Covariance functions for bivariate Gaussian

random fields

Olga Moreva1,* Martin Schlather1,**

1Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany. e-mail:*omoreva@mail.uni-mannheim.de; **schlather@math.uni-mannheim.de

Abstract: We introduce two novel bivariate parametric covariance models, the powered ex-ponential (or stable) covariance model and the generalized Cauchy covariance model. Bothmodels allow for flexible smoothness, variance, scale, and cross-correlation parameters. Thesmoothness parameter is in (0, 1]. Additionally, the bivariate generalized Cauchy model allowsfor distinct long range parameters. We also show that the univariate spherical model can begeneralized to the bivariate case only in a trivial way. The results are based on general suffi-cient conditions for the positive definiteness of 2×2-matrix valued functions. These conditionsrequire only computing the derivatives of a bivariate covariance function of order 2 and 3 inR and in R3, respectively, and calculating an infimum of a function of one variable. In a dataexample on the content of copper and lead in the top soil in a flood plain along the riverMeuse we compare the bivariate powered exponential model to the traditional linear model ofcoregionalization and the bivariate Matern model.

Keywords and phrases: matrix-valued covariance function, cross-correlation, multivariateGaussian random fields, multivariate geostatistics, bivariate powered exponential model, bi-variate stable model, bivariate Cauchy model.

1. Introduction

Spatial data often have several components, for example temperature and pressure, or the contentsof different heavy metals in the top soil. Spatial dependence within and between the components isexploited in particular when the component of interest is not exhaustively sampled and the mea-surement of other components can be easily carried out, e.g. in soil sciences ([16], [2]). To this end,a multivariate spatial covariance model is needed, and an appropriate model gives more sensibleresults for spatial interpolation, see for example [5]. In environmental and climate sciences it is im-portant to model spatial meteorological data jointly in order to reflect spatial dependence withinand between components adequately (see Discussions in [8, 3, 9]); otherwise the obtained resultsmight be unsound.

We focus on a Euclidean space, Rn, n ≤ 3. Spatial data are assumed to stem from a multivariateGaussian random field Z(x) = (Z1(x), . . . , Zm(x)), x ∈ Rn, m ∈ N, which is uniquely characterizedby its mean and covariance function. For simplicity we assume in the theoretical part of the paper,that the random field has zero mean. A covariance function C of a multivariate field is a matrix-valued function, whose diagonal elements are the marginal covariances and the off-diagonal elementsare the cross-covariances. A covariance function C is called stationary if for any x, h ∈ Rn andi, j = 1, . . . ,m it holds

cov(Zi(x+ h), Zj(x)) = Cij(h).

C is stationary and isotropic if additionally C(h1) = C(h2) whenever ‖h1‖ = ‖h2‖, i.e. the marginalcovariances and the cross-covariances depend only on the distance between the variables locations.Hereinafter we write C(r) instead of C(h) with r = ‖h‖, whenever C(h) is stationary and isotropic.

We recall that a covariance function must be positive definite, i.e. it guarantees that the varianceof an arbitrary linear combination of observations of any involved components Zi(x), i = 1, . . . ,m,taken at arbitrary spatial locations is nonnegative. That is, for any p ∈ N, a1, . . . , an ∈ Rm and

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O. Moreva, M. Schlather/Covariance functions for bivariate random fields 2

x1, . . . , xp it must holdp∑i=1

aTi C(xi − xj)aj ≥ 0.

Checking positive definiteness is difficult for many univariate functions, multivariate case is evenmore complex.

A comprehensive overview of covariance functions for multivariate geostatistics is found in [10, 30].Among these models is the linear model of coregionalization ([17], [33]). Although it is widely byamong practitioners, it lacks flexibility; its limitations are discussed in [13]. Models with compactsupport are introduced in [7], [27], [6] and [31]. [21] studies the properties of multivariate randomfields in the frequency domain. [5] develop a conditional approach for constructing multivariatemodels. In this paper we restrict our attention to stationary and isotropic bivariate models, whosecomponents stem from the same family, i.e. to models of the form

C(r) =

[σ2

1ψ11(r) ρσ1σ2ψ12(r)ρσ1σ2ψ12(r) σ2

2ψ22(r)

