Spatial Risk Measure for Max-Stable and Max-Mixture Processes
Ahmed Manaf, Véronique Maume-Deschamps, Pierre Ribereau, Céline Vial
SSPATIAL RISK MEASURE FOR MAX-STABLE ANDMAX-MIXTURE PROCESSES
M. AHMED, V. MAUME-DESCHAMPS, P.RIBEREAU, AND C.VIAL
Abstract.
In this paper, we consider isotropic and stationary max-stable, in-verse max-stable and max-mixture processes X = ( X ( s )) s ∈ R and the damagefunction D νX = | X | ν with 0 < ν < /
2. We study the quantitative behaviorof a risk measure which is the variance of the average of D νX over a region A ⊂ R . This kind of risk measure has already been introduced and studiedfor some max-stable processes in [14]. We evaluated the proposed risk measureby a simulation study. Introduction
Storms are the most destructive natural hazards in Europe. The economic andthe private sectors losses due to these extreme events are often important. For ex-ample, during December 1999, three storms hit Europe causing insured losses above10 billion e (see [9, 20, 26]). The storms may have a huge spatial component; inother words, the underlying spatial process may have a strong spatial dependenceeven at a long distance.One of the main characteristics of climate events is the spatial dependence. Manydependence structures may arise: Asymptotic dependence; Asymptotic indepen-dence or both [27]. The high impact of storm losses motivated us to propose riskmeasures taking into account the spatial dependence.In case of univariate random variables, risk measures has been widely studied in theliterature and the corresponding axiomatic formulation has been presented in [3].In [10] a collection of risk measures indexed by a network is introduced for somefinancial products. In spatial contexts, the spatial dependence plays an importantrole. For example, wind speed and rainfall amount e.g. have different spatial be-havior, so that, after normalization of their marginal distributions, the value of arisk measure should not be the same.In [12], the authors proposed to evaluate the risk on a region by a probability P ( S > s ) where S is an integrated damage function. In [14] or [13] this idea is de-veloped to define spatial risk measures taking into account the spatial dependence.In [2] the same idea is used: a risk measure constructed with the damage function D X,u = ( X − u ) + with u a fixed threshold for a Gaussian process X is studied. Inthe same spirit as [3], the authors propose a set of axioms that a risk measure inthe spatial context should verify. This point of view has been previously adoptedin [14] for some max-stable processes. Our main contributions concern the riskmeasure based on the intensity damage function D νX = | X | ν with 0 < ν < /
2, itconsists in the development of the results from [14]: further max-stable processesare involved and the computation technics are extended to max-mixture processes.We study the properties of the risk measure with respect to the parameters of each
Key words and phrases.
Risk measures, Spatial dependence, Max-stable process, Max-Mixtureprocess, Extreme value theory. a r X i v : . [ m a t h . S T ] J un M. AHMED, V. MAUME-DESCHAMPS, P.RIBEREAU, AND C.VIAL model (with a focus on the dependence parameter). We also study its axiomaticproperties.This paper is organized as follows. Section 2 recalls definitions and properties ofmax-stable and max-mixture processes. In Section 3 we consider spatial risk mea-sures and recall the axiomatic setting from [2] which derives from [14]. Section 4 isdevoted to the study of the risk measures with damage function | X | ν for max-stableand max-mixture processes. We propose forms of this risk measures and derive itsbehavior. We present in Section 5 a simulation study in order to evaluate thisspatial risk measures. Concluding remarks are discussed in Section 6.2. Spatial extreme processes
We shall focus on max-stable processes, inverse max-stable processes and max-mixtures of both, and we call these processes extreme processes. We shall em-phasize on the modelization of the dependence structure and thus assume thatthe marginal laws have been normalized to unit Fr´echet with distribution function F ( x ) = exp( − /x ), x > Max-stable model.
This is an extension of the multivariate extreme valuetheory to the spatial setting. We refer to [7, 8] for definitions and properties ofmax-stable processes. We shall consider max-stable processes on S ⊂ R with unitFr´echet marginal distributions (i.e. simple max-stable processes), for any ( s, t ) ∈ S , P (cid:0) X ( s ) ≤ x , X ( t ) ≤ x (cid:1) = G s,t ( x , x )= exp( − V s,t ( x , x )) , (2.1)where V s,t is the so-called exponent measure function. It is homogenous of order − { /x , /x } ≤ V s,t ( x , , x ) ≤ { /x + 1 /x } . These bounds imply that X is positive quadrant dependent (PQD), see [17] for def-initions and properties of PQD processes. In this paper, we consider stationary andisotropic processes. Thus, the exponent measure V s,t and the distribution function G s,t depend only on the norm h = || s − t || and will be denoted by V h and G h .In [7], it is also proved that every simple max-stable process X has the follow-ing spectral representation: X ( s ) = max i ≥ ξ i W i ( s ) s ∈ S , where { ξ i , i ≥ } is an i.i.d Poisson point process on (0 , ∞ ), with intensity dξ/ξ and { W i , i ≥ } are i.i.d copies of a positive random field W = { W ( s ) , s ∈ S } , suchthat E [ W ( s )] = 1 for all s and independent of ξ i .Many dependence measures for spatial processes X have been introduced. Theseare generally bivariate dependence measures used in a spatial context. The taildependence coefficient χ introduced in [16] is defined by(2.3) χ ( h ) = lim u → P (cid:0) F ( X ( s )) > u | F ( X ( s + h )) > u (cid:1) . If χ ( h ) = 0, the pair ( X ( s + h ) , X ( s )) is said to be asymptotically independent(AI).If χ ( h ) (cid:54) = 0, the pair ( X ( s + h ) , X ( s )) is said to be asymptotically dependent (AD).The process is said AI (resp. AD) if for all h ∈ S χ ( h ) = 0 (resp. χ ( h ) (cid:54) = 0). Theextremal coefficient Θ( h ) = V h (1 ,
