E cient Monte Carlo for Gaussian Fields and Processes

Transcription

1 E cient Monte Carlo for Gaussian Fields and Processes Jose Blanchet (with R. Adler, J. C. Liu, and C. Li) Columbia University Nov, 2010 Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

2 Outline 1 Introduction 2 Importance Sampling and E ciency 3 Example 1: Maximum of Gaussian Process 4 Example 2: Gaussian Random Fields Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

3 A Motivating Application Contamination level in a geographic area... (yellow area = Mexico City) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

4 A Motivating Application When the ground-level ozone reaches 270 parts per billion, Mexico City bans the use of a signi cant portion of cars, orders industry to reduce emissions by 50% and closes gas stations... Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

5 A Motivating Application When the ground-level ozone reaches 270 parts per billion, Mexico City bans the use of a signi cant portion of cars, orders industry to reduce emissions by 50% and closes gas stations... What s the chance that high levels are reached in a given area? What happens to other areas given that a high level has been reached somewhere? Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

6 A Motivating Application When the ground-level ozone reaches 270 parts per billion, Mexico City bans the use of a signi cant portion of cars, orders industry to reduce emissions by 50% and closes gas stations... What s the chance that high levels are reached in a given area? What happens to other areas given that a high level has been reached somewhere? X (t, s) = logarithm of ozone s concentration at position s (in yellow area) at time t... Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

7 A Motivating Application When the ground-level ozone reaches 270 parts per billion, Mexico City bans the use of a signi cant portion of cars, orders industry to reduce emissions by 50% and closes gas stations... What s the chance that high levels are reached in a given area? What happens to other areas given that a high level has been reached somewhere? X (t, s) = logarithm of ozone s concentration at position s (in yellow area) at time t... Adler, Blanchet and Liu (2010): Algorithm for conditional sampling with ε relative precision in time Polyfε 1 log[1/p(high excursion)]g Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

8 A Motivating Application Another motivating application: Queues with Gaussian input Mandjes (2007) u (b) := P max X (k) > b k0 where X () is a suitably de ned Gaussian process. Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

9 A Motivating Application Another motivating application: Queues with Gaussian input Mandjes (2007) u (b) := P max X (k) > b k0 where X () is a suitably de ned Gaussian process. Blanchet and Li (2010): Algorithm for conditional sampling with ε relative precision in time Polyfε 1 log[1/u (b) ]g Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

10 Outline 1 Introduction 2 Importance Sampling and E ciency 3 Example 1: Maximum of Gaussian Process 4 Example 2: Gaussian Random Fields Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

11 Importance Sampling Goal: Estimate u := P (A) > 0 and Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

12 Importance Sampling Goal: Estimate u := P (A) > 0 and Choose ep and simulate ω from it Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

13 Importance Sampling Goal: Estimate u := P (A) > 0 and Choose ep and simulate ω from it I.S. estimator per trial is Z = L (ω) I (ω 2 A), where L (ω) is the likelihood ratio (i.e. L (ω) = P (ω) /ep (ω)). Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

14 Importance Sampling Goal: Estimate u := P (A) > 0 and Choose ep and simulate ω from it I.S. estimator per trial is Z = L (ω) I (ω 2 A), where L (ω) is the likelihood ratio (i.e. L (ω) = P (ω) /ep (ω)). ep is called a change-of-measure Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

15 Importance Sampling Must apply I.S. with care > Can increase the variance Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

16 Importance Sampling Must apply I.S. with care > Can increase the variance Glasserman and Kou 95, Asmussen, Binswanger and Hojgaard 00 Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

17 Importance Sampling Goal: Choose ep simulatable and e ciently as P (A) & 0. Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

18 Importance Sampling Goal: Choose ep simulatable and e ciently as P (A) & 0. Strong E ciency: eez 2 /P (A) 2 = O (1). Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

19 Importance Sampling Goal: Choose ep simulatable and e ciently as P (A) & 0. Strong E ciency: eez 2 /P (A) 2 = O (1). Weak E ciency: eez 2 /P (A) 2 = O 1/P (A) ε for every ε > 0. Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

