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New Validity Conditions for the Multivariate Matérn Coregionalization Model, with an Application to Exploration Geochemistry

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Abstract

This paper addresses the problem of finding parametric constraints that ensure the validity of the multivariate Matérn covariance for modeling the spatial correlation structure of coregionalized variables defined in an Euclidean space. To date, much attention has been given to the bivariate setting, while the multivariate setting has been explored to only a limited extent. The existing conditions often imply severe restrictions on the upper bounds for the collocated correlation coefficients, which makes the multivariate Matérn model appealing for the case of weak spatial cross-dependence only. We provide a collection of sufficient validity conditions for the multivariate Matérn covariance that allows for more flexible parameterizations than those currently available, and prove that one can attain considerably higher upper bounds for the collocated correlation coefficients in comparison with our competitors. We conclude with an illustration on a trivariate geochemical data set and show that our enlarged parametric space yields better fitting performances.

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Acknowledgements

This work was supported by the National Agency for Research and Development of Chile [grants ANID/FONDECYT/REGULAR/No. 1210050 and ANID PIA AFB180004]. The authors are grateful to two anonymous reviewers for insightful comments.

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Correspondence to Xavier Emery.

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The first author acknowledges the funding of the National Agency for Research and Development of Chile, through grants ANID/FONDECYT/REGULAR/1210050 and ANID PIA AFB180004.

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Appendices

Appendix A – Technical Lemmas with their Proofs

From a notation viewpoint, we recall that, in what follows, all the matrix operations (product, division, inverse, power, exponential, integration, etc.) are taken element-wise.

Lemma 1

Let \(\varvec{A}=[a_{ij}]_{i,j=1}^p\) be a symmetric real-valued matrix. The following assertions are equivalent (Schoenberg 1938; Berg et al. 1984; Reams 1999):

  • \(\varvec{A}\) is conditionally negative semidefinite;

  • \(\exp (- t \, \varvec{A})\) is positive semidefinite for all \(t \ge 0\);

  • \([a_{ip}+a_{pj}-a_{ij}-a_{pp}]_{i,j=1}^p\) is positive semidefinite.

Lemma 2

If \(\varvec{A}=[a_{ij}]_{i,j=1}^p\) is a conditionally negative semidefinite matrix with positive entries, then \([a^{-1}_{ij}]_{i,j=1}^p\) is positive semidefinite, and \([a^{\mu }_{ij}]_{i,j=1}^p\) is conditionally negative semidefinite for any \(\mu \in (0,1]\).

The proof of this lemma can be found in Berg et al. (1984, corollary 2.10 and exercise 2.21).

Lemma 3

The multivariate Matérn model (7) is valid in \(\mathbb {R}^d\), \(d \ge 1\), if

  1. (1)

    \(\varvec{\nu }\) is conditionally negative semidefinite;

  2. (2)

    \(\mathbf{K}({\varvec{h}};\varvec{\alpha },\varvec{\nu }+\mu ,\varGamma (\varvec{\nu }+\mu ) \, \varGamma (\varvec{\nu })^{-1} \, \varvec{\sigma })\) is a valid model in \(\mathbb {R}^d\) for some positive constant \(\mu \), where the exponent \(-1\) stands for the element-wise inverse and \(\varGamma \) for the gamma function.

Proof

The proof relies on the decomposition of the Matérn covariance with smoothness parameter \(\varvec{\nu }\) as a scale mixture of Matérn covariances with smoothness parameter \(\varvec{\nu }+\mu \) (Gradshteyn and Ryzhik 2007, formula 6.592.12)

$$\begin{aligned} \mathbf{K}(\varvec{h};\varvec{\alpha },\varvec{\nu },\varvec{\sigma }) = \varGamma (\varvec{\nu }+\mu ) \, \varGamma (\varvec{\nu })^{-1} \, \int _{1}^{+\infty } t^{-\varvec{\nu }-\mu } \, (t-1)^{\mu -1} \mathbf{K}(\varvec{h}\sqrt{t};\varvec{\alpha },\varvec{\nu }+\mu ,\varvec{\sigma }) \text {d}t, \end{aligned}$$
(14)

where all operations are taken element-wise. Under Conditions (1) and (2) of Lemma 3, \(\mathbf{K}(\varvec{h};\varvec{\alpha },\varvec{\nu },\varvec{\sigma })\) appears as a sum of valid multivariate Matérn covariance functions of the form \(\mathbf{K}(\varvec{h}\sqrt{t};\varvec{\alpha },\varvec{\nu }+\mu ,\varGamma (\varvec{\nu }+\mu ) \, \varGamma (\varvec{\nu })^{-1} \, \varvec{\sigma })\) weighted by positive semidefinite matrices of the form \(t^{-\varvec{\nu }-\mu } \, (t-1)^{\mu -1}\) (Lemma 1), hence, it is a valid covariance model. \(\square \)

