Conditional simulations of max-stable processes

Transcription

1 Conditional simulations of max-stable processes C. Dombry, F. Éyi-Minko, M. Ribatet Laboratoire de Mathématiques et Application, Université de Poitiers Institut de Mathématiques et de Modélisation, Université Montpellier 2 Conditional simulations of max-stable processes Mathieu Ribatet 1 / 24

2 Goal Max-stable processes are widely used for modelling spatial extremes since they arise as the only possible (non degenerate) limit of pointwise maxima over independent replicates, i.e., max i=1,...,n X i (x) b n (x) a n (x) Z (x), n, x X R d, for some normalizing functions a n > 0 and b n and where X i are independent copies of a stochastic process X. Conditional simulations of max-stable processes Mathieu Ribatet 2 / 24

3 Goal Max-stable processes are widely used for modelling spatial extremes since they arise as the only possible (non degenerate) limit of pointwise maxima over independent replicates, i.e., max i=1,...,n X i (x) b n (x) a n (x) Z (x), n, x X R d, for some normalizing functions a n > 0 and b n and where X i are independent copies of a stochastic process X. Can we get a procedure for conditional simulations of max-stable processes (with continuous spectral measure)? Conditional simulations of max-stable processes Mathieu Ribatet 2 / 24

4 Setup Recall that any max-stable process has the spectral characterization where Z ( )=max i 1 ζ i Y i ( ), Y i ( ) are independent copies of a non negative stochastic process such that E[Y (x)]=1 for all x X ; {ζ i } i 1 are the points of a Poisson process on (0, ) with intensity dλ(ζ)=ζ 2 dζ. Given a study region X R d, we want to sample from Z ( ) {Z (x 1 )=z 1,..., Z (x k )=z k }, for some z 1,..., z k > 0 and k conditioning locations x 1,..., x k X. Conditional simulations of max-stable processes Mathieu Ribatet 3 / 24

5 Random partitions Sampling scheme of max-stable processes Conditional simulations of max-stable processes Mathieu Ribatet 4 / 24

6 Decomposition of Φ Random partitions Sampling scheme Let Φ a point process on (0, ) k whose atoms are ϕ i (x)=ζ i Y i (x), x=(x 1,..., x k ). Consider the two following point processes Φ = { ϕ Φ: ϕ(x i )<z i, for all i {1,...,k} }, (sub-extremal functions) Φ + = { ϕ Φ: ϕ(x i )=z i, for some i {1,...,k} }.(extremal functions) Clearly Φ=Φ Φ + and Φ + = {ϕ + 1,...,ϕ+ k }={ϕ+ 1,...,ϕ+ l }, a.s. (1 l k). Conditional simulations of max-stable processes Mathieu Ribatet 5 / 24

7 Decomposition of Φ Random partitions Sampling scheme Let Φ a point process on (0, ) k whose atoms are ϕ i (x)=ζ i Y i (x), x=(x 1,..., x k ). Consider the two following point processes Φ = { ϕ Φ: ϕ(x i )<z i, for all i {1,...,k} }, (sub-extremal functions) Φ + = { ϕ Φ: ϕ(x i )=z i, for some i {1,...,k} }.(extremal functions) Clearly Φ=Φ Φ + and Φ + = {ϕ + 1,...,ϕ+ k }={ϕ+ 1,...,ϕ+ l }, a.s. (1 l k). Key point #1: Conditionally on Z (x)=z, Φ and Φ + are independent. Conditional simulations of max-stable processes Mathieu Ribatet 5 / 24

8 Conditional intensity function Random partitions Sampling scheme Z (x)=max i 1 ζ i Y i (x)=max i 1 ϕ i (x) The Poisson point process {ϕ i (x)} i 1 has intensity measure Λ x (A)= 0 Pr{ζY (x) A}ζ 2 dζ, Borel set A R k. We assume that Φ is regular, i.e., Λ x (dz)=λ x (z)dz, for all x X k. Conditional simulations of max-stable processes Mathieu Ribatet 6 / 24

9 Conditional intensity function Random partitions Sampling scheme Z (x)=max i 1 ζ i Y i (x)=max i 1 ϕ i (x) The Poisson point process {ϕ i (x)} i 1 has intensity measure Λ x (A)= 0 Pr{ζY (x) A}ζ 2 dζ, Borel set A R k. We assume that Φ is regular, i.e., Λ x (dz)=λ x (z)dz, for all x X k. λ s x,z (u)= λ (s,x)(u, z) λ x (z) Key point #2: The conditional intensity function is the (regular) conditional distribution of Z (x) if we integrate w.r.t. all possible partitions of x. But not that of Z ( )!!! Conditional simulations of max-stable processes Mathieu Ribatet 6 / 24