1) satisfies χ ( h ) = 2 − Θ( h ) and (see [27])(2.4) G h ( x, x ) = exp( − Θ( h ) /x ) . PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 3
In [5] an alternative definition of the tail dependence coefficient is given.(2.5) χ ( h, u ) = 2 − log P (cid:0) F ( X ( s )) < u, F ( X ( s + h )) < u (cid:1) log P (cid:0) F ( X ( s )) < u (cid:1) , ≤ u ≤ . We have lim u → χ ( h, u ) = χ ( h ).The spectral representation is useful to construct specific max-stable processes.We present three of them: Smith, Schlater and truncated Schlater models. Smith Model
Introduced in [24]. It is defined on S = R d . Its dependence struc-ture is contained in a covariance matrix Σ. Let { ( ξ i , s i ) } be a Poisson point processon (0 , ∞ ) × R d with intensity ξ − d ξ d s and consider the d -dimensional Gaussianprobability density function ϕ d ( . ; Σ) with mean 0 and covariance matrix Σ. For all s ∈ R d , define W i ( s ) = ϕ d ( s − s i ; Σ) and X ( s ) = max i ≥ { ξ i ϕ d ( s − s i ; Σ) } . The exponent measure function is given by V h ( x , x ) = 1 x Φ (cid:18) τ ( h )2 + 1 τ ( h ) log x x (cid:19) + 1 x Φ (cid:18) τ ( h )2 + 1 τ ( h ) log x x (cid:19) ;with τ ( h ) = √ h T Σ − h and Φ( · ) the standard normal cumulative distribution func-tion.The pairwise extremal coefficient equalsΘ( h ) = 2Φ (cid:18) τ ( h )2 (cid:19) . Note that if the covariance matrix is diagonal Σ = σ I d , then the process X is isotropic as its bivariate distribution depends only on h through the function τ ( h ) = σ (cid:107) h (cid:107) . Schlather Models
This model introduced in [22] provides a class based on astationary Gaussian random field. Let W := { W ( s ) , s ∈ S } be a stationary randomfield, with E (cid:2) W + ( s ) (cid:3) = µ ∈ (0 , ∞ ) where W + ( s ) = max { , W ( s ) } . Let { ξ i , i ≥ } be a Poisson point process on (0 , ∞ ), with intensity d ξ/ξ and { W i , i ≥ } are iidcopies of W ( s ). Consider X ( s ) = µ − max i ≥ ξ i W + i ( s ) , s ∈ S , it defines a stationary max-stable process. Schlather proposed to take a stationaryGaussian process W ( s ) with correlation function ρ ( · ) and µ − = √ π . In this case,the resulting max-stable process X is called Extremal Gaussian process (EG) . Theexponent measure function is V h ( x , x ) = 12 (cid:18) x + 1 x (cid:19)(cid:20) (cid:114) − ρ ( h ) + 1) x x ( x + x ) (cid:21) . The extremal coefficient is given byΘ( h ) = 1 + (cid:18) − ρ ( h )2 (cid:19) / . We have lim h →∞ χ ( h ) (cid:54) = 0. In other words, the asymptotic dependence persistseven at infinite distances. This might be unrealistic in applications. To overcomethis problem a truncated version of W ( s ) can be used. Let { r i } be a homogenous M. AHMED, V. MAUME-DESCHAMPS, P.RIBEREAU, AND C.VIAL
Poisson point process of unit rate on S and µ − = √ π ( E [ |B| ]) − . Then, for astationary Gaussian process W i ( s ), define(2.6) X ( s ) = max i ≥ ξ i W i ( s ) B i ( s − r i ) , s ∈ S with B ⊂ S a compact random set and B i i.i.d. copies of B . The process X isa truncated extremal Gaussian process (TEG) . The exponent measure function isgiven by V h ( x , x ) = (cid:18) x + 1 x (cid:19)(cid:20) − α ( h )2 (cid:18) − (cid:114) − ρ ( h ) + 1) x x ( x + x ) (cid:19)(cid:21) . The extremal coefficient is given byΘ( h ) = 2 − α ( h ) (cid:40) − (cid:18) − ρ ( h )2 (cid:19) / (cid:41) where α ( h ) = E {|B ∩ ( h + B ) |} / E [ |B| ].Usually, B is a disk of radius r . In that case, α ( h ) = { − h/ r } + . For moredetails, see [6]. That leads to χ ( h ) = 0 , ∀ h ≥ r . In other word the process X isasymptotically independent (and thus independent because it is max-stable) for all h ≥ r .2.2. Inverse max-stable processes.
Max-stable processes are either AD or theyare independent. This behavior may be unapropriate in applications: data mayreveal asymptotic independence without being independent.In [5] the lower tail dependence coefficient χ ( h ) is proposed in order to studythe strengt of dependence in AI cases.(2.7) χ ( h ) = lim u → P (cid:0) F ( X ( s )) > u (cid:1) log P (cid:0) F ( X ( s )) > u, F ( Y ( s + h )) > u (cid:1) − , ≤ u ≤ . We have − ≤ χ ( h ) ≤ χ ( h ) = 1. Otherwise, it is asymptotically independent.We shall consider a class of asymptotically independent processes introduced in[27]: Inverse max-stable process . Let X (cid:48) be a max-stable process with unitFr´echet margin, consider X ( s ) = g ( X ( s )) = − / log { − e − /X (cid:48) ( s ) } s ∈ S . Then X is asymptotically independent with unit Fr´echet margin and bivariatesurvivor function P (cid:0) X ( s ) > x , X ( s + h ) > x (cid:1) = exp (cid:0) − V h (cid:0) g ( x ) , g ( x ) (cid:1)(cid:1) . where V h is the exponent measure function of X (cid:48) . We a slight langage abuse,we shall say that V h is the exponent measure function of X . Inverse max-stableprocesses enter in the class of processes defined in [16] which satisfy: P (cid:0) X ( s ) > x, X ( s + h ) > x (cid:1) = L h ( x ) x − /η ( h ) , x → ∞ where L h ( x ) is a slowly varying function and η ( h ) ∈ (0 ,
1] is called the taildependence coefficient . For these kind of processes, the AI is caracterized by η ( h ) < PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 5
Max-Mixture model.