], (1.1)

where σi > 0 is the variance of the field Zi, ψij(·) = ψ(·|θij , sij) is a continuous univariate stationaryand isotropic correlation function, which depends on a scale (or range) parameter sij > 0, i, j =1, 2 and another optional parameter θij = (θ1

ij , ..., θkij) with k ∈ N (e.g. smoothness, long range

behaviour). Necessarily, |ρ| ≤ 1. Note that isotropy implies ψ12(r) = ψ21(r). For instance, themultivariate Matern model ([13], [1]) is a member of this class with

ψ (r|ν, s) =21−ν

Γ(ν)(sr)νKν(sr),

where s > 0 is a scale parameter, ν > 0 is a smoothness parameter and Kν is a modified Besselfunction of the second kind.

The class given by (1.1) also can be seen as a generalization of the class of separable modelsintroduced by [24], where a multivariate covariance factorizes into a product of a covariance matrixR and a univariate correlation function ψ(·), i.e.

Cij(r) = Rijψ(r), r ≥ 0, i, j = 1, . . . ,m.

That is, a separable model assumes that all components share the same spatial correlation structureand differs only in their variances. In particular, the scale parameter is the same for both marginaland cross-covariances. The class (1.1) is more flexible allowing each field to have distinct smoothness,scale, and variance parameters and admitting flexible cross-correlation between the fields. Given theunivariate correlation function ψ, our goal is to find the parameter sets for which the function Cin (1.1) is a covariance function. Clearly, if the components are uncorrelated, i.e. ρ = 0, then C isalways a bivariate covariance function. Thus, we are interested in |ρ| > 0.

It is worth pointing out that not all univariate models can be generalized to the multivariate casein a non-trivial way. For example, the univariate spherical model, ψ(r|s) =

(1− 3

2sr + 12 (sr)3

)+,

s > 0, is widely used in geostatistics, but its bivariate generalization[σ2

1

(1− 3

2s11r + 12 (s11r)

3)

+ρσ1σ2

(1− 3

2s12r + 12 (s12r)

3)

+

ρσ1σ2

(1− 3

2s12r + 12 (s12r)

3)

+σ2

2

(1− 3

2s22r + 12 (s22r)

3)

+

], (1.2)

with σi > 0, sij > 0, |ρ| ≤ 1, i, j = 1, 2, is a valid covariance model in R3 only if s11 = s12 = s22

or ρ = 0. This follows directly from the multivariate version of Schoenberg’s theorem ([32] [34]) andthe fact that the spectral density of the spherical covariance is a pseudo periodic function with aninfinite number of zeros, see Appendix A for details.

[10] pose the question, how to characterize a parameter set of the valid multivariate powered expo-nential (or stable) model. In the next section we give a partial answer, providing sufficient conditions

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 3

for the positive definiteness of the bivariate model. In a similar way we can also formulate sufficientconditions for the positive definiteness of the bivariate generalized Cauchy model. The models areflexible, intuitive and easily interpretable: in both models three parameters characterize the smooth-ness of the covariances of process components and the cross-covariances. Further three parametersmodel the long-range behaviour in the bivariate Cauchy model. The smoothness parameters of themarginal covariances in both models are restricted to values in (0, 1].

The remainder of the paper is organized as follows. In Section 3, we discuss necessary conditionsfor positive definiteness of models from the class (1.1). In Section ??, we fit a bivariate poweredexponential model to the Meuse dataset ([26]) and compare the results with the bivariate Maternmodel and with the linear model of coregionalization.

2. Flexible bivariate models

We introduce novel bivariate covariance models of the form (1.1) and provide sufficient conditionsfor their validity. The proofs of the main results, Theorems 1 and 2 are similar and postponed toAppendix B.

2.1. Bivariate powered exponential model

The univariate powered exponential correlation function

ψ(r|α, s) = exp(−(sr)α),

s > 0, α ∈ (0, 2], contains the exponential model (α = 1) and the Gaussian model (α = 2). Itpermits the full range of allowable values for the fractal dimension ([15]). Unlike the Matern model,the univariate powered exponential correlation function does not allow for a smooth parametrizationof the differentiability of the field paths. Indeed, the paths are continuous and non-differentiable forα < 2 and infinitely often differentiable for α = 2. Nevertheless, the powered exponential covariancemay be a good alternative for non-differentiable fields, as it is easy to calculate. The univariatepowered exponential covariance is used in [29], [18], [19], and [20], for example.

