On the distribution of a max-stable process conditional on max-linear functionals

Statistics and Probability Letters 100 (2015) 158–163

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

On the distribution of a max-stable process conditional on max-linear functionals Marco Oesting 1 AgroParisTech/INRA, UMR 518 Math. Info. Appli., F-75005 Paris, France

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Article history: Received 18 September 2014 Received in revised form 27 January 2015 Accepted 2 February 2015 Available online 19 February 2015 Keywords: Conditional simulation Normalized representation Spatial extremes

Recently, Dombry and Éyi-Minko (2013) provided formulae for the distribution of a maxstable process conditional on its values at given sites and proposed a methodology for sampling from this distribution. We generalize their results by allowing for conditions stemming from max-linear functionals of the process. Furthermore, we show that the conditional distribution of the extremal functions, i.e. the spectral functions attaining the imposed conditions, is closely related to the normalized spectral representation. The results are illustrated in several examples. © 2015 Elsevier B.V. All rights reserved.

1. Introduction During the last years, max-stable processes have become frequently used models for spatial extremes, in particular for applications in environmental sciences. In the context of the prediction of these processes given some data, the question of their conditional distribution arises. The conditions considered so far are restricted to values of the process at several sites. For this case, exact formulae in terms of the exponent measure of the max-stable process have been provided (Dombry and Éyi-Minko, 2013) and explicit computations have been implemented for several subclasses (cf. Dombry et al., 2013; Oesting and Schlather, 2014, for example). In this paper, we analyze the conditional distribution allowing for more general conditions given by max-linear functionals of the process. For instance, a condition on the maximum of the process may be considered. In this case, the analysis of the conditional distribution may provide further insight in characteristics of extreme events that exceed a certain value. More precisely, we consider a max-stable process {Z (x), x ∈ K } on some compact set K ⊂ Rd which – without loss of generality – can be assumed to have unit Fréchet marginals, i.e. P(Z (x) ≤ z ) = exp(−1/z ), z > 0, for all x ∈ K . Further, we require Z to be sample-continuous, that is, all sample paths are in the space C+ (K ) of nonnegative continuous functions on K . Thus, Z possesses a spectral representation (see de Haan, 1984; Giné et al., 1990; Penrose, 1992, for example): Z (t ) = max Ui Wi (t ), i∈N

t ∈ K,

(1)

where {Ui , i ∈ N}, is a Poisson point process on (0, ∞) with intensity measure u−2 du and Wi , i ∈ N, are independent copies of a nonnegative sample-continuous stochastic process W with EW (t ) = 1 for all t ∈ K . Assume that we observe values of continuous max-linear functionals L1 , . . . , Ln : C+ (K ) → [0, ∞), i.e. Lj (max{a1 f1 , a2 f2 }) = max{a1 Lj (f1 ), a2 Lj (f2 )},

for all a1 , a2 ≥ 0, f1 , f2 ∈ C+ (K ), j = 1, . . . , n.

E-mail address: [email protected]. 1 Current address: University of Twente, Faculty of Geo-Information Science and Earth Observation, NL-7500 AE Enschede, The Netherlands. http://dx.doi.org/10.1016/j.spl.2015.02.002 0167-7152/© 2015 Elsevier B.V. All rights reserved.

M. Oesting / Statistics and Probability Letters 100 (2015) 158–163

159

An example for such max-linear functionals are L(f ) = supt ∈K ′ f (t ) for some compact set K ′ ⊂ K , including the special cases L(f ) = supt ∈K f (t ) and L(f ) = f (t0 ) for some t0 ∈ K . More generally, using continuity arguments and the compactness of K , it can be shown that every functional of this type is of the form L(f ) = sup h(t )f (t ),

f ∈ C+ (K ),

(2)

