Environmental applications are often spatial in nature so benefits can be gained by using spatial methods. For example, we can improve inference for a single location by using data from the spatial neighbourhood around the location. Spatial extremes also allows us to quantify the joint risk of multiple locations being affected by a single extreme event. Most spatial extremes models can capture only one of the two classes of extremal dependence described in Section 2.2. The most widely studied and used spatial extremes models are in the family of max-stable processes, which can model asymptotic dependence or perfect independence. Asymptotically independent models include Gaussian processes and inverted max-stable processes (Wadsworth and Tawn, 2012a), but these are not widely adopted in practice due to a general preference for conservativeness (i.e. overestimation of extremes is considered safer than underestimation). The hybrid model of Wadsworth and Tawn (2012a) is capable of modelling both asymptotic dependence and independence, but its complexity makes it difficult to use in practice. We will touch upon all of these models in the subsequent chapters of this thesis (Chapters 3 and 4 in particular), but since max-stable processes are the most widely used, we will focus our attention on them for the rest of this section. For a more comprehensive review of methods for spatial extremes, the reader is referred to Davison et al. (2012).
2.3.1 Max-stable processes
The random process Z(t) is called max-stable on Ω, if for each k = 1,2, . . ., there exist continuous functionsak(t)>0 andbk(t) such that for any functionz(t),
i.e.Z(t) and the maximum of k independent copies of {Z(t)−bk(t)}/ak(t) have the same
distribution. A consequence of this property is that{Z(t1), . . . , Z(tm)}follows a multivariate
extreme value distribution for allt1, . . . , tm ∈Ω and all m.
All max-stable processes are asymptotically dependent over all distances, i.e.η = 1, unless they are perfectly independent. Here we will show this for the bivariate extreme value (BEV) distribution. The BEV distribution function takes the form:
G(x, y) = exp{−V(x, y)}. (2.3.1)
Hence it follows that,
Pr(X > x, Y > x) = 1−2 exp −1 x + exp{−V(x, x)}, = 1−2 exp −1 x + exp −V(1,1) x , ∼ 2−V(1,1) x , as x→ ∞,
where the second line follows due to the homogeneity property of V as shown in (2.2.3). By comparison with (2.2.11), η = 1, unless V(1,1) = 2, which only occurs when X and
Y are perfectly independent. Hence, indeed the bivariate extreme value distribution is asymptotically dependent.
In the following, we will introduce four commonly used classes of max-stable processes. For the sake of simplicity of notation, here we will only present joint distributions for two-dimensional max-stable models. Higher order joint distributions have been studied by Wadsworth and Tawn (2014) and Genton et al. (2015).
Smith process
Following Smith (1990), consider a stochastic process{Z(t)}, with t∈Ω for some arbitrary index set Ω. Without loss of generality we may assume that Z(t) has standard Fr´echet
margins:
Pr(Z(t)6z) =e−1/z,∀t∈Ω.
Smith defines{(ξi, si), i>1}as the points of a Poisson process on (0,∞)×S, with intensity
measureξ−2dξ×ν(ds), whereS is an arbitrary measurable set andν is a positive measure
onS. Furthermore, let {f(s, t), s∈S, t∈Ω}denote a non-negative function for which
Z
S
f(s, t)ν(ds) = 1,∀t∈Ω.
Then Smith definesZ(t) as:
Z(t) = max
i {ξif(si, t)}, t∈Ω,
wheref(s, t) is a multivariate normal density with mean sand covariance matrix Σ,
f(s, t) =f0(s−t) = (2π)−d/2|Σ|−1exp −1 2(s−t) TΣ−1(s−t) , (2.3.2) and takesν(ds) =ds.
To motivate this representation from a practical point of view, we can think ofsi as storm
centres in Ω, with magnitude ξi, distributed over space according to a Poisson process, so
centres are uniformly distributed with intensitydsand sizes decay with a density 1/ξ2. The functionf represents the ‘shape’ of the storm. Hence, ξif(si, t) is the size of the storm at
positiontfrom a storm of sizeξi centred at locationsi. Figure 2.3.1 shows three realisations
of the process in one dimension. The underlying events are shown in black and the pointwise maximum of these is taken to obtain a one-dimensional realisation from the Smith process (shown in red).
It can be shown that this choice ofZ(t) is max-stable and that it has unit Fr´echet margins for anyt∈Ω (see Smith (1990)). Figure 2.3.2 shows two simulations from the two-dimensional Smith model with different covariance matrices. The realisation shown on the left panel is
Figure 2.3.1: Three one-dimensional realisations from the Smith process (top: σ = 0.5, middle: σ= 1, bottom: σ= 1.5), with the red line being the pointwise maximum, and the black lines the underlying events.
