Description of the Dependence Strength of Two Variogram Models of a Spatial Structure Using Archimedean Copulas

Variogram is a geostatistical tool which describes how the spatial continuity changes with a given separating distance between pairs of stations. In this paper, we study the dependence structure within a same class of bivariate spatialized archimedean copulas. Specifically, we point out properties of the gaussian variogram and the exponential one. A new measure of similarity of two copulas is computed particularly between the spatial independent copula and full dependence one.


Introduction
The main objective of geostatistical analysis is the characterization of spatial phenomenos that are incompletely known.Different definitions of geostatistic have been proposed by spatial statistics researchers.Thereby, geostatistics can be defined as a branch of statistics focusing on spatial or spatio-temporal datasets.While G. Matheron (Matheron, G., 1969)found in geostatistics the application of probabilistic methods to regionalized variables, Issaks E. H. et all (Helena, F., 2012) rather defined them as a way of describing phenomenons and provides adaptation of classical regression technics to take advantages of this continuity.Geostatistics covers mainly three subdomains of statistical studies: analysis of variogram, krieging and stochastic simulation.All of these subdomains use variogram models, so that variogram lies at the earth of every geostatistical activity.
The variogram function describes the degree of spatial dependence of a given spatial random field or stochastic process {Z(x), x ∈ D} .This tool plays with the madogram a key role in spatial modelling, estimations and inference properties given data constraints.The simple form for madogram is for all h where h is the average value of the separating distance between the two points (Shepard, R. N., 1987).
Moreover, the concept of F-madogram has been introduced by Cooley et al. (Shepard, R. N., 1987) to generalized is the λ-madogram associated to the distribution underlying the stochastic process {Z (x)} . γ In the same way, combining results of spatial statistics and multivariate dependence tools, Barro D. (Diakarya, B., 2012) used spatial extreme values copulas to characterize the λ−madogram of process distribution under a distortional assumption.Suppose H is a bivariate distribution satisfying the key assumption.If its associated multivariate EV distribution marginal are unit-Fréchet distributed, then, the λ-madogram is given by where D h is a conditional spatial measure convex defined on the unit simplex of R 2 .In spatial prevision in particular, the current problem is that one needs to replace empirical model by an theorical admissible variogram model.These models are characterized by a nugget, sill and range.In practice, for a same data, different geostatisticians can obtains different paramaters because of the their appreciations.
The main contribution of this study is to study the dependence structure between two models of variograms within a same class.A new measure of similarity of two copulas are computed in particular between the spatial independent copula and full dependence one.

Prelimaries
In this section we collect important definitions and properties on bivariate copulas, variogram tail dependence coefficient.These tools turn out to be necessary for our approach.We refer the readers to standard references for copulas analysis as Joe (1997) which provide detailed and readable introductions to copulas.

Elements of Copulas Analysis
Introduced in spatial analysis by authors as Blanchet (Shepard, R. N., 1987)and Kazianka (Kazianka, H. 2009)copulas functions constitutes the fundamental tool in dependence modeling in statistics.Particularly, every n-copula must satisfy the n-increasing property, that means that, for any rectangle In a bivariate study, the relation (2.1) is equivalent to the rectangular inequality, that is, for all (u 1 , v 1 ); Two particular copulas are used in multivariate dependence modelling: the independent or product copula Π, defined for all by: Π (u 1 , ..., and the complete dependence one, given for a parameter θ and in bivariye case by where M and W are the so called bounds of Fréchet.
Particularly, an n-dimensional copula C is an Archimedean copula, if there exists a continuous and strictly decreasing convex function ϕ, the generator of C, in the class of completely monotone functions

Setting of Variogram
The variogram is a measure of dissimilarity.Let S = } be the set of n spatial sites.In classic statistic the computation of the parameters is relatively possible because the repetition of independent data.However in spatial statistics one can study the most of the time a single observation in each site.Therefore, in isotropical erea one groups these sites according to their separed distance h and we compute the parameters of interest.In the practice, the paramaters depend on chosen step (espace unity).
Let Y (s) be the regionalized variable, then the empirical version of variogram is computed as where N (h) is the set of pair sites {s + h, s} separated by a magnitude lag h and ♯N (h) its cardinal.
Our key assumption may be assumed and strenghtened by the intrinsec hypothesis and the continuous in quadratic which mean that Next, we denote H the set of spatial fields that respect these previous conditions.