10 Random partitions? Random partitions Sampling scheme Z(x) x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 7 / 24

11 Random partitions? Random partitions Sampling scheme Z(x) ϕ 1 + x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 7 / 24

12 Random partitions? Random partitions Sampling scheme Z(x) ϕ 1 + ϕ 2 + x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 7 / 24

13 Random partitions? Random partitions Sampling scheme Z(x) ϕ 1 + ϕ 2 + ϕ 3 + x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 7 / 24

14 Random partitions? Random partitions Sampling scheme Z(x) ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 + x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 7 / 24

15 Random partitions? Random partitions Sampling scheme Z(x) ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 + x 2 x 5 x 1 x 3 x 4 Here the set {x 1,..., x 5 } is partitioned into ({x 1, x 3 },{x 2 },{x 4 },{x 5 }) x Conditional simulations of max-stable processes Mathieu Ribatet 7 / 24

16 Random partitions? Random partitions Sampling scheme Z(x) ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 + x 2 x 5 x 1 x 3 x 4 Here the set {x 1,..., x 5 } is partitioned into ({x 1, x 3 },{x 2 },{x 4 },{x 5 }) The hitting bounds {z i } i=1,...,k might be reached by several extremal functions, i.e., Φ + = {ϕ + 1,...,ϕ+ k }={ϕ+ 1,...,ϕ+ l } a.s., 1 l k. So we need to take into account all possible ways these hitting bounds are reached: the hitting scenarios x Conditional simulations of max-stable processes Mathieu Ribatet 7 / 24

17 Why should we bother about Φ? Random partitions Sampling scheme Z(x) x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 8 / 24

18 Why should we bother about Φ? Random partitions Sampling scheme Z(x) max Φ + x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 8 / 24

19 Why should we bother about Φ? Random partitions Sampling scheme Z(x) max Φ + max Φ x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 8 / 24

20 Why should we bother about Φ? Random partitions Sampling scheme Z(x) max Φ + max Φ x 2 x 5 x 1 x 3 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet 8 / 24

21 Why should we bother about Φ? Random partitions Sampling scheme Z(x) max Φ + max Φ x 2 x 5 x 1 x 3 x 4 The atoms of Φ + are only of interest if we restrict our attention to the conditioning points x; But most often one would like to get realizations at s x. The atoms of Φ are needed since it is likely that maxφ (s)>maxφ + (s)! x Conditional simulations of max-stable processes Mathieu Ribatet 8 / 24

22 A three step procedure Random partitions Sampling scheme The above key points suggest a three step sampling scheme: Step 1 Draw a random partition τ, i.e., a hitting scenario; Step 2 Given τ of size l, draw the extremal functions ϕ + 1,...,ϕ+ l independently; Step 3 Independently from Steps 1 & 2, draw the sub-extremal functions ϕ i, i 1. Conditional simulations of max-stable processes Mathieu Ribatet 9 / 24

23 Step 1: The random partitions Random partitions Sampling scheme Let P k the set of all possible partitions of the set {x 1,..., x k }. Draw a random partition τ P k with distribution π x (z,τ)= 1 C (x, z) τ j=1 λ xτ j (z τj ) }{{} density that some bounds are reached, i.e., the zτ j {u<z τ c j } λ xτ c x τ j j,z τ j (u)du, }{{} probability to lie below the remaining bounds, i.e., below the z τ c j where the normalization constant C (x, z) is given by C (x, z)= θ P k θ j=1 λ xθj (z θj ) λ xθ c x θj,z θj (u)du, {u<z θ c } j j and τ is the size of the partition τ. Conditional simulations of max-stable processes Mathieu Ribatet 10 / 24

24 Step 2: The extremal functions Random partitions Sampling scheme Given τ=(τ 1,...,τ l ), draw l independent random vectors ϕ + 1 (s),...,ϕ+ (s) from the distribution l { } [ ] Pr ϕ + j (s) dv j = 1 C j 1 {u<zτ c }λ (s,xτ c ) x τ j j j,z τ j (v j, u) du }{{} density of ϕ Φ such that ϕ(x τj )= z τj dv j, where 1 { } is the indicator function and C j = 1 {u<zτ c j }λ (s,xτ c j ) x τ j,z τ j (v j, u)dudv j. Define the random vector Z + (s)= max j=1,...,l ϕ+ j (s), s Xm. Conditional simulations of max-stable processes Mathieu Ribatet 11 / 24