In spatial contexts, specifically in environmental do-main many scenarios of dependence could arise and AD and AI might cohabite.The work by [27] provides a flexible model called max-mixture.Let X be a max-stable process, with extremal coefficient Θ( h ) and exponent mea-sure function V Xh . Let Y be an inverse max-stable process with tail dependencecoefficient η ( h ) and exponent measure function V Yh . Assume that X and Y areindependent and each of them has Fr´echet margin. Let a ∈ [0 ,
1] and define Z ( s ) = max { aX ( s ) , (1 − a ) Y ( s ) } , s ∈ S .Z has unit Fr´echet marginals. Its bivariate distribution function is given by(2.8) P (cid:0) Z ( s ) ≤ z , Z ( s + h ) ≤ z (cid:1) = e − aV Xh ( z ,z ) (cid:20) e − (1 − a ) z + e − (1 − a ) z − e − V Yh ( g a ( z ) ,g a ( z )) (cid:21) , where g a ( z ) = g ( z − a ). Its bivariate survivor function satisfies P (cid:0) Z ( s ) > z, Z ( t ) > z (cid:1) ∼ a { − Θ( h ) } z + (1 − a ) /η ( h ) z /η ( h ) + O ( z − ) , z → ∞ . If h ∗ = inf { h : Θ( h ) (cid:54) = 0 } < ∞ , then Z is asymptotically dependent up to distance h ∗ and asymptotically independent for larger distances. See [4] for more details.Of course, if a = 0 then Z is indeed an inverted max-stable process. If a = 1 then Z is a max-stable process. Moreover(2.9) χ ( h ) = a (2 − Θ( h ))and(2.10) χ ( h ) = [ h ∗ Consider a spatial process X := { X ( s ) , s ∈ S } , S ⊂ R . We use the definition ofrisk measures proposed in [2] and [14]. Given a damage function D : R −→ R + ,and A ∈ B ( R d ), the normalized aggregate loss function on A is L ( A , D ) = 1 |A| (cid:90) A D ( s )d s, where |A| stands for the volume of A . The quantity (cid:90) A D ( s )d s represents theaggregated loss over the region A . Therefore the function L ( A , D ) is the proportionof loss on a A .3.1. Definition of spatial risk measures. In this paper, we work with unitFr´echet margin processes and thus ( X − u ) + as no finite expectation nor variance.The risk measure considered in [2] is not suitable. In [14], the damage function { X>u } is considered and the subsequent risk measure is computed for Smith,Schlater and the so-called tube processes. This damage function does not takeinto account the behaviour of the process over the threshold u , this is why, we didnot consider it. We shall consider the damage function D νX ( s ) = | X ( s ) | ν , for 0 < ν < . This type of damage function is used e.g. in analyzing the negativeeffects due to the wind speed (see [21] for more details). In [14], the risk measureassociated to D νX has been computed for Smith processes.Since we work with stationary processes, the expectation of the normalized lossfunction do not take into account the dependence structure. As in [2] and [14], weshall focus on its variance. M. AHMED, V. MAUME-DESCHAMPS, P.RIBEREAU, AND C.VIAL R ( A , D X ) } = Var (cid:0) L ( A , D X ) (cid:1) . If X is a spatial process with unit Fr´echet marginal distributions, R ( A , D νX ) iswell defined provided that 0 < ν < .Remark that(3.1) R ( A , D X ) } = 1 |A| (cid:90) A×A Cov (cid:0) D X ( s ) , D X ( t ) (cid:1) d s d t. Axiomatic properties of spatial risk measures. Several authors such as[3], [15] and [25] presented an axiomatic setting for univariate risk measures. In[14] a first set of axioms for risk measures in spatial context is considered for thedamage functions: D X ( s ) = { X ( s ) >u } , D X ( s ) = X ( s ) ν where X is a max-stableprocess. In [2] the damage function D X ( s ) = ( X ( s ) − u ) + for Gaussian processeshas been investigated.Let us recall the mains axioms proposed in [14] and [2] for the real valued spa-tial risk measure R ( A , D ). Axioms 1. and 4. below have been introduced in [14],and studied for some max-stable processes. Definition 1. Let A ⊂ R be a region of the space. (1) Spatial invariance under translation Let A + v ⊂ R be the region A translated by a vector v ∈ R . Then for v ∈ R , R ( A + v, D ) = R ( A , D ) . (2) Spatial anti-monotoncity Let A , A ⊂ R , two regions such that |A | ≤ |A | , then R ( A , D ) ≤R ( A , D ) . (3) Spatial sub-additivity Let A , A ⊂ R be two regions disjointed, then R ( A ∪A , D ) ≤ R ( A , D )+ R ( A , D ) . (4) Spatial super sub-additivity Let A , A ⊂ R be two regions disjointed, then R ( A ∪A , D ) ≤ min i =1 , [ R ( A i , D )] . (5) Spatial homogeneity Let λ > and A ⊂ R then R ( λ A , D ) = λ k R ( A , D ) , that is R ishomogenous of order k , where λ A is the set { λx, x ∈ A} . In [14] the invariance by translation, the monotonicity and super sub-additivity inthe case where A , A are either disks or squares is proved for max-stable processesfor the damage function { X>u } and for the damage function X ν in the case ofthe Smith process. While in [2] the invariance by translation and sub-additivityis proved for any processes provided that D X admits an order 2 moment. Theanti-monotonicity