Main Results
In this paper the main objective is to describe the strenght of the dependence between two variogram functions in the same erea.

Concept of Class of Variograms
In this subsection, we study the dependence structure in two main classes which are exponential class and gaussian one.The spatial structure has about the intrinsic hypothesis.Then, any admissible variogram functions γ is supposed continuous at the origin, has a sill ϖ (asymptotically or not) and concave function.
Definition 2 Two variogram functions γ 1 and γ 2 are said to be of the same type if and only if, for all h ∈ [0, +∞] there exists two real quantities β > 0, α ∈ R such that Moreover, the set of variograms of the same type is called a class of variograms.
Endeed, it is easy to prove that the relation( 9) is an equivalent relation.Each of these classes is represented by its normalized version provided respectively by equation ( 10) for the exponential class and by (11) for gaussian one.
The main objective consists in describing the strenght dependence between two variogram functions in the same region via archimedean copulas, so that, we need to find the generator of each class.
Proposition 3 Let γ be an normalized exponential or gaussian variogram, then function F defined one R by is a distribution function.
proof.Let γ be an exponential or gaussian variogram normalized.Then, on [0, +∞[, F is continuous and increasing function because γ satisfies these properties.Morever, For a simplicity reason, let's use the standardized version of each class of variogram in the main assumption.There is no loss of generality in this case, this standardization does not modifie the strength spatial dependence structure.It allows to reduce all scales to one without affect the spatial dependence structure.Another advantage of this standardization is the follolling proposition.
Proposition 4 Let γ be the standardized variogram with the main asumption, their exist δ ∈ R + ∪ {+∞} such that the function Fs defined from R to R by is a distribution function.
Proof.In the main asumption, if the spatial structure has a range then δ is 1 else +∞.Suppose that the spatial structure has a range, then F becomes Since γ is a continuous and increasing function and then F is continuous and increasing function.Morever, it follows that Therefore F is a distribution function as asserted.Similarly, the case δ = +∞ has been proved.

Spatial Archimedean Copulas of a Class of Variograms
Let X s = {( X s,1 ; ...; X s,n ) , s ∈ S } be a continuous stochastic random vector observed at a finite number of locations S m = {s 1 , ..., s m } ; s i ∈ R 2 .Barro et all (2016) have introdued the defintio of spatialized generator ϕ s of Archimedean copulas as for a given s ∈ S and for all Spatial copulas are used for generating joint distributions with a variety of spatial dependence structure.This section deals with how to find a spatial copula knowing the variogram function of field.Let ϕ be a continuous and strictly decreasing function from [0, 1] to [0, +∞] such that ϕ (1) = 0.The pseudo-inverse of ϕ is the function ϕ [−1] is given as Proposition 5 Let γ be an admissible standardized variogram function and F γ its associated distribution function.The bivariate distribution function C γ defined from [0, 1] 2 to [0, 1] as is a spatial Archimedean copula, where φ γ is the Laplace transform of F γ defined as Proof.Previously we prove that F γ is a distribution function.And Nelson proved that, Laplace transform of any distribution function is a Archimedean copula generator.Consequently φ γ is a Archimedean generator.This table put out that the dependence structure in gaussian class and exponential class are respectively descibed by the inferior bound of Fréchet-Hoeffding and Clayton copula with parameter one.
Table 1.Spatial Archimedean Copulas with Generators in exponential and gaussian class.