The marginal covariances of the bivariate powered exponential model,

C11(r) = σ21 exp(−(s11r)

α11),

C22(r) = σ22 exp(−(s22r)

α22),(2.1)

are of powered exponential type with variance parameter σi, smoothness parameter αi ∈ (0, 1] andscale parameter sii > 0, i = 1, 2. The cross-covariance functions,

C12(r) = C21(r) = ρσ1σ2 exp(−(s12r)α12), (2.2)

are also a powered exponential function with colocated correlation ρ, smoothness parameter α12 ∈(0, 2] and scale parameter s12 > 0. Note that the univariate powered exponential model has α ∈ (0, 2],but for the marginal covariances we consider only the case αii ∈ (0, 1], i = 1, 2.

We define auxiliary functions q(n)α,s(r), n ∈ 1, 3 by

q(1)α,s(r) = α(sr)α − α+ 1,

q(3)α,s(r) = α2(sr)2α − 3α2(sr)α + 4α(sr)α + α2 − 4α+ 3.

Theorem 1. A matrix-valued function C given by equations (2.1) and (2.2) is positive definite inRn, n ∈ 1, 3 if

ρ2 ≤ α11α22sα1111 sα22

22

α212s

2α1212

infr>0

rα11+α22−2α12e2(s12r)α12−(s11r)

α11−(s22r)α22 q

(n)α11,s11(r)q

(n)α22,s22(r)

(q(n)α12,s12(r))2

, (2.3)

In particular, the infimum in (2.3) is positive if and only if one of the following conditions is satisfied

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 4

Fig 1: The maximum attainable |ρ| in in-equality (2.3) for the bivariate powered expo-nential covariance model in R. The parame-ters are σ1 = σ2 = 1, α11 = 0.2, α22 = 0.5,s11 = 2, s22 = 3.

Fig 2: The maximum attainable |ρ| in in-equality (2.6) for the bivariate Cauchy co-variance model in R. The parameters areσ1 = σ2 = 1, α11 = 0.5, α22 = 0.9, β11 = 2,β12 = 2.5, β22 = 2.1, s11 = 2, s22 = 2.5.

(i) α12 = α11 = α22 and sα1112 ≥

sα1111 +s

α1122

2 ,

(ii) α12 = α11 > α22 and s12 > 2−1/α11s11,(iii) α12 = α22 > α11 and s12 > 2−1/α22s22,(iv) α12 > maxα11, α22.

Moreover, if α12 < (α11 + α22)/2 the model is valid only for ρ = 0.

As inequality (2.3) provides only a sufficient but not a necessary condition for positive definiteness,zero infimum in inequality (2.3) does not imply that the model defined by (2.1) and (2.2) is not avalid covariance model.

The model is implemented in the R package RandomFields ([31]). Figure 1 provides an exampleof the maximum attainable |ρ| in inequality (2.3) that have been found numerically.

2.2. Bivariate generalized Cauchy model

The univariate generalized Cauchy model,

ψ(r|α, β, s) = (1 + (sr)α)−β/α,

has been introduced in [12, 14]. Here s > 0 is a scale parameter, α ∈ (0, 2] is a smoothness parameterand β > 0 controls the long range behaviour of the field.

The marginal covariances of the bivariate generalized Cauchy model,

C11(r) = σ21(1 + (s11r)

α11)−β11/α11 ,

C22(r) = σ22(1 + (s22r)

α22)−β22/α22 ,(2.4)

are of generalized Cauchy type with variance parameter σi, smoothness parameter αi ∈ (0, 1], longrange parameter βii > 0 and scale parameter sii > 0, i = 1, 2. Each cross-covariance,

C12(r) = C21(r) = ρσ1σ2(1 + (s12r)α12)−β12/α12 , (2.5)

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 5

is also of generalized Cauchy type with colocated correlation ρ, smoothness parameter α12 ∈ (0, 2],long range parameter β12 > 0 and scale parameter s12 > 0. Again, the univariate generalized Cauchymodel admits α ∈ (0, 2], but for the marginal covariances we consider only the case αii ∈ (0, 1], i =1, 2.