for disks and squares is proved for the damage function ( X − u ) + with X a Gaussian process. We shall study further the properties of R ( A , D X )for max-mixture processes.4. Risk measures for max-mixture processes. Let X an isotropic and stationary process, with unit Fr´echet margin, let 0 < ν < / General forms for R ( A , D νX ) . The following result shows that the compu-tation of R ( A , D X ) may reduce to smaller dimension integral. It has been provedin [13] for Smith models. Following the lines of its proof, it remains valid providedthat the damage function X ν has an order 2 moment (see Theorem 3.3 in [2]).Let f disk ( · , R ) and f square ( · , R ) be the density of the distance between two points PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 7 randomly chosen in a disk of radius R and a square of side R respectively. We have(see [18]): f disk ( h, R ) = 2 hR (cid:18) π arccos (cid:0) h R (cid:1) − hπR (cid:114) − h R (cid:19) , and f square ( h, R ) = 2 πhR − h R + 2 h R where b = h R . Lemma 4.1. Let X := { X ( s ) , s ∈ S } be an isotropic and stationary spatial processsuch that the damage function D X has finite order moment.Let Q ( h ) = Cov (cid:0) D X ( s ) , D X ( s + h ) (cid:1) .Consider A ⊂ R a disk of radius R , we have: (4.1) R ( A , D X ) = Var (cid:0) L ( A , D X ) (cid:1) = (cid:90) Rh =0 Q ( h ) f disk ( h, R )d h, Consider A ⊂ R a square of side R , we have: (4.2) R ( A , D X ) = Var (cid:0) L ( A , D X ) (cid:1) = (cid:90) √ Rh =0 Q ( h ) f square ( h, R )d h, In what follows, results are written for square regions A , but the results hold fordisks as well. Remark 1. Properties of moments of Fr´echet distributions give that if X has unitFr´echet marginal distributions, E ( L ( A , D νX )) = Γ(1 − ν ) . Proposition 4.2. Consider X := { X ( s ) , s ∈ S } an isotropic and stationary spa-tial process with unit Fr´echet margin F and pairwise distribution function G Xh = P ( X ( s ) ≤ x , X ( s + h ) ≤ x ) . Let A be a square of side R . We have (4.3) R ( A , D νX ) = (cid:90) √ Rh =0 Q ( h, ν ) f square ( h, R )d h, with Q ( h, ν ) = Cov (cid:0) D νX ( s ) , D νX ( s + h ) (cid:1) ;(4.4) Q ( h, ν ) = (cid:90) ∞ (cid:90) ∞ (cid:2) G Xh ( x /ν , x /ν ) − F ( x /ν ) F ( x /ν ) (cid:3) d x d x or equivalently (4.5) Q ( h, ν ) = ν (cid:90) ∞ (cid:90) ∞ x ν − x ν − (cid:2) G Xh ( x , x ) − F ( x ) F ( x ) (cid:3) d x d x . M. AHMED, V. MAUME-DESCHAMPS, P.RIBEREAU, AND C.VIAL Proof. Since X is a non negative process, the result follows directly from Hoeffding’sidentity ([11] and [23]):Cov (cid:0) D νX ( s ) , D νX ( s + h ) (cid:1) = (cid:90) (cid:90) R (cid:2) P (cid:0) X ( s ) ν ≤ x , X ( s + h ) ν ≤ x (cid:1) − P (cid:0) X ( s ) ν ≤ x (cid:1) P (cid:0) X ( s + h ) ν ≤ x (cid:1)(cid:3) d x d x = ν (cid:90) (cid:90) R + x ν − x ν − (cid:2) P (cid:0) X ( s ) ≤ x , X ( s + h ) ≤ x (cid:1) − P (cid:0) X ( s ) ≤ x (cid:1) P (cid:0) X ( s + h ) ≤ x (cid:1)(cid:3) d x d x . (cid:3) Explicit form for R ( A , D νX ) for TEG max-stable process X . Equation(4.3) shows that, if A is either a disk or a square, the computation of R ( A , D νX )reduces to the integration of Q ( h, ν ) f square (resp. Q ( h, ν ) f disk ). In [13], the com-putation of Q ( h, ν ) f square for the Smith model has been done. In that case, thecomputation of R ( A , D νX ) is reduced to a one dimensional integration. In thissection, we do the computation for a TEG model. Corollary 4.3. Let X := { X ( s ) , s ∈ S } be a truncated extremal Gaussian TEGmax-stable process with unit Fr´echet margin, correlation function ρ and truncatedparameter r . For < ν < / , we have Q ( h, ν ) = (cid:90) + ∞ w ν (cid:20) Γ(2(1 − ν )) T ( w, h ) T ( w, h ) ν − + Γ(1 − ν ) T ( w, h ) T ( w, h ) ν − (cid:21) d w − (cid:2) Γ(1 − ν ) (cid:3) where, (4.6) T ( w, h ) = w + 1 w (cid:20) − α ( h )2 (cid:0) − K ( w, h ) (cid:1)(cid:21) ; T ( w, h ) = (cid:20) − α ( h )2 (cid:0) − K ( w, h ) (cid:1) − α ( h )( ρ ( h ) + 1)(1 − w )2 K ( w, h )( w + 1) (cid:21) × (cid:20) w − α ( h )2 w (cid:0) − K ( w, h ) (cid:1) − α ( h )( ρ ( h ) + 1)( w − w K ( w, h )( w + 1) (cid:21) ;(4.7)(4.8) T ( w, h ) = α ( h ) (cid:20) ( ρ ( h ) + 1) K ( w, h )( w + 1) − ( ρ ( h ) + 1) ( w − K ( w, h ) ( w + 1) (cid:21) ; K ( w, h ) = (cid:20) − w ( ρ ( h ) + 1)( w + 1) (cid:21) / and α ( h ) = { − h r } + .Proof. We have,Cov (cid:0) D νX ( s ) , D νX ( s + h ) (cid:1) = E (cid:2) D νX ( s ) D νX ( s + h ) (cid:3) − (cid:2) E [ D νX ( s )] (cid:3) . PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 9 From Remark 1, E [ D νX ( s )] = Γ(1 − ν ). Moreover, E (cid:2) D νX ( s ) D νX ( s + h ) (cid:3) = (cid:90) ∞ (cid:90) ∞ x ν x ν f ( X ( s ) ,X ( s + h )) ( x , x )d x d x , where f ( X ( s ) ,X ( s + h )) ( x , x ) is the bivariate density function of the TEG model. Itrewrites: E [ D ν ( s ) D ν ( s + h )] = (cid:90) + ∞ (cid:90) + ∞ u ν +1 w ν f ( u, uw )d u d w. The bivariate density function of a TEG model is given by f ( X ( s ) ,X ( s + h )) ( u, uw ) = (cid:2) u T ( w, h ) + 1 u T ( w, h ) (cid:3) e − u T ( w,h ) where T ( w, h ), T ( w, h ) and T ( w, h ) are given in (4.6), (4.7) and (4.8). Therefore E [ D ν ( s ) D ν ( s + h )] = (cid:90) + ∞ w ν (cid:20) T ( w, h ) (cid:90) + ∞ u ν − e − u T ( w,h ) d u + T ( w, h ) (cid:90) + ∞ u ν − e − u T ( w,h ) d u (cid:21) d w. Moment properties of Fr´echet distributions give (cid:90) + ∞ u ν − e − u T ( w,h ) d u = 1 T ( w, h ) .