t ∈K

for some bounded, but not necessarily continuous function h : K → [0, ∞). In this paper, we analyze the conditional distribution of Z | L(Z ) where L(Z ) = (L1 (Z ), . . . , Ln (Z ))⊤ . In Section 2, we provide formulae for the conditional distribution in terms of the exponent measure generalizing the results of Dombry and Éyi-Minko (2013). More explicit expressions for the case of one single condition are derived in Section 3 making use of connections to the normalized spectral representations to max-stable processes. Section 4 deals with the more general case of a finite number of conditions. 2. General theory In the following, we analyze the distribution of Z conditionally on L(Z ) = z for some z = (z1 , . . . , zn )⊤ ∈ (0, ∞)n . Here, we note that, because of the max-linearity of Lj , j = 1, . . . , n, the properties of Lj (Z ) are directly connected to those of Lj (W ). By Prop. 2.3 in Oesting et al. (2013), the finiteness of L(Z ) implies that ELj (W ) < ∞ and P(Lj (Z ) ≤ z ) = exp(−E(Lj (W ))/z ), z > 0, i.e., Lj (Z ) follows a Fréchet distribution. To exclude the trivial case that the distribution of Lj (Z ) is degenerate, assume that pj = P(Lj (W ) > 0) > 0 for every j ∈ {1, . . . , n}. We consider the extended process ZL = ({Z (t ), t ∈ K }, L(Z )) on C+ (K ) × (0, ∞)n . By the max-linearity of Lj , we obtain that Lj (Z ) = maxi∈N Ui Lj (Wi ). Thus, ZL = max(ξi , L(ξi ))

(3)

i∈N

where the maximum is considered componentwise and Π = C+ (K ) × [0, ∞)n with intensity measure

Λ(A × B) =



i∈N

δ(ξi ,L(ξi )) denotes a Poisson point process on S =

∞



u−2 P(uW ∈ A, uL(W ) ∈ B) du, 0

for Borel sets A ⊂ C+ (K ) and B ⊂ [0, ∞)n (cf. Kingman, 1993), that is, ξi corresponds to the product Ui Wi in representation (1). Perceiving the conditions L1 (Z ) = z1 , . . . , Ln (Z ) = zn as conditions on the value of the process ZL at specific ‘‘sites’’ according to representation (3) the results of Dombry and Éyi-Minko (2013) can be applied on ZL , in order to derive the distribution of Z conditional on L(Z ). To this end, for every non-empty index subset J ⊂ {1, . . . , n}, we consider the J-extremal random point measure ΠJ+ and the J-subextremal random point process ΠJ− , defined by

ΠJ+ =



δξi 1{Lj (ξi )=Lj (Z ) for some j∈J } and ΠJ− =



δξi 1{Lj (ξi )
i∈N

i∈N

It can be shown that ΠJ+ and ΠJ− are well-defined point processes on C+ (K ) (see Dombry and Éyi-Minko, 2013, Lemma A.3). Further, as Π{+j} (CK+ ) = 1 a.s. (cf. Dombry and Éyi-Minko, 2013, Prop. 2.5), Π{+1,...,n} is characterized via so-called hitting

scenarios (cf. Wang and Stoev, 2011; Dombry and Éyi-Minko, 2013), i.e. partitions τ = {τ1 , . . . , τl } of {1, . . . , n} representing the situation that Π{+1,...,n} = {ξ1+ , . . . , ξl+ } s.t.

 = Lj (Z ), Lj (ξk ) < Lj (Z ), +

j ∈ τk , j ̸∈ τk

1 ≤ j ≤ n, 1 ≤ k ≤ l.

Let Θ ∈ P{1,...,n} be the random partition realized by Z where P{1,...,n} denotes the space of partitions of {1, . . . , n}. Based on the different hitting scenarios, conditional simulations of Z | L(Z ) = z can be performed via the following three-step procedure proposed by Dombry and Éyi-Minko (2013): 1. Draw a partition τ = {τk }lk=1 from the distribution of Θ | L(Z ) = z. 2. Simulate (ξk+ }lk=1 as a realization of Π{+1,...,n} | L(Z ) = z, Θ = τ . 3. Draw {ξi− }i∈N from the distribution of Π{−1,...,n} | L(Z ) = z.