25 Step 3: The sub-extremal functions Random partitions Sampling scheme Independently draw {ζ i } i 1 a Poisson point process on (0, ) with intensity ζ 2 dζ and {Y i ( )} i 1 independent copies of Y ( ) Define the random vector Z (s)=max i 1 ζ i Y i (s)1 {ζi Y i (x)<z}, s X m. Conditional simulations of max-stable processes Mathieu Ribatet 12 / 24

26 Step 3: The sub-extremal functions Random partitions Sampling scheme Independently draw {ζ i } i 1 a Poisson point process on (0, ) with intensity ζ 2 dζ and {Y i ( )} i 1 independent copies of Y ( ) Define the random vector Z (s)=max i 1 ζ i Y i (s)1 {ζi Y i (x)<z}, s X m. Then provided Φ is regular, the random vector Z (s)=max { Z + (s), Z (s) } follows the conditional distribution of Z (s) given Z (x) = z. Conditional simulations of max-stable processes Mathieu Ribatet 12 / 24

27 As an aside Random partitions Sampling scheme The conditional cumulative distribution function is { } τ Pr[Z (s) a, Z (x) z] Pr {Z (s) a Z (x)=z}= π x (z,τ) F τ, j (a), τ P k j=1 Pr[Z (x) z] }{{}}{{} Steps 1 & 2 Step 3 where F τ, j (a)= {y<z τ c,u<a} λ (s,x τ c ) x τ j j j,z τ j (u, y)dydu {y<z τ c } λ. t τ c x τ j j j,z τ j (y)dy Conditional simulations of max-stable processes Mathieu Ribatet 13 / 24

28 As an aside Random partitions Sampling scheme The conditional cumulative distribution function is { } τ Pr[Z (s) a, Z (x) z] Pr {Z (s) a Z (x)=z}= π x (z,τ) F τ, j (a), τ P k j=1 Pr[Z (x) z] }{{}}{{} Steps 1 & 2 Step 3 where F τ, j (a)= {y<z τ c,u<a} λ (s,x τ c ) x τ j j j,z τ j (u, y)dydu {y<z τ c } λ. t τ c x τ j j j,z τ j (y)dy Remark. It is clear that Z ( ) {Z (x) = z} is not max-stable. Conditional simulations of max-stable processes Mathieu Ribatet 13 / 24

29 Brown Resnick processes Random partitions Sampling scheme Example 1. If Z is a Brown Resnick process, i.e., Z (x)=max i 1 ζ i exp{ε i (x) γ(x)}, x X, then the intensity function is ( λ x (z)= C x exp 1 ) k 2 logzt Q x logz+l x logz z 1 i, z (0, ) k, i=1 and the conditional intensity function is λ s x,z (u)=(2π) m/2 Σ s x 1/2 exp { 1 } m 2 (logu µ s x,z) T Σ 1 s x (logu µ s x,z) i.e., the extremal functions are log-normal processes. i=1 u 1 i, Conditional simulations of max-stable processes Mathieu Ribatet 14 / 24

30 Schlather processes Random partitions Sampling scheme Example 2. If Z is a Schlather process, i.e., Z (x)= 2πmax i 1 ζ i max{0,ε i (x)}, x X, then the intensity function is ( ) k+ 1 λ x (z)=π (k 1)/2 Σ x 1/2 a x (z) (k+1)/2 Γ, z R k, 2 where a x (z)=z T Σ 1 x z, and the conditional intensity function is λ s x,z (u)=π m/2 (k+ 1) m/2 Σ 1/2 { 1+ (u µ)t Σ 1 (u µ) k+ 1 i.e., the extremal functions are Student processes. } ) (m+k+1)/2 Γ( m+k+1 Γ 2 ( k+1 2 ), Conditional simulations of max-stable processes Mathieu Ribatet 15 / 24

31 Computational burden Full conditional The full conditional are nice! 2. Markov chain Monte Carlo sampler (for Step 1) Conditional simulations of max-stable processes Mathieu Ribatet 16 / 24

32 Do you recognize these numbers? Computational burden Full conditional The full conditional are nice! Conditional simulations of max-stable processes Mathieu Ribatet 17 / 24