We define the auxiliary functions p(n)α,β,s(r), n ∈ 1, 3

p(1)α,β,s(r) =

(β + 1)(sr)α − α+ 1

(1 + (sr)α)β/α+2

p(3)α,β,s(r) =

(β + 1)(β + 3)(sr)2α + (sr)α(4β + 5− 3α− 3βα− α2) + (α− 1)(α− 3)

(1 + (sr)α)β/α+3

Theorem 2. A matrix-valued function C given by equations (2.4) and (2.5) is positive definite inRn, n ∈ 1, 3 if

ρ2 ≤ β11β22

β212

sα1111 sα22

22

s2α1212

infr>0

rα11+α22−2α12p

(n)α11,β11,s11

(r)p(n)α22,β22,s22

(r)

(p(n)α12,β12,s12

(r))2(2.6)

In particular,

(i) if α12 < (α11 + α22)/2 the model is valid only for ρ = 0,(ii) if β12 < (β11 + β22)/2 and βij < n, i, j = 1, 2, the model is valid only for ρ = 0,

(iii) if 2β12 < βii + n, βii < n and βjj > n for i 6= j, the model is valid only for ρ = 0,(iv) if β12 < (β11 + β22)/2, the infimum in inequality (2.6) is zero,(v) if α12 ≥ (α11 + α22)/2 and β12 ≥ (β11 + β22)/2 the infimum in inequality (2.6) is positive.

Analogously to the powered exponential model, inequality (2.6) is only a sufficient but not nec-essary condition for positive definiteness. Figure 2 provides an example of the maximum attainable|ρ| in inequality (2.6) that have been found numerically.

3. Necessary condition for positive definiteness

Condition α12 < (α11 + α22)/2 necessarily leads to the independence of the components in boththe bivariate powered exponential model and the bivariate generalized Cauchy model. The samerestriction is imposed on smoothness parameters in the full bivariate Matern model ([13]) and intheir forth section, the authors only very briefly discuss the origin of this condition. Since this kind ofrestriction is common for all models of the type (1.1) we look at it in detail and explain additionallywhy β12 < (β11 +β22)/2 in the bivariate generalized Cauchy model leads again to correlation ρ = 0.

These constraints come from the multivariate version of Schoenberg’s theorem, see [32], Chapter4 in [34], and the Tauberian theorems ([4], [22]), as we will see in the following. By Schoenberg’stheorem a stationary and isotropic matrix-valued function C, which falls rapidly enough to zero atinfinity, is a Hankel type transform of a spectral density matrix [f(u)]mi,j=1

C(r) = (2π)n/2∫ ∞

0

(ru)−(n−2)/2J(n−2)/2(ru)un−1f(u)du, r ≥ 0, (3.1)

and [f(u)]mi,j=1 is a positive semi-definite matrix for all u ≥ 0.Hereinafter by fij we denote the spectral density of a correlation function ψij , i, j = 1, 2. Then,

a matrix-valued function C in (1.1) is positive definite if and only if

f11(r)f22(r)− ρ2f12(r) ≥ 0 (3.2)

for all r > 0.Tauberian theorems link the properties of a univariate correlation function with those of its

spectral measure. We first need the notion of slow variation.

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 6

A function L : (0,∞) 7→ [0,∞) is said to be slowly varying at infinity (at zero), if for every λ > 0,

L(λr)

L(r)→ 1 as r →∞ (r → 0).

For functions f and g on [0,∞) we write f·∼ g as t → t0, t0 ∈ [0,∞], if f is asymptotically

proportional to g, i.e. f(t)/g(t)→ A, A > 0, as t→ t0.

Theorem 3 (Abelian and Tauberian theorems). Let F be a spectral measure on [0,∞) of a stationaryand isotropic univariate correlation function ψ in Rn and let L be a function varying slowly atinfinity.