µ (2 ν − , with µ (2 ν − the moment of order k = (2 ν − 1) of a Fr´echet distribution. In thesame way, we get (cid:90) + ∞ u ν − e − u T ( w,h ) d u = T ( w, h ) (2 ν − Γ(1 − ν ) . Then, E (cid:2) D νX ( s ) D νX ( s + h ) (cid:3) = (cid:90) + ∞ w ν (cid:20) T ( w, h ) T ( w, h ) ν − Γ2( ν − 1) + T ( w, h ) T ( w, h ) (2 ν − Γ(1 − ν ) (cid:21) d w, and the result follows. (cid:3) Corollary 4.3 shows that the risk measure for a TEG process may be computedefficiently, since it reduces to a one dimensional integration involving a Gammafunction.4.3. Behavior of R ( λ A , D νX ) with respect to λ for max-mixture processes. In what follows, we consider an isotropic and stationary max-mixture spatial processwith unit Fr´echet margin F . We denote X and V Xh the process and the exponentmeasure function corresponding to the max-stable part and Y and V Yh the processand the exponent measure function corresponding to the inverse max-stable pro-cess Y . Let a ∈ [0 , Z = max( aX, (1 − a ) Y ). We shall study the behavior of R (cid:0) λ A , D νZ (cid:1) with respect to λ . Of course, the case a = 1 gives results for max-stable processes and a = 0 gives results for inverse max-stable processes. Recallthat the bivariate distribution function is given by G Zh ( x , x ) = e − aV Xh ( x ,x ) (cid:20) e − (1 − a ) x + e − (1 − a ) x − e − V Yh ( g a ( x ) ,g a ( x )) (cid:21) , where g ( z ) = − − e − z ) and g a ( z ) = g ( z − a ).Lemma 4.1 and Proposition 4.2 are a keystone to describe the behaviour of R (cid:0) λ A , D νZ (cid:1) .As in Lemma 3.4 in [2], we get for any λ > R ( λ A , D νZ ) = (cid:90) √ Rh =0 f square ( h, R ) Q ( λh, ν ) d h. Corollary 4.4. Let Z be an isotropic and stationary max-mixture spatial processas above. Assume that the mappings h (cid:55)→ V Xh ( x , x ) and h (cid:55)→ V Yh ( x , x ) are nondecreasing for any ( x , x ) ∈ R . Let A ⊂ S be either a disk or a square, then themapping λ (cid:55)→ R ( λ A , D νZ ) is non-increasing.Proof. We use (4.9) and from Proposition 4.2, Q ( h, ν ) = ν (cid:90) ∞ (cid:90) ∞ x ν − x ν − (cid:2) G Zh ( x , x ) − F ( x ) F ( x ) (cid:3) d x d x . Since h (cid:55)→ V Xh ( x , x ) and h (cid:55)→ V Yh ( x , x ) are non decreasing, h (cid:55)→ G Zh ( x , x ) isnon increasing and the result follows. (cid:3) Remark 2. For a spatial max-stable or inverse max-stable process X , the factthat h (cid:55)→ V Xh ( x , x ) is non decreasing implies that the dependence between X ( t ) and X ( t + h ) decreases as h increases, which seems reasonable in applications. Onanother hand, if in addition, V Xh ( x , x ) goes to x + x as h goes to infinity, thismeans that X ( t ) , X ( t + h ) tend to behave independently as h goes to infinity. Corollary 4.5. Let Z be an isotropic and stationary max-mixture spatial processas above. Assume that the mappings h (cid:55)→ V Xh ( x , x ) and h (cid:55)→ V Yh ( x , x ) are nondecreasing for any ( x , x ) ∈ R . Moreover, we assume that V Xh ( x ; x ) −→ x + 1 x as h → ∞ and V Yh ( x , x ) −→ x + 1 x as h → ∞∀ x , x ∈ R + . Let A ⊂ S be either a disk or a square, we have lim λ →∞ R ( λ A , D νZ ) = 0 . If there exists V (resp. V ) an exponent measure function of a non independentmax-stable (resp. inverse max-stable) bivariate random vector, such that V Xh −→ V (resp. V Yh −→ V ) as h → ∞ , then lim λ →∞ R ( λ A , D νZ ) > . Proof. In the case of A a square of side R , we use Q ( h, ν ) = ν (cid:90) ∞ (cid:90) ∞ x ν − x ν − (cid:2) G Zh ( x , x ) − F ( x ) F ( x ) (cid:3) d x d x . If V Wh ( x ; x ) is non decreasing to x + x as h → ∞ for W = X or W = Y , then G Zh ( x , x ) is non increasing to F ( x ) F ( x ) and we conclude by using the monotoneconvergence theorem. (cid:3) Corollary 4.6. Let Z be an isotopic and stationary max-mixture as above. Assumethat h (cid:55)→ V Wh ( x , x ) is non increasing, with W = X or W = Y . Let A and A be either disks or squares such that |A | ≤ |A | then R ( A , D νZ ) ≤ R ( A , D νZ ) . Proof. Since the risk measure R ( A , D νZ ) is invariant by translation, we may assumethat A = λ A for some λ ≥ 1. Then, Equation (4.9) gives the result. (cid:3) PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 11 Numerical study In this section, we will study the behavior of the spatial covariance damagefunction and its spatial risk measure corresponding to a stationary and isotropicmax-stable, inverse max-stable and max-mixture processes. We shall use the cor-relation functions introduced in [1].