Then, maxlk=1 ξk+ ∨ maxi∈N ξi− is a realization of Z | L(Z ) = z. The distributions involved in the algorithm are given in the following theorem which summarizes Theorems 3.1 and 3.2 in Dombry and Éyi-Minko (2013). First, we need some more notation. For a non-empty index subset J ⊂ {1, . . . , n}, let RJ : [0, ∞)n → [0, ∞)|J | be the projection on the components belonging to J, i.e. RJ (z) = zJ where zJ = (zj )j∈J for z = (zj )nj=1 . Further, let µ be the exponent measure of Z , i.e. µ(A) = Λ(A × [0, ∞)n ), A ⊂ C+ (K ), and µJ the exponent

160

M. Oesting / Statistics and Probability Letters 100 (2015) 158–163

measure of (Lj (Z ))j∈J , i.e. 1 µJ (B) = Λ(C+ (K ) × R− J (B)),

B ⊂ [0, ∞)|J | Borel.

|J | The tail function of µJ is denoted by µ ¯ J (zJ ) = µJ ([0, zJ )c ), zJ ∈ [0, ∞)|J | . Furthermore, let{PJ (zJ ; ·), zJ ∈ [0, ∞)   } be a regular version of the measure µ(df ) conditional on (Lj (f ))j∈J = zJ . That is, PJ (zJ ; ·) satisfies A 1B (L(f )) µ(df ) = B A PJ (z; df ) µJ (dzJ ) for all Borel sets A ⊂ C+ (K ) and B ⊂ [0, ∞)|J | .

Theorem 1 (Adapting Dombry and Éyi-Minko, 2013). With the above notation, the following holds true: P(Θ =τ , L(Z )∈ dz) , ′ τ ′ ∈P{1,...,n} P(Θ =τ , L(Z )∈ dz)

1. P(Θ = τ | L(Z ) = z) = 

P(Θ = τ , L(Z ) ∈ dz) = exp −µ ¯ {1,...,n} (z)



where the fraction is understood as a Radon–Nikodym derivative and

l  



Pτk (zτk ; {(Lj (f ))j∈τ c < zτ c })µτk (dzτk ) . k

k

k=1

2. P(ξk+ ∈ dfk | L(Z ) = z, Θ = τ ) = 3. Conditionally on L(Z ) =

Π{+1,...,n} .

z, Π{−1,...,n}

1{L c (f )
l

k

.

k

is a Poisson point process with intensity measure 1{L(f )
Remind that, for the conditional simulation of Z , we do not need the whole Poisson point process Π{−1,...,n} , but only the

pointwise maximum Z − (t ) = maxi∈N ξi− (t ), t ∈ K . By the third part of Theorem 1, simulation of Z − is closely related to unconditional simulation of Z . The only difference is that those points that violate the condition L(Z ) = z are neglected. A specific spectral representation of Z proves beneficial for an exact simulation: By de Haan and Ferreira (2006), Cor. 9.4.5, any sample-continuous max-stable process with unit Fréchet margins can be written as Z (t ) = max Ui Vi (t ), i∈N

t ∈ K,

where Vi , i ∈ N, are independent copies of some stochastic process V with supt ∈K V (t ) = c a.s. for some uniquely determined c > 0. Recently, Oesting et al. (2013) revisited this representation – which they call the normalized spectral representation – and showed that the law of the process V can be expressed in terms of the law of the process W in representation (1) via the equality

˜ (t ))−1 · W ˜, V =d c · (sup W t ∈K

˜ is given by P(W ˜ ∈ A) = c −1 supt ∈K w(t ) P(W ∈ dw), A ⊂ C+ (K ), with c = E(supt ∈K W (t )). For this where the law of W A representation, only a finite number of tuples (Ui , Vi ) – those tuples that satisfy cUi > inft ∈K Z (t ) – can contribute to the maximum Z . This fact allows for an exact simulation of Z in finite time and can also be used for an exact simulation in the third step of the procedure for conditional simulation described above. Further, as we will see in what follows, the distribution of the normalized spectral functions is closely connected to the conditional distribution of the extremal functions. 

3. Conditioning on one max-linear functional We first consider to the special case that we have one condition L(Z ) = z for some z > 0, only. In this case, there is one extremal function ξ + satisfying L(ξ + ) = z and there is no need to consider different hitting scenarios. The second part of Theorem 1 allows us to calculate the distribution of ξ + :

P(ξ

+

∈ A | L(Z ) = z ) = P{1} (z ; A) =

d dz

E L(W )1{zW /L(W )∈A}



Λ(A × [0, z ]) =

EL(W )

 ,

(4)