• If 0 < α < 2, then1− ψ(r)

·∼ rαL(1/r) as r → 0+ (3.3)

if and only if

1− F (u)·∼ L(u)

uα(3.4)

If α = 2, relation (3.3) is equivalent to∫ r

0

u[1− F (u)]du·∼ L(r) as r →∞

or to ∫ r

0

u2F (du)·∼ L(r) as r →∞.

If α = 0, the relation (3.3) implies the asymptotic equivalence (3.4). Conversely, (3.4) implies(3.3) with α = 0 if [1− F (u)] is convex for u sufficiently large, but not in general.

• Let 0 < β < n. Ifψ(r)

·∼ L(r)r−β as r →∞,

then

F (u)·∼ L

(1

u

)uβ as u→ 0 + .

Remark 1. Let θij = (αij , βij) in equation (1.1) parametrize the behaviour of ψ(r|θij , sij) at theorigin and at infinity respectively, i.e.

1− ψ(r|θij)·∼ rαij as r → 0+,

ψ(r|θij)·∼ r−βij as r →∞,

αij ∈ (0, 2), βij > 0, βij < n, i, j = 1, 2. Here we assume for simplicity that sij = 1, i, j = 1, 2,because distinct scales would not change the asymptotics. Additionally, we took L(1/r) to be equalto some constant, which is indeed the case for the examples below. Then for the spectral densities itholds

fij(r)·∼ r−αij−n as r →∞,

fij(r)·∼ rβij−n as r → 0 + .

From inequality (3.2) it follows, that unless ρ = 0 a matrix-valued function C in (1.1) with α12 <(α11 + α22)/2 or β12 < (β11 + β22)/2 cannot be positive definite.

Example 1 (Powered exponential correlation). Let ψ(r) = e−rα

, r > 0, α ∈ (0, 2), then observe

that 1−ψ(r)·∼ rα as r → 0+ and we have L(r) = 1. The corresponding measure F varies at infinity

as follows1− F (u)

·∼ u−α as u→∞.

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 7

Since the function ψ decreases rapidly enough at infinity, the density f of F exists and

f(u)·∼ u−α−n as u→∞.

The latter matches with the series representation of f in [25]. Thus, the bivariate powered exponentialmodel requires necessarily α12 ≥ (α11 + α22)/2 unless ρ = 0.

Example 2 (Generalized Cauchy correlation). Let ψ(r) = (1+ rα)−β/α, r, β > 0, β < n, α ∈ (0, 2).

Then 1 − ψ(r)·∼ βαr

α as r → 0+, so in this case L(r) = βα . The spectral density f of ψ decays at

infinity as followsf(u)

·∼ u−α−n as u→∞.This matches the series representation for the spectral density of the Cauchy covariance in [23].Analogously, the spectral density f behaves at the origin as

f(u)·∼ uβ−n as u→ 0.

Thus, the bivariate generalized Cauchy model requires necessarily α12 ≥ (α11 + α22)/2 and β12 ≥(β11 + β22)/2 unless ρ = 0.

Appendix A: Bivariate spherical model

We first prove the following lemma.

Lemma 1. Let uk, k ∈ N be the positive roots of the equation

u = tan(u). (A.1)

Then, for any s < 1, there exists a k0 ∈ N such that uk0 is not the root of the equation

u

s= tan

(us

). (A.2)

Proof. We prove the lemma by contradiction. Suppose that uk is a root of the equations (A.1) and(A.2) for all k ∈ N. First note that uk ↑ π

2 + πk as k → ∞. Thus, every uk lies inside the interval(π2 + π(k − 1), π2 + πk

)and there exists a decreasing sequence of εk ∈ (0, 1) such that

uk =π

2+ πk − πεk.

Then we haveuks

=π

2s+πk

s− π

sεk.

Since uk is the root of (A.2), there exist lk > k, such that uks = ulk .

For any s < 1, there exist n ∈ N0 and 0 ≤ c < 1 such that πs = πn− πc.

Consider the following cases.

(i) c = 0, n = 2m, m ∈ N. Then

uks

= πm+ 2πkm− 2πmεk

=π

2+ πm(2k + 1)− π

(2mεk +

1

2

)

We choose k large enough, so that(2mεk + 1

2

)< 1. Since

π

2+ π(2mk +m− 1) < ulk <

π

2+ πm(2k + 1),

then lk = 2km+m and ε2km+m =(2mεk + 1

2

). But then it follows that ε2km+m > εk, which

cannot be true, since (εk)k∈N is a decreasing sequence.