20 Importance Sampling Goal: Choose ep simulatable and e ciently as P (A) & 0. Strong E ciency: eez 2 /P (A) 2 = O (1). Weak E ciency: eez 2 /P (A) 2 = O 1/P (A) ε for every ε > 0. Polynomial E ciency: eez 2 /P (A) 2 = O (log (1/P (A)) q ) for some q > 0. Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

21 Importance Sampling Goal: Choose ep simulatable and e ciently as P (A) & 0. Strong E ciency: eez 2 /P (A) 2 = O (1). Weak E ciency: eez 2 /P (A) 2 = O 1/P (A) ε for every ε > 0. Polynomial E ciency: eez 2 /P (A) 2 = O (log (1/P (A)) q ) for some q > 0. Strong Weak Polynomial Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

22 Importance Sampling Suppose we choose ep () = P ( j A) L (ω) = P (ω) I (ω 2 A) P (ω) I (ω 2 A) /P (A) = P (A) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

23 Importance Sampling Suppose we choose ep () = P ( j A) L (ω) = P (ω) I (ω 2 A) P (ω) I (ω 2 A) /P (A) = P (A) Estimator has zero variance, but requires knowledge of P (A) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

24 Importance Sampling Suppose we choose ep () = P ( j A) L (ω) = P (ω) I (ω 2 A) P (ω) I (ω 2 A) /P (A) = P (A) Estimator has zero variance, but requires knowledge of P (A) Lesson: Try choosing ep () close to P ( j A)! Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

25 Outline 1 Introduction 2 Importance Sampling and E ciency 3 Example 1: Maximum of Gaussian Process Target Bridge Sampling Brownian Motion E ciency Analysis Numerical Example 4 Example 2: Gaussian Random Fields Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

26 Example 1: Maximum of Gaussian Process Let X = (X k : k 0) be a Gaussian process with Var (X k ) = σ 2 k and EX k = µ k < 0. Want to e ciently estimate (as b % ) Assume u(b) = P[ max k1 X k > b] σ k c σ k H σ, jµ k j c µ k H µ, 0 < H σ < H µ < so u(b)! 0 as b!. No assumption on correlation structure. Asymptotic regime known as "large bu er scaling". Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

27 Related Asymptotics Rich literature on asymptotics for u (b) Pickands (1969), Berman (1990), Du eld and O Connell (1995), Piterbarg (1996), Husler and Piterbarg (1999), Dieker (2006), Husler (2006), Likhanov and Mazumdar (1999), Debicki and Mandjes (2003)... Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

28 Related Simulation Algorithms Common basic idea is to sample Gaussian process by mean tracking the most likely path given the over ow, time-slot by time-slot. X t b X 0 T (b) t Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

29 Target Bridge Sampling X t 1 Identifying the target set T ; b X 0 T (b) t Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

30 Target Bridge Sampling X t 1 Identifying the target set T ; 2 Targeting; b X 0 T (b) t Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

31 Target Bridge Sampling X t 1 Identifying the target set T ; 2 Targeting; 3 Bridging; b X 0 T (b) t Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

32 Target Bridge Sampling X t 1 Identifying the target set T ; 2 Targeting; 3 Bridging; 4 T (b) = inffk : X k > bg; X T (b) b X 0 T (b) t Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

33 Target Bridge Sampling X t 1 Identifying the target set T ; 2 Targeting; 3 Bridging; 4 T (b) = inffk : X k > bg; 5 Deleting X 0 X T (b) T (b) b t Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

34 Likelihood of a Path = ep(x 1,..., X T (b) ) Z ep (τ = k) P(X 1,..., X T (b) jx k )ep (dx k ). k=t (b) b Select ep (τ = k) P (X k > bj T (b) < ) P (X k > b) Given τ = k sample ep (X k 2 ) = P (X k 2 j X k > b) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

35 Likelihood Ratio = = = ep(x 1,..., X T (b) ) Z ep (τ = k) P(X 1,..., X T (b) jx k )ep (dx k ) k=t (b) b k=t (b) P (X k > b) Z j=1 P (X j > b) b P(X 1,..., X T (b), X k > b) k=t (b) j=1 P (X j > b) = P(X 1,..., X T (b) ) j=1 P (X j > b) P (dx P(X 1,..., X T (b) jx k ) k ) P (X k > b) P X k > bjx 1,..., X T (b) k=t (b) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

