ n ď ε a ] ě 99% Central Limit Theorem: ^µ n «N(µ, σ 2 /n), suggesting S (2.58 ) W 2 ˆ 1.2 ˆ ^σ n = ε a ^σ 2 = sample variance using n σ samples option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray µ = E(Y) =?, where Y complicated» Y(4) %random # generator ans = » meanmc_clt(y,.2,.1,1) ans = 1. +/-.2» meanmc_clt(y,.2,.1,1) ans = 1. +/-.2» meanmc_clt(y,.2,.1,1) ans = 1. +/-.2 for n σ = 1, rather than 1 hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 2 / 15

7 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n nÿ Y i, Y 1, Y 2,... Y Monte Carlo = sampling with computers Given ε a, want n such that P[ µ ^µ n ď ε a ] ě 99% Central Limit Theorem: ^µ n «N(µ, σ 2 /n), suggesting S (2.58 ) W 2 ˆ 1.2 ˆ ^σ n = ^σ 2 ε a = sample variance option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray µ = E(Y) =?, where Y complicated» Y(4) %random # generator ans = PDF of Y y hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 2 / 15

8 Guaranteed Adaptive Monte Carlo Simulation Adaptive Algorithm meanmc_g. (Choi et al., ) Given i) IID random generator for Y 1, Y 2,..., ii) absolute error tolerance, ε a, and iii) n σ P N and C ą 1, Step 1. Bound the Variance. Compute the sample variance, ^σ 2, based on n σ samples of Y. We know P[C 2^σ 2 ě var(y)] ě 99.5% for kurt(y) := E[(Y µ) 4 ]/ var(y) 2 ď κ max (n σ, C) by Cantelli s inequality. Step 2. Estimate the Mean. Use a Berry-Esseen Inequality to determine the sample size, n, needed for the sample mean by solving Φ (?nε a /(C^σ) ) loooooooooomoooooooooon + n (? loooooooooooomoooooooooooon nε a /(C^σ), κ max ) ď.25. CLT part Berry-Esseen extra part Compute the sample mean, ^µ n, of an independent sample of size n. Theorem. (H. et al., 214) Assuming ď, we must have P[ µ ^µ n ď ε a ] ě 99%. The computational cost is n σ + n = O(var(Y)/ε 2 a) with high probability, even though var(y) is unknown a priori. hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 3 / 15

9 Examples for meanmc_g Asian Geometric Mean Call Option d = 1, 2, 4,..., 64 time steps Failure.8 c d ż e x 2 R d cos( x ) dx = 1 d = 1, 2, 3, 4 (Keister, 1996) Failure.8 Time (seconds) Time (seconds) Success Error 1 3 Success Error «1% success «1% success Need n σ ě 1 3 to get a reasonable κ max meanmc_g is conservative, see Bayer et al. (214) for a heuristic alternative 1 3 hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 4 / 15

10 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d.75 Given ε a, want n such that µ ^µ n ď ε a a fixed-width confidence interval We may sample the X i as IID hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

11 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d Given ε a, want n such that µ ^µ n ď ε a We may sample the X i as IID Or by highly stratified sampling, also known as quasi-monte Carlo sampling or low discrepancy sampling hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

12 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d Given ε a, want n such that µ ^µ n ď ε a We may sample the X i as IID Or by highly stratified sampling, also known as quasi-monte Carlo sampling or low discrepancy sampling hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

13 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d Given ε a, want n such that µ ^µ n ď ε a We may sample the X i as IID Or by highly stratified sampling, also known as quasi-monte Carlo sampling or low discrepancy sampling hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

14 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d Given ε a, want n such that µ ^µ n ď ε a We may sample the X i as IID Or by highly stratified sampling, also known as quasi-monte Carlo sampling or low discrepancy sampling hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

15 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d Given ε a, want n such that µ ^µ n ď ε a We may sample the X i as IID Or by highly stratified sampling, also known as quasi-monte Carlo sampling or low discrepancy sampling hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

16 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d Given ε a, want n such that µ ^µ n ď ε a We may sample the X i as IID Or by highly stratified sampling, also known as quasi-monte Carlo sampling or low discrepancy sampling hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

