Simulation of Max Stable Processes

Transcription

1 Simulation of Max Stable Processes Whitney Huang Department of Statistics Purdue University November 6, / 30

2 Outline 1 Max-Stable Processes 2 Unconditional Simulations 3 Conditional Simulations 4 Simulations in R 2 / 30

3 Max-Stable Processes Definition: Max-stable processes A stochastic process {Y (x)} x X is max-stable if there exist functions {a n (x)} x X > 0 and {b n (x)} x X such that { M n(x) b n (x) d } x X = {Y (x)}x X, n 1 (1) a n (x) where M n (x) = max i=1,,n Y i (x), Y 1,, Y n be the independent copies of Y Remarks: X is a finite set Y is a multivarite extreme value distribution For any fixed x X Y is Generalized Extreme Value distribution (GEV) with parameters {µ(x), σ(x), ξ(x)} 3 / 30

4 Max-Stable Processes Max-stable Processes Without loss of generality we assume each margin follow unit Fréchet distribution i.e. P(Z z) = exp( 1 z ) z > 0 this can be achieved by applying the monotone transformation Z (x) = [ 1 + ξ(x)(y (x) µ(x)) σ(x) ] 1 ξ(x) x X In such case, a n (x) = n and b n (x) = 0 x X i.e. { n i=1 Z i(x)} x X d = {nz (x)}x X, n 1 4 / 30

5 Max-Stable Processes Max-stable Processes Without loss of generality we assume each margin follow unit Fréchet distribution i.e. P(Z z) = exp( 1 z ) z > 0 this can be achieved by applying the monotone transformation Z (x) = [ 1 + ξ(x)(y (x) µ(x)) σ(x) ] 1 ξ(x) x X In such case, a n (x) = n and b n (x) = 0 x X i.e. { n i=1 Z i(x)} x X d = {nz (x)}x X, n 1 4 / 30

6 Max-Stable Processes Max-stable Processes Without loss of generality we assume each margin follow unit Fréchet distribution i.e. P(Z z) = exp( 1 z ) z > 0 this can be achieved by applying the monotone transformation Z (x) = [ 1 + ξ(x)(y (x) µ(x)) σ(x) ] 1 ξ(x) x X In such case, a n (x) = n and b n (x) = 0 x X i.e. { n i=1 Z i(x)} x X d = {nz (x)}x X, n 1 4 / 30

7 Max-Stable Processes Spectral Representations I Theorem (de Haan, 1984) Let {(ζ i, U i )} i 1 be the points of a Poisson process on (0, ] R d with intensity dλ(ζ, u) = ζ 2 dζν(du), ν a σ-finite measure on R d. Let {Z (x)} x R d be a max-stable process with unit Fréchet margins, then there exist non-negative continuous functions {f x (y) : x, y R d } such that R d f x (y) ν(dy) = 1 x R d, then {Z (x)} x R d d = ζ i f x (U i ) i 1 x R d 5 / 30

8 Max-Stable Processes Storm Interpretation de Haan s spectral representation has a storm interpretation : {ζ i } i 1 storm ferocities {U i } i 1 storm center ζ i f x (U i ) quantity of rain at x for the i th storm 6 / 30

9 Max-Stable Processes Storm Interpretation cont d de Haan s spectral representation has a storm interpretation : {ζ i } i 1 storm ferocities {U i } i 1 storm center ζ i f x (U i ) quantity of rain at x for the i th storm 7 / 30

10 Max-Stable Processes Spectral Representations II Theorem (Schlather, 2002) Let {ζ i } i 1 be the points of Poisson process on (0, ] with intensity dλ(ζ) = ζ 2 dζ and Y 1, Y 2, be independent copies of a non-negative stochastic process {Y (x)} x R d such that E[Y (x)] = 1 x R d. Suppose {ζ i } i 1 independent of Y i s. Then d {Z (x)} x R d = ζ i Y i (x) i 1 x R d is max-stable process with unit Fréchet marginals 8 / 30

11 Max-Stable Processes Schlather s spectral representation There is no storm" interpretation in Schlather s representation It allows a random shapes for the storms" since {Y (x)} x R d is a stochastic process 9 / 30

12 Max-Stable Processes Schlather s spectral representation cont d There is no storm" interpretation in Schlather s representation It allows a random shapes for the storms" since {Y (x)} x R d is a stochastic process 10 / 30

13 Max-Stable Processes Smith Model The first max-stable model in spatial extremes literature was introduced by R. Smith by utilizing de Haan s spectral representation: {Z (x)} x R 2 = max i 1 ζ iφ(x U i ; 0, Σ), x R 2 where φ( ; 0, Σ) is the bivariate normal density with mean 0 and covariance matrix Σ 11 / 30

14 Max-Stable Processes Schlather Model The second model in spatial extremes literature was due to M. Schlather: {Z (x)} x R 2 = max ζ i 2[Yi (x)] +, x R 2 i 1 where {Y i (x)} x R 2 i is the independent copies of a standard Gaussian Process with correlation function ρ 12 / 30

