The Discretized Case

CRP S_K,m -0.347 -0.413 -0.332 -0.341 M AEK,m 0.487 0.526 0.466 0.469

Table 5.2: Results of the simulation study for the Brown-Resnick process withN = 1000 and K= 100.

The results are presented in Table 5.2. Here, all the methods have almost the same ac-curacy, at least ifε= 10⁻⁶ (m= 1a). However, the last method runs much faster than the others. Furthermore, we notice the difference betweenCRP SK,1aandCRP S_K,1b in-dicating that considering approximate intersections of at least three curves yields worse results. This is because these intersections involve incorrect shape functions. Further-more, they lead to degenerated conditional distributions which are not supposed to occur in the case of the Brown-Resnick process. Thus, in this case, the approximation of the mixed moving maxima process seems to be appropriate only if we do not consider intersections of three or more curves.

5.8 The Discretized Case

By now, we have considered the general model (5.1). The procedure we proposed is exact in the case of a finite number of shape functions which are sufficiently smooth.

However, as the example of the Brown-Resnick process in Section5.7illustrates, we may run into problems if these assumptions are violated.

Now, we modify our general model (5.1) and use a discretized version Z(t) = max

i∈N

U_i^(p)·F_i^(p)(t−S_i^(p))

, t∈pZ^d, (5.37)

where (S_i^(p), U_i^(p), F_i^(p))i∈Nare the points of a simple Poisson point process Π^(p)onpZ^d× (0,∞)×Gwithp >0 and a countable setG⊂(0,∞)^pZd. The intensity measure of Π^(p) is given by

Λ({s} ×B× {g}) = X

z∈Z^d

δpz(ds)× Z

B

u⁻²du×PF({g})

where PF is a probability measure belonging to a G-valued random variable F with E(P

z∈Z^dF(pz)) = 1.

Using the same notation as before, we get the same results as in Section 5.2. However, all the calculations can be done explicitly without any further assumptions on f ∈ G.

We get the following results:

Proposition 5.22. Let B ={i} ⊂ {1, . . . , n}, z∈Rⁿ and

Di(z) ={(x, f)∈pZ^d×G: (x, y, f)∈I_{i}(z) for some y∈R}. Then, we have

Λ(I_{^(m)_i} (z)) = 1 2^m

X

(x,f)∈Di(z)

f(t_i−x)

z_i PF({f}) +o(2^−m).

5. Conditional Sampling of Mixed Moving Maxima Processes

Proposition 5.23. Let B ∈2^{1,...,n}\ ∅, |B|>1 and z∈Rⁿ such that I_A(z) ={(x_j, y_j, f_j), j= 1, . . . , l(z)}

withl(z)>0. Then, form large enough, we have Λ(I_B^(m)(z)) =

Xl(z) j=1

1

y_jPF({f_j})·

mini∈A

2^mz_i

j_m(z_i) −max

i∈A

2^mz_i j_m(z_i) + 1

In particular, Λ(I_B^(m)(z))∈ O(2^−m), butΛ(I_B^(m)(z))∈ O/ (2^−m(1+ε)) for any ε >0.

By these formulae, all the scenario probabilities can be calculated. As the intensity of each intersection set has the same rate of convergence, only scenarios with minimal Π(K_t,z) occur.

We note that our model is very close to the model investigated by Wang and Stoev (2011). To see this, we calculate that

P(Z(t₁)≤z₁, . . . , Z(t_n)≤z_n) = exp



−X

f∈G

X

z∈Z^d i=1,...,nmax

f(t_i−pz)P_F({f}) z_i



. Therefore, we get that

Z= max^d

z∈Z^dmax

f∈F

f(· −pz)PF({f})Z_f^(z) ,

where the random variables Z_f^(z), z ∈ Z^d, f ∈ G, are independently standard Fr´echet distributed.

This means, the model (5.37) is a max-linear model and the exact conditional distribution can be calculated via the algorithm of Wang and Stoev (2011) if G is finite and the support of each f ∈ G is finite. However, the algorithm of conditional sampling of the Poisson point process and the algorithm of Wang and Stoev (2011) do not work in exactly the same way. According to the latter algorithm one samples from each random variable Z_f^(z). This procedure corresponds to simulating the largest point of Π in (0,∞)×{pz}×{f}for eachz∈Z,f ∈G. The first algorithm includes the simulation of points in Π until a terminating condition given in Theorem 4 of Schlather (2002) is met. Computational experiments show that both algorithms yield identical results. Also the technical results are related.

For example, the occurrence of a scenario J ⊂Z^d×G (in the notation of Wang/Stoev) corresponds to the event that Π₂ consists of |J| points (pz,min_i=1,...,nf(t^Z(ti)

i−pz), f) with (z, f)∈J. By this correspondence, the statements

• Π(Kt,z) is minimal a.s.

• an occurring hitting scenario J satisfies |J| = r(J(A,x)) a.s. (Wang and Stoev, 2011)

are equivalent, both claiming that the number of points generating the observation (t,z) is minimal. Hence, although the approaches look quite different, there are similar observations and results inWang and Stoev (2011) and in this section.