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 8

(ii) c = 0, n = 2m+ 1, m ∈ N0

uks

=1

2(2m+ 1)π + (2m+ 1)kπ − (2m+ 1)πεk

=π

2+ π(2km+m+ k)− π(2m+ 1)εk.

Analogously to the previous case, for large enough k we have lk = 2km+m+k and ε2km+m+k =(2m+ 1)εk. Thus ε2km+m+k > εk and this is again a contradiction.

(iii) c > 0. Consider now the root uk+1 = π2 + π(k + 1) − πεk+1. Since uk+1 is the root of (A.2),

there exist lk+1 > k + 1, such that us = ulk+1

. Then we have

uk+1

s=uk + π − π(εk+1 − εk)

s

=uks

+π

s− π

s(εk+1 − εk)

= ulk + πn− πc− π(n− c)(εk+1 − εk)

=π

2+ πlk − πεlk + πn− πc− π(n− c)(εk+1 − εk)

=π

2+ (lk + n)π − π(εlk + (n− c)(εk+1 − εk) + c).

Choose k large enough, so that 0 < εlk + (n− c)(εk+1 − εk) + c < 1. Then, analogously to theprevious cases lk+1 = lk + n and εlk+n = εlk + (n − c)(εk+1 − εk) + c. Note, that εlk+n → cwhen k →∞, which is a contradiction, since c > 0.

Theorem 4. The matrix-valued function (1.2) is a valid covariance model if and only if ρ = 0 ors11 = s12 = s22.

Proof. The spectral density of the univariate spherical correlation functions is

f(u) =3s

π2u6(u cos(u/2s)− 2s sin(u/2s))2

Clearly, f is pseudo periodic and takes infinitely many zeros on u > 0. We denote by uk, k ∈ N thezeros of the function f(u) = u − tan(u) on u > 0. Then the zeros of the spectral density fij are2sijuk, k ∈ N, i, j = 1, 2. Without loss of generality we assume s11 ≤ s22. We consider three cases:

• s12 > s11, then f11(2s11u1) = 0 and f12(2s11u1) 6= 0, since 2s11u1 is less than the first zero off12, and we have

f11(2s11u1)f22(2s11u1)− ρ2f212(2s11u1) < 0.

Thus, matrix [fij(u)]2i,j=1 is not positive definite at u = 2s11u1.

• s12 < s11, then by Lemma 1 there exists k0 such that s11s12uk0 6= tan

(s11s12uk0

), thus

f11(2s11uk0)f22(2s11uk0)− ρ2f212(2s11uk0) < 0.

• s12 = s11 < s22, this case is treated analogously.

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 9

Appendix B: Sufficient conditions for positive definiteness

[28] provide the following construction principle for multivariate covariance models.

Theorem 5. A. Let (Ω,F , µ) be a measure space and E be a linear space. Assume that thefamily of matrix-valued functions A(x, u) = [Aij(x, u)] : E×Ω 7→ Rm×m satisfies the followingconditions:

(a) for every i, j = 1, . . . ,m and x ∈ E, the functions Aij(x, ·) belong to L1(Ω,F , µ);

(b) A(·, u) is a positive definite matrix-valued function for µ-almost every u ∈ Ω.

Let

C(x) :=

∫Ω

A(x, u)dµ(u) =

[∫Ω

Aij(x, u)dµ(u)

]mi,j=1

, x ∈ E.

Then C is a positive definite matrix-valued function.B. Conditions (a) and (b) in part A. are satisfied when A(x, u) = k(x, u)g(x, u), where the maps

k(x, u) : E × Ω 7→ R and g(x, u) = [gij(x, u)]mi,j=1 : E × Ω 7→ Rm×m satisfy the following

conditions:

(a) for every i, j = 1, . . . ,m and x ∈ E, the functions k(x, ·)gij(x, ·) belong to L1(Ω,F , µ);

(b) k(·, u) is positive definite for µ-almost every u ∈ Ω;

(c) g(·, u) is a positive definite matrix-valued function or g(·, u) = g(u) is a positive definitematrix for µ-almost every u ∈ Ω.