36 Consequently, the importance sampling estimator for u (b) generated by P is simply L = dp X 1,..., X T (b) d ep j=1 P (X j > b) =. j=t (b) P X j > bj X 1,..., X T (b) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

37 Special Case: Brownian Motion Consider the case when X t = B (t) t, is a standard Brownian motion, with negative linear drift µ t = t. Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

38 Special Case: Brownian Motion Consider the case when X t = B (t) t, is a standard Brownian motion, with negative linear drift µ t = t. The likelihood ratio: L = R R 0 P(B t t > b)dt T (b) P B t t > bj B T (b) = b dt is a constant! = R 0 P(B t t > b)dt R 0 P(B t t 0)dt, Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

39 Special Case: Brownian Motion Consider the case when X t = B (t) t, is a standard Brownian motion, with negative linear drift µ t = t. The likelihood ratio: L = R R 0 P(B t t > b)dt T (b) P B t t > bj B T (b) = b dt = R 0 P(B t t > b)dt R 0 P(B t t 0)dt, is a constant! Zero-variance importance sampler: eel L = P max B t t0 t > b = exp( 2b) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

40 Special Case: Brownian Motion Consider the case when X t = B (t) t, is a standard Brownian motion, with negative linear drift µ t = t. The likelihood ratio: L = R R 0 P(B t t > b)dt T (b) P B t t > bj B T (b) = b dt = R 0 P(B t t > b)dt R 0 P(B t t 0)dt, is a constant! Zero-variance importance sampler: eel L = P max B t t0 t > b = exp( 2b) In this case TBS outputs a Brownian motion with drift +1. Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

41 E ciency Analysis The e ciency analysis involves ee (L 2 ) P(max k1 X k b) 2 k=1 P(X k b) 2 max k1 P(X k b) k=0 P(X k > b) 2 P(max k1 X k b) Does not involve the correlation structure... Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

42 E ciency Analysis Theorem If (σ k : k 1) and (µ k : k 1) have power law type tails with power indices 0 < H σ < H µ < respectively, then we have that 1 L is an unbiased estimator of u(b) 2 Let h(b) = λ H µ, H σ b Hµ Hσ Hµ, then u(b) = O(h (b) 1/(H µ H σ ) exp( h(b))). 3 ee L 2 u(b) 2 = O(b2/H µ ); Polynomially e cient Strongly e cient in many source scaling Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

43 Summary of Performance Analysis (Updated) Method Many Sources Large Bu er Cost of each replication Single twist x x O(b 3 ) Cut-and-twist weakly x O(b 4 ) Random twist weakly x O(b 3 ) Sequential twist weakly x O(nb 3 ) Mean shift x x O(b 3 ) BMC x x O(b 3 ) TBS strongly polynomial O(b 3 ) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

44 Numerical Example: Fractional Brownian Noise Test the performance of our Target Bridge Sampler and compare it against other existing methods in the many sources setting. Suppose that fx k g are driven by fractional Brownian noises, that is, Cov(X k, X j ) = (k 2H σ + j 2H σ jk jj 2H σ )/2 and µ k = k. The numerical result is compared against what was reported in Dieker and Mandjes (2006). Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

45 Numerical Example b = 300 Cost of each Estimator Simulation Runs Time replication Naive O(b 3 ) s Single twist O(b 3 ) ~60s Cut-and-twist O(b 4 ) ~80s Random twist O(b 3 ) ~50s Sequential twist O(nb 3 ) ~100s TBS O(b 3 ) s Benchmark Table: Simulation result of Example 2 with n = 300, b = 3, H σ = 0.8. Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

46 Outline 1 Introduction 2 Importance Sampling and E ciency 3 Example 1: Maximum of Gaussian Process 4 Example 2: Gaussian Random Fields Discrete Version Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

47 Discrete Version: Monte Carlo Strategy Discrete version: (X 1,..., X d ) multivariate Gaussian: Consider d P max X i > b i=1 d P (X 1,..., X d ) 2 j max X i > b i=1 Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