˜ L /L(W ˜ L ), where the law of W ˜ L is given by the Radon–Nikodym derivative for A ⊂ C+ (K ). Thus, we can write ξ + = z W −1 ˜ P(WL ∈ dw)/P(W ∈ dw) = cL L(w) with cL = EL(W ). Remark 2. Note that there is a connection between the law of the extremal function ξ + and a representation similar to the normalized spectral representation. Generalizing the functional f → supt ∈K f (t ) by a max-linear functional L : C+ (K ) → [0, ∞) as above, we obtain the L-normalized spectral representation Z (t ) = max Ui ViL (t ), i∈N

t ∈ K,

where ViL , i ∈ N, are independent copies of a stochastic process VL satisfying VL = B

cL

˜ L) P(L(W ) > 0)L(W

˜ L + (1 − B)W0 W

(5)

M. Oesting / Statistics and Probability Letters 100 (2015) 158–163

161

˜ L has the same law as in (4) and W0 has where B is some Bernoulli random variable with parameter p = P(L(W ) > 0), W the distribution of W | L(W ) = 0. Thus, the spectral functions ViL , i ∈ N, of the L-normalized spectral representation are characterized by the fact that L(ViL ) = p−1 cL with probability p and L(ViL ) = 0 with probability 1 − p. From (5), it can be seen that the law of extremal function ξ + conditional on L(Z ) = z equals the law of cL−1 pzVL | L(VL ) > 0, confirming Eq. (4). ˜ L , which is needed for the simulation of the extremal function, and the process In practice, the simulation of the process W ˜ , which can be used for the exact simulation of the subextremal functions, is not straightforward. Here, remind that W ˜ is W ˜ L for the choice L(f ) = maxt ∈K f (t ). One possible way of simulation is simulation via MCMC methods. For a special case of W instance, one could use the Metropolis–Hastings algorithm with the distribution of W as proposal distribution. We now present two examples for sampling of max-stable processes conditionally on the max-linear functional L(f ) = maxt ∈K f (t ). Example 3 (Smith Process). We consider Smith’s (1990) process of moving maxima type Z (t ) = max U˜ i fσ (t − Si ), i∈N

t ∈ K,

on some finite interval K = [−r , r ], where

 −2 du ds and i∈N δ(U˜ i ,Si ) is a Poisson point process on (0, ∞)× R with intensity u √   fσ (x) = ( 2π σ )−1 exp −x2 /(2σ 2 ) , x ∈ R, denotes the normal density with standard deviation σ > 0. As we consider

a condition on the overall maximum of Z , we can make use of the normalized spectral representation of the Smith process which was calculated explicitly in Oesting et al. (2013): Z (t ) =d max Ui i∈N

cf (t − S˜i ) f (max{0, |S˜i | − r })

,

t ∈ K,

(6)

s where S˜i , i ∈ N, are independent copies of some random variable S˜ whose law is given by P(S˜ ≤ s) = c −1 −∞ f (max{0, |x|− r }) dx and c = 2rf (0) + 1. Thus, the extremal function ξ + can be written as



ξ + (t ) = zf (t − S˜ )/f (max{0, |S˜ | − r }),

t ∈ K.

As the spectral functions in representation (6) can be simulated easily, we also use them to simulate the subextremal functions contributing to Z | L(Z ) = z exactly. An example is shown in Fig. 1. Example 4 (Brown–Resnick Process). The second example we consider are Brown–Resnick processes (Brown and Resnick, 1977; Kabluchko et al., 2009) on some compact set K ⊂ Rd : Z (t ) = max Ui exp Bi (t ) − σ 2 (t )/2 ,





i∈N

t ∈ K,

where Bi , i ∈ N, are independent copies of some centered Gaussian process {B(t ), t ∈ Rd } with stationary increments, variance σ 2 (·) and variogram γ (h) = Var (B(h) − B(0)). Kabluchko et al. (2009) showed that the process Z , extended to Rd , is stationary and its law depends on the variogram γ only. Fig. 1 shows five realizations of a Brown–Resnick process Z on [−5, 5] conditional on maxt ∈[−5,5] Z (t ) = 4. Here, for the simulation as well of the extremal as of the subextremal functions, we use a Metropolis–Hastings algorithm. However, for the subextremal functions, also other (approximative) methods for the simulation of Brown–Resnick processes can be used, see Oesting et al. (2012). 4. Conditioning on a finite number of max-linear functionals In this section, we use the formulae given in Theorem 1 to explicitly calculate the distribution of the random partition Θ and the extremal functions ξk+ in the general case of n conditions L1 (Z ) = z1 , . . . , Ln (Z ) = zn . Here, we restrict to the case that the distribution of the random vector L(W ) is absolutely continuous and has density fL with respect to the Lebesgue measure. Then, applying Theorem 1, yields the following proposition. Proposition 5. Under the above assumptions, we obtain the following results. 1. For any partition τ = {τ1 , . . . , τl } ∈ P{1,...,n} , we have