Corollary 1

If \(\mathbf{K}({\varvec{h}};\varvec{\alpha },\nu \varvec{1},\varvec{\sigma })\) is a valid model in \(\mathbb {R}^d\) for some \(\nu >0\), then so is \(\mathbf{K}({\varvec{h}};\varvec{\alpha },\mu \varvec{1},\varvec{\sigma })\) for \(\mu \in (0,\nu )\).

Proof

This result stems from the fact that the constant matrix \(\nu \varvec{1}\) is conditionally negative semidefinite; hence, it satisfies Condition (1) of Lemma 3. \(\square \)

Lemma 4

The multivariate Gaussian model (9) is valid in \(\mathbb {R}^d\), \(d \ge 1\), if

  1. (1)

    \(\varvec{\beta }^{-1}\) is conditionally negative semidefinite;

  2. (2)

    \(\varvec{\sigma } \, \varvec{\beta }^{-d/2}\) is positive semidefinite.

Proof

The matrix-valued spectral density of the p-variate Gaussian covariance (9) in \(\mathbb {R}^d\) is (Eq. (6))

$$\begin{aligned} \widetilde{\mathbf{G}}(\varvec{\omega };\varvec{\beta },\varvec{\sigma }) = \frac{1}{(4\pi )^{d/2}} \left[ \varvec{\sigma } \, \varvec{\beta }^{-d/2}\right] \, \left[ \exp \left( -\frac{\Vert \varvec{\omega } \Vert ^2}{4\varvec{\beta }}\right) \right] , \quad \varvec{\omega } \in \mathbb {R}^d. \end{aligned}$$
(15)

Condition (1) and Lemma 1 ensure that the last matrix into brackets in the right-hand side member of (15) is positive semidefinite, while Condition (2) ensures the positive semidefiniteness of the first matrix into brackets. Accordingly, based on the Schur product theorem, the spectral density matrix \(\widetilde{\mathbf{G}}(\varvec{\omega };\varvec{\beta },\varvec{\sigma })\) is positive semidefinite for any frequency vector \(\varvec{\omega }\) in \(\mathbb {R}^d\), which in turn ensures that the associated covariance model is valid. \(\square \)

Remark 1

A necessary condition for the element-wise inverse matrix \(\varvec{\beta }^{-1}\) to be conditionally negative semidefinite (Condition (1) in Lemma 4) is that \(\varvec{\beta }\) is positive semidefinite (Lemma 2). Likewise, a necessary condition for Condition (2) to hold is that \(\varvec{\sigma }\) is positive semidefinite, insofar as it corresponds to the covariance matrix between the components of the p-variate random field at the same spatial location.

Appendix B – Transitive Upgrading (Montée) of a Covariance Function

The transitive upgrading, also known as “montée” or the radon transform (Matheron 1965), of order 1 of an absolutely integrable real-valued function \(\varphi _d\) defined in \(\mathbb {R}^d\) is the function \(\varphi _{d-1}\) of \(\mathbb {R}^{d-1}\) obtained by integration of \(\varphi _d\) along the last coordinate axis

$$\begin{aligned} \varphi _{d-1}(s_1,\cdots ,s_{d-1}) = \int _{-\infty }^{+\infty } \varphi _{d}(s_1,\cdots ,s_{d-1},s_d) \text {d} s_d, \quad (s_1,\cdots ,s_{d-1}) \in \mathbb {R}^{d-1}. \end{aligned}$$
(16)

The Fourier transforms of \(\varphi _d\) and its montée are related by

$$\begin{aligned} \widetilde{\varphi }_{d-1}(\omega _1,\cdots ,\omega _{d-1})=\widetilde{\varphi }_d(\omega _1,\cdots ,\omega _{d-1},0), \quad (\omega _1,\cdots ,\omega _{d-1}) \in \mathbb {R}^{d-1}. \end{aligned}$$
(17)