15 Unconditional Simulations Recall the spectral representation: Z (x) = max i 1 ζ iy i (x), x X R 2 In order to simulate Z (x) x X we need to simulate Points of Poisson process with intensity ζ 2 dζ on (0, ] Y (x) at some fixed locations We need infinite number of these! 13 / 30

16 Unconditional Simulations Recall the spectral representation: Z (x) = max i 1 ζ iy i (x), x X R 2 In order to simulate Z (x) x X we need to simulate Points of Poisson process with intensity ζ 2 dζ on (0, ] Y (x) at some fixed locations We need infinite number of these! 13 / 30

17 Unconditional Simulations Recall the spectral representation: Z (x) = max i 1 ζ iy i (x), x X R 2 In order to simulate Z (x) x X we need to simulate Points of Poisson process with intensity ζ 2 dζ on (0, ] Y (x) at some fixed locations We need infinite number of these! 13 / 30

18 Unconditional Simulations Recall the spectral representation: Z (x) = max i 1 ζ iy i (x), x X R 2 In order to simulate Z (x) x X we need to simulate Points of Poisson process with intensity ζ 2 dζ on (0, ] Y (x) at some fixed locations We need infinite number of these! 13 / 30

19 Unconditional Simulations Notice that the points of Poisson process with intensity ζ 2 dζ on (0, ] can be simulated as a strictly decreasing sequence by { } d 1 i.i.d. {ζ i } i 1 = i j=1 E, E i Exp(1) i It implies that if {Y (x)} x X is uniformly bounded by C (0, ) then ζ i Y i (x) will not contribute to Z (x) for i i 0 we only need to simulate i 0 (finite number) of these! i 1 14 / 30

20 Unconditional Simulations 15 / 30

21 Unconditional Simulations 16 / 30

22 Unconditional Simulations 17 / 30

23 Unconditional Simulations 18 / 30

24 Conditional Simulations Goal: Given {Z (x 1 ) = z 1,, Z (x n ) = z n } want to get the (regular) conditional distribution of Z (x) x X It plays an important role for interpolating the spatial extremal surface If {Z (x)} x X is a Gaussian process, then the conditional simulation can be thought as the interpolation of spatial mean surface Other than Gaussian process, the conditional distribution is difficult 19 / 30

25 Conditional Simulations Max-Linear Approximation for Max-Stable Processes Approximate {Z (x)} x X with a Max-Linear models {Z (x)} x X p a i,j Z j := A Z Z j i.i.d. α-fréchet, a i,j deterministic (known) such that a i,j 0 j=1 Max-Linear models approximate arbitrary max-stable processes with p sufficiently large 20 / 30

26 Conditional Simulations Conditional sampling for Max-Linear models Wang & Stoev, 2011 derive the explicit formula of conditional distribution for max-linear models The structure of max-linear models imposes some constraints, namely z i Z j min i=1,,n a i,j := ẑ j, inequality constraints Z J = ẑ J for some J {1,, p}, hitting constraints 21 / 30

27 Conditional Simulations A Toy Example on Max-Linear models Suppose n = p = 3 and z 1 = Z 1 i.e. z 2 = Z 1 Z2 z 3 = Z 1 Z2 Z3 A = / 30

28 Conditional Simulations A Toy Example cont d (i) If z = (z 1, z 2, z 3 ) T = (1, 2, 3) T Z = (1, 2, 3) T The hitting constraints hold for Z 1, Z 2, and Z 3, hence J = (1, 2, 3), (ii) If z = (z 1, z 2, z 3 ) T = (1, 1, 3) T Z = (Z 1 = 1, Z 2 1, Z 3 = 3) T The hitting constraints hold for Z 1, Z 3, while the inequality constraint hold for Z 2 i.e. 0 < Z 2 1, hence J = (1, 3) 23 / 30

29 Conditional Simulations Algorithm 1 Compute ẑ j, j = 1,, p 2 Determine all hitting scenarios J, and the distribution of Z J for each hitting scenarios 3 Sample the conditional distribution from the mixture distribution of each hitting scenarios Remarks: To identify all hitting scenarios is NP-hard By the max-linear structure, with probability 1, the hitting scenarios has a nice structure which leads to an efficient algorithm 24 / 30

30 Simulations in R Package SpatialExtremes SpatialExtremes is a R package written by M. Ribatet Statistical models for spatial extremes based on latent variables, on copulas and on max-stable processes are implemented It also provides unconditional/conditional simulations on max-stable processes 25 / 30

31 Simulations in R rmaxstab: Unconditional simulations on max-stable processes (a) Smith Model (b) Schlather Model 26 / 30

32 Z Z Simulations in R x y x y 27 / 30

33 Simulations in R condrmaxlin: Conditional sampling for Max-Linear models Figure: Four conditional simulations for Smith model, p = / 30

34 Simulations in R 29 / 30

35 References de Haan, L. A spectral representation for max-stable processes The Annals of Probability, , 1984 Schlather, M. Models for stationary max-stable random fields Extremes, 5(1), 33:44, 2002 Smith, R. L. Max-stable processes and spatial extremes Unpublished manuscript, 1990 Wang, Y. & Stoev, S. A. Conditional sampling for spectrally discrete max-stable random fields Adv. Appl. Prob., 43, , / 30