(1) Spherical correlation function: ρ sphθ ( h ) = (cid:20) − . (cid:18) hθ (cid:19) + 0 . (cid:18) hθ (cid:19) (cid:21) { h>θ } . (2) Cubic correlation function : ρ cubθ ( h ) = (cid:20) − (cid:18) hθ (cid:19) + 352 (cid:18) hθ (cid:19) − (cid:18) hθ (cid:19) + 35 (cid:18) hθ (cid:19) (cid:21) { h>θ } . (3) Exponential correlation functions: ρ expθ ( h ) = exp (cid:2) − hθ (cid:3) , (4) Gaussian correlation functions: ρ gauθ ( h ) = exp (cid:2) − (cid:0) hθ (cid:1) (cid:3) ;(5) Mat´ern correlation function: ρ mat ( h ) = 1Γ( κ )2 κ − ( h/θ ) κ K κ ( h/θ ) , where Γ is the gamma function, K κ is the modified Bessel function of second kindand order κ > κ is a smoothness parameter and θ is a scaling parameter.5.1. Analysis of the covariance damage function Q ( h, ν ) . The covariancedamage function plays a central role in the study of the risk measure R ( A , D νX ).5.1.1. Analysis of Q ( h, ν ) for max-stable processes. We study the behavior of Q ( h, ν )and R ( λ A , D νX ) for X a TEG spatial max-stable process, with trunacted parame-ter r , non-negative correlation function ρ and correlation length θ . We shall denoteby Q θ,r ( h, ν ) the covariance damage function in order to emphasize the dependenceof the parameters. Five different models with different correlation functions (ex-ponential, Gaussian, spherical, cubic and Matern) introduced above are considered.The behavior of Q θ,r ( h, ν ) is shown in Figure 1.(a). We set the power coefficient ν = 0 . r = 0 . 25 and θ = 0 . 20. We have that, Q θ,r ( h, ν ) = 0 for any h ≥ r ; thedecreasing speed changes according to the different dependence structures.For the behavior of (cid:0) D νY ( s ) , s ∈ S (cid:1) with respect to θ is shown in Figure 1(b).Figure 1(c) shows the behavior of Q θ,r ( h, ν ) with respect to the truncated param-eter r . We set ν = 0 . h = 0 . 25 and θ = 0 . ν . We set h = 0 . θ = 0 . 20 and r = 0 . 25. Figure1.(d)shows that the covariance between the damage functions D νY ( · ) and D νY ( · + h )increases with ν . Remark 3. The global behavior of Q θ,r ( h, ν ) for an inverse TEG is the same asfor the TEG with the same parameters. Figure 1. shows the behavior of Q θ,r ( h, ν ) with respect to thepower coefficient ν , the correlation length θ , the distance h andtruncated parameter r . Plain lines correspond to TEG and dashedlines correspond to inverse TEG. Five non-negative correlationfunctions (exponential, Gaussian, spherical, cubic and Mat´ern with κ = 1) have been examined. The graphs (a), (b) ,(c) and (d) showthe behavior of Q · , · ( · , · ) with respect to: (a) the distance h , when ν = 0 . θ = 0 . r = 0 . 25; (b) the correlation length θ , when ν = 0 . r = 0 . 25 and h = 0 . 25; (c) the truncated parameter r ,when ν = 0 . θ = 0 . 20 and h = 0 . 25; (c) the power coefficient ν ,when θ = 0 . r = 0 . 25 and h = 0 . Analysis of Q ( h, ν ) for max-mixture processes. Max-mixture models withTEG max-stable part, denoted X and inverse TEG for the inverse max-stable part- denoted by Y - cover all possible dependence structures in one model (asymptoticdependence at short distances, asymptotic independence at intermediate distancesand independence at long distances). We have simulated five max-mixture modelsaccording to the correlation functions above, X and Y have the same correlationfunctions with different correlation lengths. r X and r Y denote the respective trun-cation parameter of X and Y , ρ X and ρ Y denote the respective correlation functionsof X and Y , θ X and θ Y denote the respective correlation length. The mixing pa-rameter is denoted by a .We set the parameters a = 0 . r X = 0 . θ X = 0 . r Y = 0 . θ Y = 0 . ν = 0 . 2. In this model, the damage functions D νY ( · ) and D νY ( · + h ) areasymptotically dependent up to distance h < r X . The decreasing speed dependson the correlation function, as shown in Figure 2.Figure 3 shows the behavior of Q ( h, ν ) with respect to each parameter. When itis not varying, each parameter is fixed to a = 0 . h = 0 . ν = 0 . θ X = 0 . θ Y = 0 . r X = 0 . 15 and r Y = 0 . 35. Graph (a) shows the behavior of Q with re-spect to the mixing parameter a . The graphs from (b) to (f) shows the behavior of PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 13 Figure 2. shows the behavior of Q ( h, ν ) with respect to thedistance h . Five non-negative correlation functions (exponential,Gaussian, spherical, cubic and Mat´ern with κ = 1) have beenexamined when a = 0 . ν = 0 . X is a TEG max-stableprocess with θ X = 0 . 15 and r X = 0 . Y is an inverted TEGprocess with θ Y = 0 . 