17 Stopping the Simulation When the Error is Small Enough option price probability of process failure pixel intensity ^µ n := 1 n option payoff under random scenario random failure or not, i.e., 1 or intensity of random incident light ray ż µ = E(Y) = f (x) dx =?, where Y = f (X), X U[, 1] d [,1] d nÿ Y i = 1 nÿ 1 f (X i ), n X 1, X 2,... U[, 1] d Given ε a, want n such that µ ^µ n ď ε a We may sample the X i as IID Or by highly stratified sampling, also known as quasi-monte Carlo sampling or low discrepancy sampling hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 5 / 15

18 Highly Stratified Sampling ż f (x) dx 1 nÿ f (X i ) loooooomoooooon [,1) n ď ε a, d µ=e[f (X)] highly stratified n = 2 m n =? samples needed X 1, X 2,... dependent, can be deterministic or random (Owen, 2) Sobol sequences are a popular choice (Dick and Pillichshammer, 21) Typical error analysis requires L p -norms of mixed partial derivatives of f (Niederreiter, 1992; H., 1998) hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 6 / 15

19 Highly Stratified Sampling ż f (x) dx 1 nÿ f (X i ) loooooomoooooon [,1) n ď d µ=e[f (X)] highly stratified ÿ k(κ)pdual net p fκ n = 2 m Walsh functions & coefficients f (x) = ÿ κ= ( 1) xk(κ),xy p fκ hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 6 / 15

20 Monte Carlo Simulation Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References Highly Stratified Sampling n ÿ 1ÿ pfκ ď C(m)Sm (f ) ď εa, f (X i ) ď f (x) dx n d [,1) loooooomoooooon k(κ)pdual net ż µ=e[f (X)] highly stratified 1 Walsh functions & coefficients f (x) = ÿ n = 2m ( 1)xk(κ),xypfκ 1 5 Sm (f ) = data based sum of approximate p fκ w/ high κ H. and Jiménez Rugama (214) construct an algorithm that ^ n ď εa guarantees µ µ fˆκ κ= 1 1 error bound S12 (f) κ conditions: p fκ does not decay erratically as κ Ñ hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 6 / 15

21 Monte Carlo Simulation Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References When Does Our Error Bound Work? May Fail f (x) f (x) Succeeds x x fˆκ 1 fˆκ error bound S12 (f) κ hickernell@iit.edu κ Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 7 / 15

22 Examples for meanmc_g and cubsobol_g Asian Geometric Mean Call Option, d = 1, 2, 4,..., 64 time steps meanmc_g Failure cubsobol_g Failure Time (seconds) Success Error Time (seconds) Success Error «1% success «1% success hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 8 / 15

23 Examples for meanmc_g and cubsobol_g c d ż e x 2 R d cos( x ) dx = 1 (Keister, 1996) d = 1,..., 4 d = 1,..., 4 meanmc_g Failure cubsobol_g Failure Time (seconds) Time (seconds) Success Error 1 3 Success Error «1% success «1% success 1 3 hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 9 / 15

24 Examples for meanmc_g and cubsobol_g c d ż e x 2 R d cos( x ) dx = 1 (Keister, 1996) d = 1,..., 4 d = 1,..., 2 meanmc_g Failure cubsobol_g Failure Time (seconds) Time (seconds) Success Error 1 3 Success Error «1% success «97% success 1 3 hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 9 / 15

25 Observations on meanmc_g and cubsobol_g Highly stratified sampling (quasi-monte Carlo) can provide great efficiency gains. cublattice_g uses a integration lattices to achieve a high degree of stratification (Jiménez Rugama and H., 214). Y or f must lie in, which describes how nasty they are allowed to be. Bigger =ñ more robust algorithms with greater cost. One may not be able to verify if Y or f lies in a priori, but the theorems explain what might go wrong and provide guidance on the choice of algorithm parameters. Focus on (instead of other shapes) of Y or f because our " * " * problems are homogeneous. error bounds positively hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 1 / 15