33 Do you recognize these numbers? Computational burden Full conditional The full conditional are nice! These are the first 20 Bell numbers. Remark. Recall that Bell(k) is the number of partitions of a set with k elements. Hence with our terminology we have # hitting scenarios=card(p k )=Bell(k). Conditional simulations of max-stable processes Mathieu Ribatet 17 / 24

34 Computational burden Computational burden Full conditional The full conditional are nice! In Step 1, we need to sample from a discrete distribution whose state space is P k. Combinatorial explosion Hence we cannot compute C (x, z) in π x (z,τ)= 1 C (x, z) τ j=1 λ xτ j (z τj ) λ xτ c x τ {u<z τ c } j j,z τ j (u)du. j Conditional simulations of max-stable processes Mathieu Ribatet 18 / 24

35 Computational burden Computational burden Full conditional The full conditional are nice! In Step 1, we need to sample from a discrete distribution whose state space is P k. Combinatorial explosion Hence we cannot compute C (x, z) in π x (z,τ)= 1 C (x, z) τ j=1 λ xτ j (z τj ) λ xτ c x τ {u<z τ c } j j,z τ j (u)du. j Use of MCMC samplers to sample from the target π x (z, ). Remark. We will use a Gibbs sampler since the full conditional are especially convenient. Conditional simulations of max-stable processes Mathieu Ribatet 18 / 24

36 Full conditional Computational burden Full conditional The full conditional are nice! Want to sample from Pr[θ θ j = τ j ], θ π x (z, ) where τ j is the restriction of τ P k to the set {x 1,..., x k } \ {x j }. The number of possible states for θ is b + = { l if {x j } is a partitioning set of τ, l+1 otherwise. Example 3. For τ=({x 1, x 2 },{x 3 }) we have l=2 and Restriction Possible states τ 2 = τ 2 τ 3 = τ 3 ({x 1, x 2 },{x 3 }) ({x 1, x 2 },{x 3 }) ({x 1 },{x 2, x 3 }) ({x 1, x 2, x 3 }) ({x 1 },{x 2 },{x 3 }) Conditional simulations of max-stable processes Mathieu Ribatet 19 / 24

37 The full conditional are nice! Computational burden Full conditional The full conditional are nice! For all τ P k such that τ j = τ j, Pr[θ= τ π x (z,τ ) θ j = τ j ]= π x (z, τ)1 { τ j =τ j } τ P k where w τ, j = λ xτ j (z τj ) {u<z τ c } λ x τ c x τ j j j,z τ j (u)du. τ j=1 w τ, j τ j=1 w τ, j, In particular at most 4 weights w, need to be evaluated and the Gibbs sampler is especially convenient! Conditional simulations of max-stable processes Mathieu Ribatet 20 / 24

38 Conditional simulations of max-stable processes Mathieu Ribatet 21 / 24

39 Precipitation around Zurich in Figure 1: From left to right, maps on a grid of the pointwise 0 025, 0 5 and sample quantiles for rainfall (mm) obtained from conditional simulations of Brown Resnick processes having semi variogram γ(h)=(h/38) The rightmost panel plots the ratio of the width of the pointwise confidence intervals with and without taking estimation uncertainties into account. The squares show the conditional locations. Conditional simulations of max-stable processes Mathieu Ribatet 22 / 24

40 Temperature anomalies for the 2003 European heatwave Basel (316) Zurich (556) Santis (2490) Oeschberg (483) Neuchatel (485) Bad Ragaz (496) Engelberg (1035) Bern (565) Chateau d Oex (985) Montreux (405) Montana (1508) Gd St Bernard (2472) (m) Locarno Monti (366) Lugano (273) Davos (1590) Arosa (1840) (km) ( C) Figure 2: Left: Topographical map of Switzerland showing the sites and altitudes in metres above sea level of 16 weather stations for which annual maxima temperature data are available. Right: Map of temperature anomalies ( C), i.e., the difference between the pointwise medians obtained from conditional simulations and unconditional medians estimated from the fitted Schlather process. As expected the largest deviations occur in the plateau region of Switzerland The differences range between 2 5 C and 4 75 C Conditional simulations of max-stable processes Mathieu Ribatet 23 / 24

41 THANK YOU! ANY QUESTIONS? Dombry, C. Éyi-Minko, F. and Ribatet, M. (2012) Conditional simulation of max-stable processes To appear in Biometrika.