35 and r Y = 0 . Q with respect to the other parameters. The behavior is the same as for max-stableprocesses.5.2. Numerical computation of R ( A , D ν ) . In this study, we compute R ( A , D ν )for different max-stable processes X , inverse max-stable processes Y and max-mixture processes Z . We considered X a TEG with parameters r X and θ X , Y aSmith process with parameter σ Y . The process Z is a max-mixture between X and Y . Max-stable and inverse max-stable models are achieved for a = 1 and a = 0,respectively. We compute R ( A , D ν ) using (4.3) and (4.5) i.e. a 3 dimensionalintegration. For these models, the reduction to a one dimensional integration seemnot possible. We shall compare this computed value with the Monte Carlo esti-mation obtained by simulating the process Z . In this simulation study, the TEGhas parameters: r X = 0 . 25, non-negative exponential correlation function with θ X = 0 . 20. The inverse max-stable Y , is given by a Smith max-stable process Y (cid:48) with σ Y (cid:48) = 1. The process Z is simulated with n = 50 locations on a grid overa square A = [0 , . We set the power coefficient ν := { . , . , . , . , . } and mixing parameter a := { , . , . , . , } .The intuitive Monte-Carlo computation (M1), is obtained by generating a m = 1000sample of Z on the grid. Then, L j ( A , D + Z,u ) = 1 |A| (cid:20) Rn − (cid:21) n − (cid:88) i =1 Z ∗ ( s ij ) j = 1 , ..., m, Figure 3. (a) shows the behavior of Q ( h, ν ) with respect to themixing parameter a , the power coefficient ν , the correlation lengths θ X , θ Y , and truncation parameters r X , r Y . Five non-negativecorrelation functions (exponential, Gaussian, spherical, cubic andMat´ern with κ = 1) have been examined. For a = 0 . h = 0 . ν = 0 . θ X = 0 . θ Y = 0 . r X = 0 . 15 and r Y = 0 . 35, the graphs(a),(b),(c),(d),(e) and (f) show the behavior of Q ( · , · ) with respectto: (a) the mixing parameter a ; (b) the power coefficient ν ; (c) thetruncation parameter r X ; (d) the truncated parameter r Y ; (f) thecorrelation length θ X ; (e) the correlation length θ Y .where, Z ∗ ( s ij ) = | Z ( s ij ) | ν E M [ L ( A , D νZ )] = 1 m m (cid:88) j =1 L j ( A , D νZ )and(5.1) Var M ( L ( A , D νZ )) = 1 m − m (cid:88) j =1 ( L j ( A , D νZ ) − E M [ L ( A , D νZ )]) . Boxplots in Figure 4. represent the relative errors over 100 (M1) simulations withrespect to the 3 dimensional integration. It shows that the considered risk measuresare hardly estimated by Monte Carlo for ν greater than 0 . 30. Let us emphasizethat in the 3 dimensional integration, we used (4.5). Using (4.4) creates numericalissues when ν approaches 0 . Behavior of R ( λ A , D νX ) . We are going to study the behavior of R ( λ A , D νX )with respect to λ for A = [0 , a square and several models. We fixed ν = 0 . 20, and a = 0 . 50 for max-mixture models and also we will evaluate R ( λ A , D νX ) with respectto the mixing parameter a .We considered two models for X : TEG with r = 0 . θ = 0 . σ = 0 . 6. We considered two inverse max-stable processes Y : Inverse PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 15 Figure 4. The boxplots represent the relative errors ofthe Monte Carlo estimation of Var( L ( A , D νZ )) with respect tothe 3 dimensional integration for different power coefficient ν := { . , . , . , . , . } and mixing parameter a := { , . , . , . , } with parameters r X = 0 . 25 and θ X = 0 . X and with σ = 1 cor-responding to the inversed Smith process Y over a square A =[0 , .TEG and inverse Smith. The process Z is the max mixture Z = max( aX, (1 − a ) Y ).The different chosen parameters are listed below. • MM1: X is TEG with the parameters as for the TEG max-stable processabove and Y is inverse TEG with r Y = 0 . 45 and non-negative exponentialcorrelation function with correlation length θ = 0 . • MM2: X is TEG max-stable with the same parameters as for MM1 and Y is inverse Smith with σ Y = 0 . R ( λ A , D νX ) for max-stable and inverse max-stable processes are verysimilar. Their behavior mimics also the one of χ ( h ) in the max-stable case, or χ ( h )in the inverse max-stable case. In this picture, we have chosen h = 0 . R ( λ A , D νX ) for the max-mixture model MM1.It shows a relatively high value for R ( λ A , D νX ) up to 0 . λ < r X . Figure 6.(b) isdevoted to the model MM2. The global behavior is the same for the two models.We remark that the rupture parameter r Y is hardly identified on these graphs.Figure 6.(b) shows the behavior of R ( λ A , D νX ) with respect to the max-mixturemodel MM2. We can see the same behavior of the asymptotic dependence part inMM1 when 0 . λ < r X and the decrease to zero from 0 . λ ≥ r X . The speed ofdecrease to zero depends the chosen model.Figures 7. (a) and (b) shows the behavior of R ( λ A , D νX ) with respect to a . Figure 5. The graphs represent the behavior of R ( λ A , D νX )with respect to λ for ν = 0 . 20, a square A = [0 , and thecorresponding to tail and lower tail dependence coefficients. Fourmodels are considered:(a) TEG model with truncated parameter r X = 0 . 