26 Observations on meanmc_g and cubsobol_g cont d Exist general sufficient conditions under which adaption provides no advantage (Bahadur and Savage, 1956 ; Bakhvalov, 197 ; Traub et al., 1988, Chapter 4, Theorem ; Novak, 1996). To violate those conditions we consider nonconvex of Y or f. Can accommodate µ ^µ n ď max(ε a, ε r µ ). These algorithms featured in our Guaranteed Automatic Integration Library (GAIL) code.google.com/p/gail/ (Choi et al., ). >> Y =... %define your random # generator Y >> tic, muiid = meanmc_g(y,.1,), toc muiid = 1.2 Elapsed time is seconds. >> f =... %define your integrand f >> tic, musobol = cubsobol_g(f,... ), toc musobol = 1.3 Elapsed time is.3778 seconds. hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 11 / 15

27 Monte Carlo Simulation Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References Guaranteed Automatic Integration Library (GAIL) Version 2.1 code.google.com/p/gail/ GAIL 3. coming this fall. Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 12 / 15

28 Future Work In progress Control variates (subtle for quasi-monte Carlo (H. et al., 25)) Adaptive importance sampling Multilevel and related methods for -dimensional problems Porting to R and other languages Need further expertise or time Y P t, 1u, i.e., Bernoulli random variables (Jiang and H., 214) Markov Chain Monte Carlo Variety of use cases, e.g., Bayesian inference hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 13 / 15

29 References I Bahadur, R. R. and L. J. Savage The nonexistence of certain statistical procedures in nonparametric problems, Ann. Math. Stat. 27, Bakhvalov, N. S On the optimality of linear methods for operator approximation in convex classes of functions (in Russian), Zh. Vychisl. Mat. i Mat. Fiz. 1, English transl.: USSR Comput. Math. Math. Phys. 11 (1971) Bayer, C., H. Hoel, E. von Schwerin, and R. Tempone On nonasymptotic optimal stopping criteria in Monte Carlo Simulations, SIAM J. Sci. Comput. 36, A869 A885. Choi, S.-C. T., Y. Ding, F. J. H., L. Jiang, Ll. A. Jiménez Rugama, X. Tong, Y. Zhang, and X. Zhou GAIL: Guaranteed Automatic Integration Library (versions ). Dick, J. and F. Pillichshammer. 21. Digital nets and sequences: Discrepancy theory and quasi-monte Carlo integration, Cambridge University Press, Cambridge. Eckhardt, R Stan Ulam, John von Neumann, and the Monte Carlo method, Los Alamos Science, H., F. J A generalized discrepancy and quadrature error bound, Math. Comp. 67, H., F. J., L. Jiang, Y. Liu, and A. B. Owen Guaranteed conservative fixed width confidence intervals via Monte Carlo sampling, Monte Carlo and quasi-monte Carlo methods 212, pp hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 14 / 15

30 References II H., F. J. and Ll. A. Jiménez Rugama Reliable adaptive cubature using digital sequences. submitted for publication, arxiv: [math.na]. H., F. J., C. Lemieux, and A. B. Owen. 25. Control variates for quasi-monte Carlo, Statist. Sci. 2, Jiang, L. and F. J. H Guaranteed conservative confidence intervals for means of Bernoulli random variables. submitted for publication, arxiv: Jiménez Rugama, Ll. A. and F. J. H Adaptive multidimensional integration based on rank-1 lattices. submitted for publication, arxiv: Keister, B. D Multidimensional quadrature algorithms, Computers in Physics 1, Niederreiter, H Random number generation and quasi-monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia. Novak, E On the power of adaption, J. Complexity 12, Owen, A. B. 2. Monte Carlo, quasi-monte Carlo, and randomized quasi-monte Carlo, Monte Carlo, quasi-monte Carlo, and randomized quasi-monte Carlo, pp Traub, J. F., G. W. Wasilkowski, and H. Woźniakowski Information-based complexity, Academic Press, Boston. hickernell@iit.edu Fixed-Width Monte Carlo Confidence Intervals Journées de Statistique, 6/1/15 15 / 15