Starting from known functions k and gij , [27] and [6], see also [31], construct new compactlysupported multivariate covariance functions. Our approach, inspired by [11], is different; we considerthe model (1.1) as a candidate for a multivariate covariance function and then find the correspond-ing gij , which depend on parameters sij , θij , and the parameter set which guarantees its positivedefiniteness.

The following theorem provides sufficient conditions for positive definiteness of the bivariatemodel C in (1.1).

Theorem 6. A matrix-valued function C defined by equation (1.1) is positive definite

a) in R if ψij , i, j = 1, 2, are twice continuously differentiable in (0,∞) and the following condi-tions hold

(i) ψ′′ii(r) ≥ 0 for all r > 0, i = 1, 2,

(ii) ψij(r)→ 0, rψ′ij(r)→ 0 as r →∞, i, j = 1, 2,

(iii) rψ′′ij(r) is integrable in R+, i, j = 1, 2,

(iv) ρ2 ≤ infr>0ψ′′

11(r)ψ′′22(r)

(ψ′′12(r))2

b) in R3 if ψij, i, j = 1, 2 are three times continuously differentiable in (0,∞) and the followingconditions hold

(i) ψ′′ii(r)− rψ′′′ii (r) ≥ 0 for all r ≥ 0,

(ii) ψij(r)→ 0, rψ′ij(r)→ 0, r2ψ′′ij(r)→ 0 as r →∞, i, j = 1, 2,

(iii) rψ′′ij(r)− r2ψ′′′ij (r) is integrable in R+, i, j = 1, 2,

(iv) ρ2 ≤ infr>0(ψ′′

11(r)−rψ′′′11(r))(ψ′′

22(r)−rψ′′′22(r))

(ψ′′12(r)−rψ′′′

12(r))2 .

Proof. In Theorem 5 B. we take E as a Euclidean space Rn and µ as the Lebesgue measure. Wefirst prove the assertion in R. We take k(r, u) =

(1− r

u

)+

, gii(u) = σ2i uψ

′′ii(u), i = 1, 2 and g12(u) =

g21(u) = ρσ1σ2uψ′′12(u). Condition (iii) guarantees, that the first condition in Theorem 5 B. is

satisfied. Clearly, k(·, u) is a positive definite function in R for u > 0. The third condition in Theorem

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 10

5 B. is satisfied due to conditions (i) and (iv). Then the following matrix-valued function is positivedefinite [

σ21

∫∞0

(1− r

u

)+uψ′′11(u)du ρσ1σ2

∫∞0

(1− r

u

)+uψ′′12(u)du

ρσ1σ2

∫∞0

(1− r

u

)+uψ′′12(u)du σ2

2

∫∞0

(1− r

u

)+uψ′′22(u)du

]. (B.1)

To simplify (B.1) we use partial integration and the conditions in (ii).∫ ∞0

(1− r

u

)+uψ′′ij(u)du

=

∫ ∞r

(u− r)ψ′′ij(u)du

=

∫ ∞r

(u− r)dψ′ij(u)

= (u− r)ψ′ij(u)∣∣∞r−∫ ∞r

ψ′ij(u)du

= ψij(r).

Thus, (B.1) and (1.1) are the same matrices. The proof for R3 is analogous with k(r, u) =(1− r

u

)+−

r2u

(1− r2

u2

)+

and gii(r) = 13σ

2i (rψ′′ij(r)− r2ψ′′′ij (r)), i = 1, 2, g12(r) = 1

3ρσ1σ2(rψ′′12(r)− r2ψ′′′12(r)).

Remark 2. Condition (iii) in a) and b) of Theorem 6 can be replaced

(a) in R if ψ′′ij(r) ≥ 0 for r > 0 by limr→0 rψ′ij(r) <∞, i, j = 1, 2,

(b) in R3 if ψ′′ij(r) ≥ rψ′′′ij (r) for r > 0 by limr→0 rψ′ij(r) <∞ and limr→0 r

2ψ′′ij(r) <∞, i, j = 1, 2.