25 and exponential correlation function with correlationlength θ X = 0 . 20; (b) inverse TEG max-stable with the same pa-rameters as in (a); (c) Smith max-stable process with σ = 0 . h = 0 . Figure 6. shows the behavior of R ( λ A , D νX ), χ ( h ) and χ ( h ) fortwo max-mixture models.6. Conclusion We have developped the study of the risk measure R ( A , D ν ) for spatial processesallowing asymptotic dependence and asymptotic independence. This risk measure PATIAL RISK MEASURE FOR MAX-STABLE AND MAX-MIXTURE PROCESSES 17 Figure 7. shows the behavior of R ( λ A , D νX ) with respect tomixing parameter a for two max-mixture models : (a) MM1 model;(b) MM2 model.takes into account the spatial dependence structure over a region. It satisfies theaxioms from [2] and [14] for isotropic and stationary max-mixture processes. Asimulation study emphasized the behavior of the risk measure with respect to thevarious parameters. Finally, the sensitivity of spatial risk measures with differentdependence structures is studied for two different models.Acknowledgements: This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit´e de Lyon, within the program ”Investissements d’Avenir”(ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). References [1] P. Abrahamsen, A review of gaussian random fields and correlation functions , Norsk Regne-sentral/Norwegian Computing Center, 1997.[2] M. Ahmed, V. Maume-Deschamps, P. Ribereau, and C. Vial, Spatial risk measure for gaussianprocesses , arXiv preprint arXiv:1612.08280 (2016).[3] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, Coherent measures of risk , Mathematicalfinance (1999), no. 3, 203–228.[4] J-N. Bacro, C. Gaetan, and G. Toulemonde, A flexible dependence model for spatial extremes ,Journal of Statistical Planning and Inference (2016), 36–52.[5] S. Coles, J. Heffernan, and J. Tawn, Dependence measures for extreme value analyses , Ex-tremes (1999), no. 4, 339–365.[6] A.C. Davison and M. Gholamrezaee, Geostatistics of extremes , Proceedings of the royal so-ciety of london a: Mathematical, physical and engineering sciences, 2012, pp. 581–608.[7] L. De Haan, A spectral representation for max-stable processes , The annals of probability(1984), 1194–1204.[8] L. De Haan and A. Ferreira, Extreme value theory: an introduction , Springer Science &Business Media, 2007.[9] M.G. Donat, T. Pardowitz, G.C. Leckebusch, U. Ulbrich, and O. Burghoff, High-resolutionrefinement of a storm loss model and estimation of return periods of loss-intensive stormsover germany , Natural Hazards and Earth System Sciences (2011), no. 10, 2821–2833.[10] H. F¨ollmer, Spatial risk measures and their local specification: The locally law-invariant case ,Statistics & Risk Modeling (2014), no. 1, 79–101.[11] P. Hougaard, Analysis of multivariate survival data , Springer Science & Business Media,2012.[12] C. Keef, J. Tawn, and C. Svensson, Spatial risk assessment for extreme river flows , Journalof the Royal Statistical Society: Series C (Applied Statistics) (2009), no. 5, 601–618. [13] E. Koch, Tools and models for the study of some spatial and network risks:application toclimate extremes and contagion in france , Ph.D. Thesis, ISFA, University of Claude BernardLyon1, 2014.[14] , Spatial risk measures and applications to max-stable processes , To appear in Ex-tremes (2015).[15] P. A. Krokhmal, Higher moment coherent risk measures , Quantitative Finance (2007),no. 4, 373–387.[16] A.W. Ledford and J. A. Tawn, Statistics for near independence in multivariate extremevalues , Biometrika (1996), no. 1, 169–187.[17] E. L. Lehmann, Some concepts of dependence , The Annals of Mathematical Statistics (1966),1137–1153.[18] D. Moltchanov, Distance distributions in random networks , Ad Hoc Networks (2012),no. 6, 1146–1166.[19] P. Naveau, A. Guillou, D. Cooley, and J. Diebolt, Modelling pairwise dependence of maximain space , Biometrika (2009), no. 1, 1–17.[20] M. Re, Winterstorms in europe ii–analysis of 1999 losses and loss potentials , Publication ofMunich Re (2001).[21] , Natural catastrophes 2012 analyses, assessments, positions 2013 issue , Topics Geo(2013), 1–66.[22] M. Schlather, Models for stationary max-stable random fields , Extremes (2002), no. 1, 33–44.[23] P. K. Sen, The impact of wassily hoeffding’s research on nonparametrics , The collected worksof wassily hoeffding, 1994, pp. 29–55.[24] R.L. Smith, Max-stable processes and spatial extremes , Unpublished manuscript, Univer(1990).[25] A. Tsanakas and E. Desli, Risk measures and theories of choice , British Actuarial Journal (2003), no. 04, 959–991.[26] U. Ulbrich, M. Fink A.H.and Klawa, and J.G. Pinto, Three extreme storms over europe indecember 1999 , Weather (2001), no. 3, 70–80.[27] J. A. Wadsworth J. L.and Tawn, Dependence modelling for spatial extremes , Biometrika (2012), no. 2, 253–272.(2012), no. 2, 253–272.