The same relation applies to the Fourier transforms of continuous and absolutely integrable covariance functions: the montée operator consists of canceling out the last coordinate of the Fourier transform (spectral density) of the d-dimensional covariance to find out that of the \((d-1)\)-dimensional covariance. In the isotropic setting where the covariance and its spectral density are radial functions, the montée leaves unchanged the radial part (a function of the distance to the origin); i.e., the montée of an isotropic covariance defined in \(\mathbb {R}^d\) is an isotropic covariance in \(\mathbb {R}^{d-1}\) whose spectral density has the same radial part as that of the original covariance (Matheron 1965). For example, the montée of the isotropic Matérn covariance \(k(\cdot ; \alpha ,\nu )\) as in Equation (1) in \(\mathbb {R}^d\) is, up to a multiplicative factor, the Matérn covariance \(k(\cdot ; \alpha ,\nu +1/2)\) in \(\mathbb {R}^{d-1}\).

By repeated application of the montée of order 1, one obtains a montée of any integer order. In particular, the montée of integer order \(\vartheta \) of the isotropic Matérn covariance \(k(\cdot ; \alpha ,\nu )\) in \(\mathbb {R}^d\) is proportional to the Matérn covariance \(k(\cdot ; \alpha ,\nu +\vartheta /2)\) in \(\mathbb {R}^{d-\vartheta }\), the spectral densities of both Matérn covariances having, up to a multiplicative factor, the same radial parts (Eq. (3)). By using the formalism of Hankel transforms instead of that of Fourier transforms, it is possible to generalize the montée to fractional orders (Matheron 1965).

Appendix C – Proofs

Proof of Theorem 1

Owing to Corollary 1, it suffices to establish the result for \(\nu \) being an integer or a half-integer. We propose a proof based on the representation of the exponential and Matérn covariance as scale mixtures of compactly supported Askey and Wendland covariances, respectively. The Wendland covariance with range \(a>0\) and smoothness parameter \(m \in \mathbb {N}\) and its spectral density in \(\mathbb {R}^d\) are (Chernih et al. 2014)

$$\begin{aligned}&w({\varvec{h}};a,m,d) = \frac{2^{-\nu -m}\varGamma (\mu +1)}{ \varGamma (\mu +m+1)}\left( 1-\frac{\Vert \mathbf {h}\Vert ^2}{a^2}\right) _{+}^{\mu +m} \nonumber \\&\quad \times {}_2F_1\left( \frac{\mu }{2},\frac{\mu +1}{2};\mu +m+1;1-\frac{\Vert \mathbf {h}\Vert ^2}{a^2}\right) , \quad \varvec{h} \in \mathbb {R}^d \end{aligned}$$
(18)
$$\begin{aligned}&\widetilde{w} (\varvec{\omega }; a,m,d) = \frac{2^m \, a^{d} \varGamma (\mu +1)\varGamma \left( \frac{1+d+2m}{2}\right) }{\pi ^{(d+1)/2}\varGamma (\mu +d+2m+1)} \nonumber \\&\quad \times {}_1F_2\left( \frac{1+d+2m}{2}; \frac{1+d+2m+\mu }{2}, 1+\frac{d+2m+\mu }{2}; -\left( \frac{a\Vert \varvec{\omega }\Vert }{2}\right) ^2\right) , \nonumber \\&\quad \varvec{\omega } \in \mathbb {R}^{d}, \end{aligned}$$
(19)

with \(\mu = \left\lfloor \frac{d}{2}+m+1 \right\rfloor \), \((\cdot )_+\) the positive part function, \({}_2F_1\) the Gauss hypergeometric function, and \({}_1F_2\) a generalized hypergeometric function. The Askey covariance corresponds to the particular case when \(m=0\).

Suppose now that \(\nu \) is an integer or a half-integer and let us decompose the univariate exponential covariance in \(\mathbb {R}^{d+2\nu -1}\), \(d \ge 1\), as a scale mixture of Askey covariances in \(\mathbb {R}^{d+2\nu -1}\) with exponent \(\mu =\left\lfloor \frac{d+1}{2}+\nu \right\rfloor \):

$$\begin{aligned} k({\varvec{h}};\alpha ,1/2) = \int _{0}^{+\infty } w({\varvec{h}};t,0,d+2\nu -1) \phi (t; \alpha ,\nu ,d) \text {d}t, \quad \varvec{h} \in \mathbb {R}^{d+2\nu -1}, \end{aligned}$$
(20)

that is,

$$\begin{aligned} \exp (-\alpha \Vert \varvec{h}\Vert ) = \int _{\Vert \varvec{h}\Vert }^{+\infty } \left( 1 - \frac{\Vert \varvec{h}\Vert }{t}\right) ^\mu \phi (t;\alpha ,\nu ,d) \text {d}t, \quad \varvec{h} \in \mathbb {R}^{d+2\nu -1}. \end{aligned}$$
(21)