P (Θ = τ | L(Z ) = z) ∼

l   k=1

∞

v |τk | 0

 

0,v zτ c k

 fL

  v zτk , yτkc dyτkc dv.

162

M. Oesting / Statistics and Probability Letters 100 (2015) 158–163

Fig. 1. Examples of samples from the distribution of max-stable processes conditional on max-linear functionals containing five realizations each. The dots indicate the hitting of the condition. Left: A Smith process Z with σ = 1 conditional on maxt ∈[−5,5] Z (t ) = 4. Middle: A Brown–Resnick process Z associated to the variogram γ (h) = 2|h|1.3 conditional on maxt ∈[−5,5] Z (t ) = 4. Right: A Smith process Z with σ = 1 conditional on L1 (Z ) = maxt ∈[−4,−1] Z (t ) = 2 and L2 (Z ) = maxt ∈[1,4] Z (t ) = 3.

2. Conditional on Θ = τ for a partition τ as in the first part, the distribution of the extremal functions ξk+ , k = 1, . . . , l, is given by

P(ξk+ ∈ A | L(Z ) = z, Θ = τ )

 ∼

∞

v |τk |







0

0,v zτ c





P zj W /Lj (W ) ∈ A | L(W ) = v zτk , yτ c k

   fL v zτk , yτ c dyτ c dv, k k

k

for any Borel set A ⊂ C+ (K ) and an arbitrary index j ∈ τk . In the case τ = {{1, . . . , n}}, the inner integrals are read as a single evaluation of the integrand. Alternatively to representation (1) and the corresponding spectral process W , we can also consider the L-normalized spectral representation (5). Let V (j) be the spectral process of the Lj -normalized spectral representation conditional on Lj > 0, i.e. Lj (V (j) ) = c (j) a.s. for some c (j) > 0. Assuming that all the (n − 1)-dimensional vectors (L1 (V (j) ), . . . , Lj−1 (V (j) ), Lj+1 (V (j) ), . . . , Ln (V (j) ))⊤ , j = 1, . . . , n, possess densities fL,j w.r.t. the Lebesgue measure on (0, ∞)n−1 , Proposition 5 can be reformulated in terms of these lower-dimensional densities. Proposition 6. For any partition τ = {τ1 , . . . , τl }, let jk,τ be an element of τk , k = 1, . . . , l. Then, the following statements hold true. 1. The conditional distribution of the random partition Θ has the form

P(Θ = τ | L(Z ) = z) ∝



 l  (c (jk,τ ) )|τk | |τ |+1

k=1

 fL,j



(j ) 0,c k,τ zτ c /zj

zjk,τk

k

c

z (jk,τ ) τk \{jk,τ } zjk,τ

k,τ

 ,y

τkc

dyτ c . k

2. The distribution of the extremal functions ξk+ , k = 1, . . . , l, conditional on Θ = τ for τ = {τ1 , . . . , τl }, is given by

P(ξk ∈ A | L(Z ) = z , Θ = τ ) ∝ +



 

(j ) 0,c k,τ zτ c /zj k

 ×P

 fL,j k,τ

zjk,τ V (jk,τ ) c (jk,τ )

c

z (jk,τ ) τk \{jk,τ } zjk,τ

 , yτkc

    zτk ( j ) ( j ) ∈ A  L(V k,τ ) = c k,τ , yτkc dyτ c . k z jk,τ

In general, sampling from the distribution of the extremal functions may be quite sophisticated even in the case of few conditions. We finally consider an example that allows for conditional simulation in the case of two conditions. Example 7 (Smith Process, cf. Example 3). We consider Smith’s (1990) process {Z (t ), t ∈ K } on some real interval K conditional on L1 (Z ) = z1 and L2 (Z ) = z2 for some z1 , z2 > 0. Here, we consider conditions of the type L1 (f ) = maxt ∈A1 f (t )