Interpretation
Let C I and C D denote respectively the dependence and the independent copulas.On one hand, if the two variograms in a class (gaussian or exponential class) are totally independent, then the resulting copula will be independent (i.e C I (u, v) = uv).That means also that using one variogram rather than another could cause an important deviation of information.On other hand, if the two variograms of gaussian (or exponential) class was totally dependent, the resulting copula is rather dependent one (i.eC D (u, v) = min (u, v)).That situation would mean that using one variogram rather than another could not affect deeply the informations.The dependence structure of our interest classes are neither that of C I nor C D .Therefore, we need to know the proximity degree between our particular classes from each of C I nor C D .
Visually, by the bands of levels lines forms, the figures 2.4 and 2.5 in appendix 3, show that the associated copula of gaussian variogram class is more similar with the independent copula than the dependente copula.But this visual diagnotic does not allow to see the similarity of associated copula of exponential variogram with neither.We need to add another method for measuring the similarity between the resulting copulas and the reference copulas.That one is discussed in the following subsection.

Similarity between the Resulting Copulas and Reference Copulas
In this subsection we study the similarity between two copulas by their matrix.Many measures of similarity can be found in literature.Following Shepard in (Issaks, E. H., 1989), we defined the similarity between two copulas.In the following, let denote Ñ and M respectively the set of the n and the m first naturals.
is similarity measure.In particular that proves that s obeys to the equal of self-similarity.
Moreover, since a i j − b i j = b i j − a i j , then s (A, B) = s (B, A) , that is, the measure s is symmetric.
And then A B means that, there exists an pair (i, j) ∈ Ñ × M such as a i j b i j .
Then, exp In other word, for all A B we are s (A, B) < s (A, A).
Then s respect the minimality condition.The last criterion is that of triangle inequality.
Therefore s such is a measure of similarity.Next we use it to study the similarity between the matrix engender by two copulas.For simplicity, we will say that two matrix A et B in M n,m are simular when s (A, B) = 1.
Definition 7 Let C 1 and C 2 be two copulas defined from [0, 1] 2 to [0, 1].These copulas will say simular if and only if for all pair (n, m) ∈ N * × N * their respective engendered matrix ) for all 0 ≤ u i , v j ≤ 1, are simular.
Let remark that the couple Proposition 9 For all 0 ≤ u, v ≤ 1, the associated copula satisfies to the following properties.
Proof.To prove proposition 9 consists in proving that, for all 0 ≤ u, v ≤ 1, In fact, C γ G et C D are Fréchet-Hoeffding bound, they are considered as universal bounds for copulas.Then for any copula, particulary C γ E and C I , we are When at less one of u and v is zero, we are That result shows that the dependence structure in the both classes is more similar with that of described by the independent copula.

Simulation
The theorical results are illustrated by a similation.With statistic sofware R (Shepard, R. N. 1987), we plot the similarities

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proposition, we compute the generators of Archimedean copulas associated to gaussian variogram [see Appendix 1] and associated to gaussian variogram [see Appendix 2].The results are summarized in the following table.
valid similarity measure obeys to four axioms [see 11].

First
the other words a i j = b i j , with i ∈ N and i ∈ M.
1 are the positions of the grid points discretization of [0, 1] 2 .Denote C γ E and C γ G the copulas associated to exponential variogram and gaussian variogram respectively.Likely, C I and C D are respectively the independent copula and dependent copula.Proposition 8 Two copulas C 1 and C 2 are simular if and only if for all 0≤ u, v ≤ 1, C 1 (u, v) and C 2 (u, v) are simular.Proof.Deduce from the simularity of two copula definition.As far as concerned the simularity between C I , C D , C γ E and C γ G , we are the following proposition.
w + 1/a) t] dt = (aw + 1) −1 Appendice 3: The Figures In the Figure 2.4 and Figure 2.5, the levels lines are described colored bands.The colors goes from blue (mean level zero) two black (mean level one).

Figure 1 .
Figure 1.Images contour of: the Gaussian variogram class associated copula and Exponential variogram class associated copula