To determine \(\phi \), we differentiate \((\mu +1)\) times under the integral sign, which is permissible owing to the dominated convergence theorem, to obtain

$$\begin{aligned} \phi (t;\alpha ,\nu ,d) = \frac{\alpha ^{\mu +1} \, t^\mu }{\varGamma (\mu +1)} \exp (-\alpha \, t), \quad t>0. \end{aligned}$$
(22)

One recognizes the gamma probability density with shape parameter \(\mu +1\) and rate parameter \(\alpha \), a result that reminds of the representation in \(\mathbb {R}^3\) of the exponential covariance as a gamma mixture of spherical covariances (Emery and Lantuéjoul 2006). In terms of spectral density, (22) translates into

$$\begin{aligned} \widetilde{k}(\varvec{\omega }; \alpha ,1/2) = \int _{0}^{+\infty } \widetilde{w}(\varvec{\omega }; t,0,d+2\nu -1) \, \phi (t;\alpha ,\nu ,d) \text {d}t, \quad \varvec{\omega } \in \mathbb {R}^{d+2\nu -1}. \end{aligned}$$
(23)

To generalize these results to the Matérn covariance of integer or half-integer parameter \(\nu \), let us consider a transitive upgrading (“montée”) of order \(2\nu -1\), which provides a covariance model with the same radial spectral density in a space whose dimension is reduced by \(2\nu -1\) (Appendix B), i.e., in \(\mathbb {R}^d\). In particular, up to a multiplicative constant, the upgrading of \(k(\cdot ;\alpha ,1/2)\) and \(w(\cdot ;t,0,d+2\nu -1)\), both covariances being defined in \(\mathbb {R}^{d+2\nu -1}\), yields \(k(\cdot ;\alpha ,\nu )\) and \(w(\cdot ;t,\nu -1/2,d)\), both defined in \(\mathbb {R}^{d}\), respectively. Based on (23), the upgraded spectral densities in \(\mathbb {R}^{d}\) are found to be related by

$$\begin{aligned}&\frac{\varGamma (\nu )}{\alpha ^{2\nu -1} \, \varGamma (1/2)} \widetilde{k}(\varvec{\omega }; \alpha ,\nu ) \nonumber \\&\quad = \int _{0}^{+\infty } \frac{t^{2\nu -1}}{2^{\nu -1/2}} \, \widetilde{w}(\varvec{\omega }; t,\nu -1/2,d) \phi (t;\alpha ,\nu ,d) \text {d}t, \quad \varvec{\omega } \in \mathbb {R}^{d}, \end{aligned}$$
(24)

and the covariance functions therefore satisfy the following identity

$$\begin{aligned}&\frac{\varGamma (\nu )}{\alpha ^{2\nu -1} \, \varGamma (1/2)} {k}(\varvec{h}; \alpha ,\nu ) \nonumber \\&\quad = \int _{0}^{+\infty } \frac{t^{2\nu -1}}{2^{\nu -1/2}} \, {w}(\varvec{h}; t,\nu -1/2,d) \phi (t;\alpha ,\nu ,d) \text {d}t, \quad \varvec{h} \in \mathbb {R}^{d}. \end{aligned}$$
(25)

Based on this scale mixture representation, the multivariate Matérn covariance in \(\mathbb {R}^d\) with parameters \(\varvec{\alpha }\), \(\nu \varvec{1}\), and \(\varvec{\sigma }\) can be written as

$$\begin{aligned}&\mathbf{K}({\varvec{h}};\varvec{\alpha },\nu \varvec{1},\varvec{\sigma }) \nonumber \\&\quad = \varvec{\sigma } \, \int _{0}^{+\infty } \frac{\varGamma (1/2) (\varvec{\alpha }\,t)^{2\nu +\mu } }{{2}^{\nu -1/2} \, \varGamma (\nu )\,\varGamma (\mu +1)\, t} \, \exp (-\varvec{\alpha } t) \, w(\varvec{h};t,\nu -1/2,d) \text {d}t, \quad \varvec{h} \in \mathbb {R}^{d},\nonumber \\ \end{aligned}$$
(26)