48 Discrete Version: Monte Carlo Strategy Discrete version: (X 1,..., X d ) multivariate Gaussian: Consider d P max X i > b i=1 d P (X 1,..., X d ) 2 j max X i > b i=1 Monte Carlo strategy: Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

49 Discrete Version: Monte Carlo Strategy Discrete version: (X 1,..., X d ) multivariate Gaussian: Consider d P max X i > b i=1 d P (X 1,..., X d ) 2 j max X i > b i=1 Monte Carlo strategy: 1 Pick i-th coordinate with probability proportional to P (X i > b) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

50 Discrete Version: Monte Carlo Strategy Discrete version: (X 1,..., X d ) multivariate Gaussian: Consider d P max X i > b i=1 d P (X 1,..., X d ) 2 j max X i > b i=1 Monte Carlo strategy: 1 Pick i-th coordinate with probability proportional to P (X i > b) 2 Sample X i given X i > b Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

51 Discrete Version: Monte Carlo Strategy Discrete version: (X 1,..., X d ) multivariate Gaussian: Consider d P max X i > b i=1 d P (X 1,..., X d ) 2 j max X i > b i=1 Monte Carlo strategy: 1 Pick i-th coordinate with probability proportional to P (X i > b) 2 Sample X i given X i > b 3 Sample the remaining values given X i ep ((X 1,..., X d ) 2 ) = d P (X i > b) P ((X 1,..., X d ) 2 jx i > b) i=1 d j=1 P (X j > b) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

52 Discrete Version: Importance Sampling Distribution Prob. measure ep ((X 1,..., X d ) 2 ) = = d P (X i > b) P ((X 1,..., X d ) 2 jx i > b) i=1 d j=1 P (X j > b) d P ((X 1,..., X d ) 2 ; X i > b) i=1 d j=1 P (X j > b) Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

53 Discrete Version: Likelihood Ratio Likelihood ratio I ( d max i=1 X i > b) dp d ep (X 1,..., X d ) = I ( d max i=1 X i > b) d j=1 P (X j > b) d j=1 I (X j > b) d P (X j > b). j=1 Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

54 Total Variation Approximation Theorem (Adler, Blanchet and Liu (2008)) If Corr (X i, X j ) < 1 then as b % and therefore sup A as b %. P( max d d X i > b) = P (X j > b) (1 + o (1)) i=1 j=1 d jp((x 1,..., X d ) 2 Aj max X i > b) ep ((X 1,..., X d ) 2 A) j! 0 i=1 Jose Blanchet (Columbia) Monte Carlo for Gaussian Fields Nov, / 29

55 Efficient Monte Carlo for High Excursions of Gaussian Random Fields Jose Blanchet Columbia University Joint work with Robert Adler (Technion Israel Institute of Technology ) Jingchen Liu (Columbia University) 1

56 Continuous Gaussian Random Fields on Compacts Gaussian random field, where T R d is a compact set. 2

57 High Excursion Probability Interesting probability as b. More generally, where Γ( ) is suitable functional. 3

58 Asymptotic results Under very mild conditions Sharp asymptotics for mean zero and constant variance random fields (under conditions) k depends on dimension of T and continuity of f(t). Pickands (1969), Piterbarg (1995), Sun (1993), Adler (1981), Azais and Wschebor (2005), Taylor, Takemura and Adler (2005). 4

59 The change of measure mes is the Lebesgue measure A γ is the excursion set A change of measure on C(T) 5

60 Simulation from this change of measure 6

61 The estimator The estimator and its variance The second moment 7

62 The Estimator and Its Variance 8

63 The choice of γ γ = b results in infinite variance γ = b a/b 9

64 The area of excursion set, mes(a b-1/b ), given high excursion 10

65 Efficiency results the general case 11

66 Efficiency results the homogeneous case 12

67 Implementation discretize the continuous filed Discretize the space T, {t 1,,t n } It is sufficient to choose n = (b/ε) α such that Adler, Blanchet and L. (2010) 13

68 Summary Non-exponential change-of-measure for Gaussian processes and fields (efficiency properties & conditional sampling) Polynomialy efficient for general Hölder continuous fields Strong efficiency for homogeneous fields 14