Indeed, if ψ′′ij(r) ≥ 0 for r > 0, then we have∫ ∞0

|uψ′′ij(u)|du =

∫ ∞0

uψ′′ij(u)du =

∫ ∞0

udψ′ij(u) = uψ′ij(u)∣∣∞0−∫ ∞

0

ψ′ij(u)du.

and limr→∞ rψ′ij(r) = 0, limr→∞ ψij(r) = 0 by (ii) and ψij(0) = 1. Thus,∫∞

0|uψ′′ij(u)|du <∞.

Analogously, in R3, if ψ′′ij(r) ≥ rψ′′′ij (r), then we have.

∫ ∞0

|uψ′′ij(u)− u2ψ′′′ij (u)|du =

∫ ∞0

(uψ′′ij(u)− u2ψ′′′ij (u))du

=

∫ ∞0

uψ′′ij(u)du−∫ ∞

0

u2dψ′′ij(u)

=

∫ ∞0

uψ′′ij(u)du− u2ψ′′ij(u)∣∣∞0

+ 2

∫ ∞0

uψ′′ij(u)du

= 3

∫ ∞0

udψ′ij(u)− u2ψ′′ij(u)∣∣∞0

= 3 uψ′ij(u)∣∣∞0− 3

∫ ∞0

ψ′ij(u)du− u2ψ′′ij(u)∣∣∞0

and limr→∞ r2ψ′′ij(r) = 0, limr→∞ rψ′ij(r) = 0, limr→∞ ψij(r) = 0 by (ii) and ψij(0) = 1. Thus,∫∞0|uψ′′ij(u)− u2ψ′′′ij (u)|du <∞.

Both ψ′′ij(r) ≥ 0 and ψ′′ij(r) ≥ rψ′′′ij (r) hold true for completely monotone ψij(r).

The infimum in inequality (iv) in R and R3 is taken over all r > 0 with ψ′′12(r) 6= 0 and ψ′′12(r)−rψ′′′12(r) 6= 0 respectively. Note that the condition (i) must hold only for diagonal covariances, but not

O. Moreva, M. Schlather/Covariance functions for bivariate random fields 11

for the cross-covariance, because g12(u) ≥ 0 is not required. This allows α12 to take values in (0, 2]in the bivariate powered exponential model and the bivariate generalized Cauchy model. Moreover,α12 can be larger than two.

Condition (iii) in R and R3 is not restrictive and fulfilled by many model classes, including theMatern model. Condition (iv) is crucial for the positive definiteness of C in (1.1).

The functions k(r, u) are equal to Euclid’s hat function, k(r, u) = hn(r/u), n = 1, 3 ([11]). Thus,Theorem 6 can be generalized to higher dimensions with corresponding functions hn, but it requiresthe calculation of higher order derivatives.

Proof of Theorem 1 . Functions ψij(r|αij , sij), i, j = 1, 2 of the bivariate powered exponential modelsatisfy the requirements of Theorem 6. Inequality (2.3) follows directly from the inequalities (iv) inTheorem 6. All factors of the right-hand side of inequality (2.3) are positive for r > 0. That meansthat the infimum can be zero only at r = 0 or r = ∞. Clearly, for the parameters values given in(i) − (iv), the infimum is positive and it is zero otherwise. The case α12 <

α11+α22

2 is discussed inSection 3.

Proof of Theorem 2 . Analogously to the bivariate powered exponential model, correlation func-tions ψij(r|αij , βij , sij),i, j = 1, 2 of the bivariate generalized Cauchy model satisfy the requirementsof Theorem 6. Inequality (2.6) follows directly from the inequalities (iv) in Theorem 6. The cases (i)and (ii) are discussed in Section 3. The assertion in (iii) follows from the inequality (3.2) and the

fact that f12(r)·∼ rβ12−n, fii(r)

·∼ rβii−n as r → 0 and fjj(r)·∼ C for some C > 0 as r → 0, i 6= j.

The assertions in (iv) and (v) follows directly from the inequality (2.6). Analogously to the bivariatepowered exponential model, all factors of the right-hand side of inequality (2.6) are positive forr > 0. That means that the infimum can be zero only at r = 0 or r =∞. Clearly, for the parametervalues given in (iv) the infimum is zero at infinity, and positive for parameters in (v).

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