0 2 4 6 8 10 0 2 4 6 8 10 x y 0 2 4 6 8 10 0 2 4 6 8 10 x y
Figure 2.3.2: Two simulations from the Smith model with different covariance matrices on a 100×100 grid. Left panel: σ11=σ22= 1.5 andσ12= 0; right panel: σ11=σ22= 1.5 and σ12= 1.
isotropic whereas the one on the right is not. Both realisations are very smooth due to the underlying Gaussian densities.
For two sites (d= 2), say the origin oand a location h, the joint distribution is given by
Pr(Z(o)6z1, Z(h)6z2) = exp{−Vh(z1, z2)}
whereVh is the exponent measure defined as
Vh(z1, z2) =z1−1Φ a(h) 2 +a −1(h) log z2 z1 +z2−1Φ a(h) 2 +a −1(h) log z1 z2 , (2.3.3)
where Φ is the standard normal distribution function, anda(h) =
√
hTΣ−1his the Mahalan-
obis distance betweenhand the origin. Higher order joint distributions become increasingly complicated.
The extremal coefficient for the Smith model is θ(h) = 2Φ{a(h)/2}, for which θ(h) → 2 as khk → ∞, and θ(h) → 1 as khk → 0, spanning the range of possible asymptotic dependencies.
0 2 4 6 8 10 0 2 4 6 8 10 x y 0 2 4 6 8 10 0 2 4 6 8 10 x y
Figure 2.3.3: Two simulations from the Schlather model with different range parameters on a 100×100 grid. Left panel: range = 10; right panel: range = 3.
Schlather process
Following Schlather (2002), let {Sj}∞j=1 be the points of a Poisson process on R+ with
intensityds/s2. Let{Wj(x)}∞
j=1 be independent replicates of a stationary Gaussian process W(x) on Rd satisfying E[max{0, Wj(o)}] = 1, where o denotes the origin. Then Schlather
defines
Z(x) = max
j Sjmax{0, Wj(x)}. (2.3.4)
and proves thatZ(x) is a stationary max-stable process onRd with unit Fr´echet marginals. The exponent measure for this model in the bivariate case is given by
Vh(z1, z2) = 1 2 1 z1 + 1 z2 × 1 + 1−2(ρ(h) + 1)z1z2 (z1+z2)2 1/2! ,
where ρ is a valid correlation function. The most commonly used correlation functions are isotropic, i.e. ρ(h) = ρ(khk), and include the Whittle-Matern, Cauchy and powered exponential correlation functions. As for the Smith model, higher order forms are difficult to express analytically. Figure 2.3.3 shows two realisations of the two-dimensional Schlather process with powered exponential correlation functions with different range parameters. Note that the Schlather model realisations are less smooth than the Smith realisations.
The extremal coefficient isθ(h) = 1 +{[1−ρ(h)]/2}1/2. Because of the requirement that ρ(h) be a positive definite function for h ∈ R2, and W
j(x) be stationary and isotropic,
θ(h) < 1.838. So this means that the model cannot account for extremes that become independent whenkhk → ∞.
Brown-Resnick process
Let (x) be an isotropic fractional Brownian process with semivariogram γ(h) ∝ khkα, 0< α62 and(0) = 0 almost surely. Then W(x) in (2.3.4) can be taken as:
W(x) = exp{(x)−γ(x)}.
This process was introduced by Brown and Resnick (1977). When is a Brownian process and α = 2, W(x) corresponds to the Smith model. The bivariate extremal coefficient is θ(h) = 2Φ{a(khk)/2}, as for the Smith process, but a takes a different value (a =
{2γ(khk)}1/2). Therefore, θ(khk) → 2 as khk → ∞, so the process captures complete
independence for large distances.
Extremal-t process
The extremal-t process was first proposed by Demarta and McNeil (2005) and it assumes the following representation ofW(x) in (2.3.4):
W(x) =√π2−ν/2+1Γ ν+ 1 2 −1 max{(x),0}ν,
where µ > 1, Γ is the Gamma function, and (x) is a Gaussian random field with mean zero and correlation function ρ(h). The case when ν = 1 corresponds to the Schlather process. The bivariate extremal coefficient isθ(h) = 2Tν+1(
p
(ν+ 1)[1−ρ(h)]/[1 +ρ(h)]), whereTν denotes the cumulative distribution function of a student-t random variable with
0 2 4 6 8 10 0 2 4 6 8 10 x y 0 2 4 6 8 10 0 2 4 6 8 10 x y
Figure 2.3.4: Two simulations from the extremal-t model with different range parameters on a 100×100 grid. Left panel: range = 1; right panel: range = 3.
ν degrees of freedom, and correlation function ρ(h). Figure 2.3.4 shows two realisations of the extremal-t process with different dependence structures. These realisations appear similar in roughness to the simulations from the Schlather proces.