and L2 (f ) = maxt ∈A2 f (t ) for disjoint closed intervals A1 , A2 ⊂ K . Using the notation f˜σ ,A (x) = maxt ∈A fσ (t − x) and cA =

f (x) dx, R σ ,A



˜

the spectral function V (j) of the Lj -normalized spectral representation can be written as V (j) (t ) =

cAj fσ (t − Xj )/f˜σ ,Aj (Xj ), t ∈ K , j = 1, 2, where Xj is a random variable with Lebesgue density cA−j 1 f˜σ ,Aj (cf. Oesting et al.,

2013). Thus, the distribution of Lk (V (j) ), j ̸= k, is given by

P(Lk (V (j) ) ∈ ·) =

 R

1{cA f˜σ ,A (x)/f˜σ ,A (x)∈·} cA−j 1 f˜σ ,Aj (x) dx. j k j

(7)

M. Oesting / Statistics and Probability Letters 100 (2015) 158–163

163

By Proposition 6, we obtain

P(Θ = {{1}, {2}}) = P(Θ = {{1, 2}}) =

cA1 cA2 z12 z22

cA21 z13





P L2 (V

(1)

)≤

P L2 (V (1) ) ∈ d

cA1 z2

cA1 z2 z1

 

z1

 =

P L1 (V

cA22 z23



(2)

)≤

cA2 z1 z2

P L1 (V (2) ) ∈ d



,

cA2 z1 z2



,

where the right-hand sides can be calculated by solving the integral in (7) numerically. The extremal functions belonging to the partition {{1}, {2}} can be simulated easily by simulating V (1) and V (2) conditional on L2 (V (1) ) ≤ cA1 z2 /z1 and L1 (V (2) ) ≤ cA2 z1 /z2 , respectively. Sampling the extremal function associated to {{1, 2}} is more involved as it includes the law of V (j) | Lk (V (j) ). However, in many situations, V (j) is uniquely determined by the condition Lk (V (j) )/Lj (V (j) ) = zk /zj . Fig. 1 shows five realizations of a Smith process Z conditional on L1 (Z ) = 2 and L2 (Z ) = 3. Acknowledgments This work has been supported by the ANR project McSim. The author thanks Liliane Bel and Christian Lantuéjoul for fruitful discussions on the simulation of normalized spectral functions and an anonymous referee for valuable comments and remarks. References Brown, B.M., Resnick, S.I., 1977. Extreme values of independent stochastic processes. J. Appl. Probab. 14, 732–739. Dombry, C., Éyi-Minko, F., 2013. Regular conditional distributions of continuous max-infinitely divisible random fields. Electron. J. Probab. 18, 1–21. Dombry, C., Éyi-Minko, F., Ribatet, M., 2013. Conditional simulation of max-stable processes. Biometrika 100, 111–124. de Haan, L., 1984. A spectral representation for max-stable processes. Ann. Probab. 12, 1194–1204. de Haan, L., Ferreira, A., 2006. Extreme Value Theory: An Introduction. Springer, Berlin. Giné, E., Hahn, M., Vatan, P., 1990. Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Related Fields 87, 139–165. Kabluchko, Z., Schlather, M., de Haan, L., 2009. Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37, 2042–2065. Kingman, J.F.C., 1993. Poisson Processes. Oxford University Press, New York. Oesting, M., Kabluchko, Z., Schlather, M., 2012. Simulation of Brown–Resnick processes. Extremes 15, 89–107. Oesting, M., Schlather, M., 2014. Conditional sampling for max-stable processes with a mixed moving maxima representation. Extremes 17, 157–192. Oesting, M., Schlather, M., Zhou, C., 2013. On the normalized spectral representation of max-stable processes on a compact set. Available from http://arxiv.org/abs/1310.1813. Penrose, M.D., 1992. Semi-min-stable processes. Ann. Probab. 20, 1450–1463. Smith, R., 1990. Max-stable processes and spatial extremes. Unpublished manuscript. Wang, Y., Stoev, S.A., 2011. Conditional sampling for spectrally discrete max-stable random fields. Adv. Appl. Probab. 43, 461–483.