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6 Conditional Sampling of Brown-Resnick and Extremal Gaussian Processes

Besides mixed moving maxima processes, there are basically two further models for max-stable processes which are frequently used in applications: so-called extremal Gaussian processes (Schlather 2002; for applications seeBlanchet and Davison 2011,Davison and Gholamrezaee 2011, for example) and Brown-Resnick processes (Kabluchko et al. 2009;

for applications see Buishand et al. 2008 and Eckert et al. 2011), which we already considered in Chapter 4.

In this chapter, we will analyse the problem of conditional sampling for both types of processes. As both models are also based on Poisson point processes, we will adapt arguments and methods from Chapter 5 to this framework. However, as the finite-dimensional marginal distributions of extremal Gaussian and Brown-Resnick processes are absolutely continuous w.r.t. the Lebesgue measure, formulae for conditional distribu-tions can be derived more directly using derivatives of the exponent measure. Recently, this way has been taken successfully by Dombry et al.(2011) and Dombry and Ribatet (2012) based on the work of Dombry and Eyi-Minko (2011). But, as we have seen in Chapter 5, the results of Dombry and Eyi-Minko (2011) cannot be applied directly if the finite-dimensional marginals of the max-stable random field are not absolutely con-tinuous w.r.t. the Lebesgue measure. Therefore, we will independently derive the same results as Dombry et al. (2011) and Dombry and Ribatet (2012), using the arguments from Chapter5 which can be used in a more general context.

After a short introduction to a general model pooling both types of processes, we con-sider a partition of the underlying Poisson point process (Section 6.1). Following the approach of Chapter 5, in Section 6.2, we analyse “blurred” intersection sets to calcu-late probabilities for different scenarios. Finally, we deal with the distribution of those points of the underlying Poisson point process which generate the observations within a scenario (Section6.3).

Let ˜Π = P

i∈NδU_i be a Poisson point process on (0,∞) with intensity u⁻²du defined on a probability space (Ω,A,P). Independently, let{W_i(t), t∈R^d}i∈N be independent copies of a Gaussian random field{W(t), t∈R^d}, which will be specified separately for the two types of processes.

Then, the extremal Gaussian process (Schlather,2002) is defined by Z_S(t) = max

i∈N

U_i·maxn√

2πW_i(t),0o

, t∈R^d, (6.1)

whereW is a stationary zero mean Gaussian random field with standard variance.

The Brown-Resnick process is given by ZBR(t) = max

i∈N

Ui·exp

Wi(t)−σ²(t) 2

, t∈R^d, (6.2)

6. Conditional Sampling of Brown-Resnick and Extremal Gaussian Processes

whereW is a zero mean Gaussian random field with stationary increments and variance σ²(·). Recall that the distribution does not depend on the variance σ²(·), but only on the variogramγ of the underlying Gaussian processW, i.e.

Z_BR(t)= max^d

i∈N

U_i·exp

W_i(t)−W_i(0)−γ(t) 2

, t∈R^d,

(cf. Thm.4.1). Henceforth, we assumeW(0) = 0 a.s. for the Brown-Resnick process.

Both the extremal Gaussian process and the Brown-Resnick process define stationary max-stable processes with standard Fr´echet margins. Pooling both models, we write

Z(t) = max

i∈N (U_i·f(t, W_i(t))), t∈R^d, (6.3) where the link functionf :R^d×R→[0,∞) either has the formf(t, y) = max{√

2πy,0} (for the extremal Gaussian process) orf(t, y) = exp(y−γ(t)/2) (for the Brown-Resnick process).

This chapter aims to sample from Z(·) conditional on its values at some fixed loca-tions t₁, . . . , t_n ∈ R^d. In the following, we will assume that the joint distribution of (W(t1), . . . , W(tn)) is non-degenerated and thatW(·) has continuous sample paths. Un-der these assumptions, we will determine the distribution of the Poisson point process

Π =X

i∈N

δ_(Ui,Wi)

on (0,∞)×C(R^d) conditional onZ(t₁), . . . , Z(t_n). The intensity measure Λ of Π is given by

Λ(A×B) = Z

A

u⁻²du·PW(B), A⊂(0,∞), B ∈ C(R^d), (6.4) whereP_W is the probability measure belonging toW(·) andC(R^d) denotes the Borel-σ-algebra onC(R^d) w.r.t. uniform convergence on compact sets (see Section 3.3).

6.1 Random Partition of Π

ForA∈2^{1,...,n}, we consider the random measure ΠA defined by ΠA(·) = Π · ∩

(u, w)∈(0,∞)×C(R^d) : uf(ti, w(ti)) =Z(ti), i∈A, uf(t_j, w(t_j))< Z(t_j), j /∈A . We will show that the random sets Π_Aare point processes.

Proposition 6.1. For any A∈2^{1,...,n}, the random set Π_A is a point process.

Proof. First, we note that the mapping

Ψ : (0,∞)×C(R^d)→C(R^d), (u, w(·))7→uf(·, w(·))

is well-defined as W(·) has continuous sample paths. This also implies the continuity of γ(·). Furthermore, Ψ is ((B ∩(0,∞))× C(R^d),C(R^d))-measurable since the mapping

98

6.2. Blurred Sets

Im Dokument Spatial Interpolation and Prediction of Gaussian and Max-Stable Processes (Seite 103-107)