where the products, exponent, exponential, and integration are taken element-wise. Under Condition (A.1) of Theorem 1, \(\exp (-t \varvec{\alpha })\) is positive semidefinite (Lemma 1). If, furthermore, Condition (A.2) also holds, then \(\varvec{\sigma } \, \varvec{\alpha }^{2\nu +\mu } \, \exp (-\varvec{\alpha } t)\) is positive semidefinite for any \(t>0\), as the element-wise product of positive semidefinite matrices. The matrix-valued Matérn covariance function \(\mathbf{K}({\varvec{h}};\varvec{\alpha },\nu \varvec{1},\varvec{\sigma })\) appears as a mixture of valid Wendland covariances in \(\mathbb {R}^{d}\) weighted by positive semidefinite matrices; thus, it is a valid model in \(\mathbb {R}^{d}\), which completes the proof. \(\square \)

Proof of Theorem 2

We first prove (A). The matrix-valued spectral density of the multivariate Matérn covariance \(\mathbf{K}({\varvec{h}}; \varvec{\alpha }, \varvec{\nu },\varvec{\sigma })\) in \(\mathbb {R}^d\) is (Eq. (3))

$$\begin{aligned} \widetilde{\mathbf{K}}(\varvec{\omega }; \varvec{\alpha }, \varvec{\nu },\varvec{\sigma }) = \left[ \frac{{\varvec{\sigma }} \, \varGamma (\varvec{\nu }+d/2)}{\pi ^{d/2} \,\varvec{\alpha }^{d} \, \varGamma (\varvec{\nu })} \right] \, \left[ \exp \left( -\left( \varvec{\nu }+\frac{d}{2}\right) \ln \left( 1+\frac{\Vert \varvec{\omega } \Vert ^2}{\varvec{\alpha }^2}\right) \right) \right] , \quad \varvec{\omega } \in \mathbb {R}^d. \end{aligned}$$
(27)

Since the composition of two Bernstein functions is still a Bernstein function and based on the fact that \(x \mapsto \ln (1+x)\) is a Bernstein function (Schilling et al. 2010, corollary 3.8 and chapter 16.4), the matrix

$$\begin{aligned} \left[ \left( \varvec{\nu }+\frac{d}{2}\right) \ln \left( 1+\frac{\Vert \varvec{\omega } \Vert ^2}{\varvec{\alpha }^2}\right) \right] \end{aligned}$$

turns out to be the element-wise product of two Bernstein matrices with the same supporting points under Conditions (A.1) and (A.2) of Theorem 2; hence, it is conditionally negative semidefinite (see example following (11)). If Condition (A.3) also holds, then the spectral density matrix \(\widetilde{\mathbf{K}}(\varvec{\omega }; \varvec{\alpha }, \varvec{\nu },\varvec{\sigma })\) is positive semidefinite for any \(\varvec{\omega } \in \mathbb {R}^d\), based on Lemma 1 and the Schur product theorem, and \(\mathbf{K}(\varvec{h}; \varvec{\alpha }, \varvec{\nu },\varvec{\sigma })\) is therefore a valid covariance function in \(\mathbb {R}^d\).

We now prove (B). Consider the p-variate Matérn covariance (7), in which each entry is a scale mixture of Gaussian covariances of the form (4). Let \(\varvec{\psi }=[\psi _{ij}]_{i,j=1}^p\) be a matrix with positive entries. The change of variable \(u = \psi ^{-1}_{ij} v\) in the scale mixture representation of the cross-covariance \(k_{ij}(\varvec{h})\) yields the following expression for the p-variate Matérn covariance model, where the exponent \(-1\) indicates the element-wise inverse

$$\begin{aligned} \mathbf{K}({\varvec{h}};\varvec{\alpha },\varvec{\nu },\varvec{\sigma }) = \int _{0}^{+\infty } g(\varvec{h}; \varvec{\psi }^{-1} v) \, \varvec{\sigma } \, \mathbf{F}(\varvec{\psi }^{-1} v; \varvec{\alpha },\varvec{\nu }) \, \varvec{\psi }^{-1} \text {d}v. \end{aligned}$$
(28)

Based on Lemma 4, sufficient conditions for this model to be valid in \(\mathbb {R}^d\) are as follows:

  1. (1)

    \(\varvec{\psi }\) is conditionally negative semidefinite;

  2. (2)

    \(\varvec{R}(v)= \varvec{\sigma } \mathbf{F}(\varvec{\psi }^{-1} v; \varvec{\alpha }, \varvec{\nu }) \varvec{\psi }^{-1} (\varvec{\psi }^{-1} v)^{-d/2}\) is positive semidefinite for any \(v \ge 0\).

The latter matrix can be rewritten as

$$\begin{aligned} \begin{aligned} \varvec{R}(v)&= \frac{1}{v^{1+d/2}} \, \frac{\varvec{\sigma } \varvec{\psi }^{d/2}}{ \varGamma (\varvec{\nu })} \left( \frac{\varvec{\alpha }^2 \, \varvec{\psi }}{4 v} \right) ^{\varvec{\nu }} \exp \left( -\frac{\varvec{\alpha }^2\, \varvec{\psi }}{4 v}\right) \\&= (4t(v))^{1+d/2} \, \frac{\varvec{\sigma } \varvec{\psi }^{d/2}}{ \varGamma (\varvec{\nu })} \left( t(v) \varvec{\alpha }^2\, \varvec{\psi } \right) ^{\varvec{\nu }} \exp \left( -t(v) \varvec{\alpha }^2\, \varvec{\psi }\right) , \end{aligned} \end{aligned}$$

with \(t(v) = \frac{1}{4v} > 0\). To prove that, under Conditions (B.2) to (B.4) of Theorem 2, this matrix is positive semidefinite for any positive value of t(v), we distinguish two cases:

  • \(t(v) \le 1\). Up to a positive scalar factor, \(\varvec{R}(v)\) can be decomposed as

    $$\begin{aligned} \varvec{R}(v)&= (4t(v))^{1+d/2} \left[ \exp ((\ln t(v)+1-t(v)) \varvec{\nu })\right] \nonumber \\&\quad \left[ \exp \left( -t(v) \left( \varvec{\alpha }^2\, \varvec{\psi }-\varvec{\nu }\right) \right) \right] \, \left[ \frac{\varvec{\sigma } \varvec{\psi }^{d/2}}{\varGamma (\varvec{\nu })} \left( \varvec{\alpha }^{2}\, \varvec{\psi }\right) ^{\varvec{\nu }} \exp (-\varvec{\nu }) \right] ,\nonumber \\ \end{aligned}$$
    (29)

    with \(\ln t(v) + 1- t(v) < 0\) and \(-t(v) < 0\). Together with Lemma 1, Conditions (B.2), (B.3), and (B.4) of Theorem 2 ensure the positive semidefiniteness of the first, second, and third matrices into brackets in the second member of (29), respectively. Accordingly, \(\varvec{R}(v)\) is positive semidefinite based on the Schur product theorem.

  • \(t(v) > 1\). One has the decomposition

    $$\begin{aligned} \begin{aligned} \varvec{R}(v) =&(4t(v))^{1+d/2} \, \left[ \exp \left( -\left( \ln t(v) + 1\right) \, \left( \varvec{\alpha }^2\, \varvec{\psi }- {\varvec{\nu }}\right) \right) \right] \\&\left[ \exp \left( \left( \ln t(v) +1- t(v)\right) \, \varvec{\alpha }^2\, \varvec{\psi }\right) \right] \, \left[ \frac{\varvec{\sigma } \varvec{\psi }^{d/2}}{\varGamma (\varvec{\nu })} \left( \varvec{\alpha }^{2}\, \varvec{\psi }\right) ^{\varvec{\nu }} \exp (-\varvec{\nu }) \right] , \end{aligned} \end{aligned}$$
    (30)

    with \(-(\ln t(v) + 1)<0\) and \(\ln t(v)+ 1- t(v)<0\). The positive semidefiniteness of \(\varvec{R}(v)\) follows from Conditions (B.2), (B.3), and (B.4) of Theorem 2, Lemma 1, and the Schur product theorem. In particular, one uses the fact that Conditions (B.2) and (B.3) imply the conditional negative semidefiniteness of \(\varvec{\alpha }^2\,\varvec{\psi }\).

\(\square \)

Proof of Theorem 3

Conditions (A) and (B) are particular cases of Theorem 2B, with \(\varvec{\psi }=\varvec{\nu } \, \varvec{\alpha }^{-2}\) and \(\varvec{\psi }=\varvec{1}/\beta \), respectively. \(\square \)

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Emery, X., Porcu, E. & White, P. New Validity Conditions for the Multivariate Matérn Coregionalization Model, with an Application to Exploration Geochemistry. Math Geosci 54, 1043–1068 (2022). https://doi.org/10.1007/s11004-022-10000-6

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