Covariance functions for multivariate Gaussian fields evolving temporally over planet earth

Abstract

The construction of valid and flexible cross-covariance functions is a fundamental task for modeling multivariate space–time data arising from, e.g., climatological and oceanographical phenomena. Indeed, a suitable specification of the covariance structure allows to capture both the space–time dependencies between the observations and the development of accurate predictions. For data observed over large portions of planet earth it is necessary to take into account the curvature of the planet. Hence the need for random field models defined over spheres across time. In particular, the associated covariance function should depend on the geodesic distance, which is the most natural metric over the spherical surface. In this work, we propose a flexible parametric family of matrix-valued covariance functions, with both marginal and cross structure being of the Gneiting type. We also introduce a different multivariate Gneiting model based on the adaptation of the latent dimension approach to the spherical context. Finally, we assess the performance of our models through the study of a bivariate space–time data set of surface air temperatures and precipitable water content.

Appendices

Appendix 1: Background on positive definite functions

We start by defining (semi) positive definite functions, that arise in statistics as the covariances of Gaussian RFs as well as the characteristic functions of probability distributions. Let ${\mathcal {E}}$ be a non-empty set and $m\in {\mathbb {N}}$. We say that the matrix-valued function ${\mathbf{F}}:{\mathcal {E}}\times {\mathcal {E}} \rightarrow {\mathbb {R}}^{m\times m}$ is (semi) positive definite if for all integer $n\ge 1$, $\{e_1,\ldots ,e_n\} \subset {\mathcal {E}}$ and $\{\varvec{a}_1,\ldots ,\varvec{a}_n\}\subset {\mathbb {R}}^m$, the following inequality holds:

$$\begin{aligned} \sum _{\ell =1}^{n} \sum _{r=1}^n \varvec{a}_{\ell }^\top {\mathbf{F}}(e_\ell ,e_r) \varvec{a}_{r} \ge 0. \end{aligned}$$

(6.1)

We denote as ${\mathcal {P}}^m({\mathcal {E}})$ the class of such mappings ${\mathbf{F}}$ satisfying Eq. (6.1).

Next, we focus on the cases where ${\mathcal {E}}$ is either ${\mathbb {R}}^d$, ${\mathbb {S}}^d$ or ${\mathbb {S}}^d \times {\mathbb {R}}^k$, for $d,k\in {\mathbb {N}}$. For a clear presentation of the results, Table 7 summarizes the notation introduced along this “Appendices 1, 2 and 3”.

1.1 Matrix-valued positive definite functions on Euclidean spaces: the classes $\Phi ^{m}_{d,{\mathcal {S}}}$ and $\Phi ^m_{d,{\mathcal {I}}}$

This section summarizes matrix-valued positive definite functions on the Euclidean space ${\mathcal {E}} = {\mathbb {R}}^d$. Specifically, we expose some characterizations for the stationary and Euclidean isotropic members of the class ${\mathcal {P}}^m({\mathbb {R}}^d)$.

Table 7 Summary of the notation used along the “Appendices 1, 2 and 3”

We say that ${\mathbf{F}} \in {\mathcal {P}}^m({\mathbb {R}}^d)$ is stationary if there exists a mapping $\tilde{\varvec{\varphi }}:{\mathbb {R}}^d \rightarrow {\mathbb {R}}^{m\times m}$ such that

$$\begin{aligned} {\mathbf{F}}(\varvec{x},\varvec{y}) = \tilde{\varvec{\varphi }}(\varvec{x}-\varvec{y}) = [\tilde{\varphi }_{ij}(\varvec{x}-\varvec{y})]_{i,j=1}^m, \quad \varvec{x},\varvec{y}\in {\mathbb {R}}^d. \end{aligned}$$

(6.2)

We call $\Phi _{d,{\mathcal {S}}}^m$ the class of continuous mappings $\tilde{\varvec{\varphi }}$ such that ${\mathbf{F}}$ in (6.2) is positive definite. Cramér’s Theorem (Cramer 1940) establishes that $\tilde{\varvec{\varphi }}\in \Phi _{d,{\mathcal {S}}}^m$ if and only if it can be represented through

$$\begin{aligned} \tilde{\varvec{\varphi }}(\varvec{h}) = \int _{{\mathbb {R}}^d} \exp \{ \imath \varvec{h}^\top \varvec{\omega } \} \text {d}\tilde{\varvec{\Lambda }}_d(\varvec{\omega }), \quad \varvec{h}\in {\mathbb {R}}^d, \end{aligned}$$

(6.3)

where $\imath = \sqrt{-1} \in {\mathbb {C}}$ and $\tilde{\varvec{\Lambda }}_d: {\mathbb {R}}^d\rightarrow {\mathbb {C}}^{m\times m}$ is a matrix-valued mapping, with increments being Hermitian and positive definite matrices, and whose elements, $\tilde{\Lambda }_{d,ij}(\cdot )$, for $i,j=1,\ldots ,m$, are functions of bounded variation (see Wackernagel 2003). In particular, the diagonal terms, $\tilde{\Lambda }_{d,ii}(\varvec{\omega })$, are real, non-decreasing and bounded, whereas the off-diagonal elements are generally complex-valued. Cramer’s Theorem is the multivariate version of the celebrated Bochner’s Theorem (Bochner 1955). If the elements of $\tilde{\varvec{\Lambda }}_d(\cdot )$ are absolutely continuous, then Eq. (6.3) simplifies to

$$\begin{aligned} \tilde{\varvec{\varphi }}(\varvec{h}) = \int _{{\mathbb {R}}^d} \exp \{ \imath \varvec{h}^\top \varvec{\omega } \} \tilde{\varvec{\lambda }}_d(\varvec{\omega })\text {d}\varvec{\omega }, \quad \varvec{h}\in {\mathbb {R}}^d, \end{aligned}$$

with $\tilde{\varvec{\lambda }}_d(\varvec{\omega })=[\tilde{\lambda }_{d,ij}(\varvec{\omega })]_{i,j=1}^m$ being Hermitian and positive definite, for any $\varvec{\omega }\in {\mathbb {R}}^d$. The mapping $\tilde{\varvec{\lambda }}_d(\varvec{\omega })$ is known as the matrix-valued spectral density and classical Fourier inversion yields

$$\begin{aligned} \tilde{\varvec{\lambda }}_d(\varvec{\omega }) = \frac{1}{(2\pi )^d} \int _{{\mathbb {R}}^d} \exp \{ - \imath \varvec{h}^\top \varvec{\omega } \} \tilde{\varvec{\varphi }}(\varvec{h}) \text {d}\varvec{h}, \quad \varvec{\omega }\in {\mathbb {R}}^d. \end{aligned}$$

Finally, the following inequality between the elements of $\tilde{\varvec{\varphi }}$ is true

$$\begin{aligned} |\tilde{\varphi }_{ij}(\varvec{h})|^2 \le \tilde{\varphi }_{ii}(\varvec{0}) \tilde{\varphi }_{jj}(\varvec{0}),\quad {\text { for all }} \varvec{h}\in {\mathbb {R}}^d. \end{aligned}$$

However, the maximum value of the mapping $\tilde{\varphi }_{ij}(\varvec{h})$, with $i\ne j$, is not necessarily reached at $\varvec{h}=\varvec{0}$. In general, $\tilde{\varphi }_{ij}$ is not itself a scalar-valued positive definite function when $i\ne j$.

Consider an element ${\mathbf{F}}$ in ${\mathcal {P}}^m({\mathbb {R}}^d)$ and suppose that there exists a continuous and bounded mapping $\varvec{\varphi }:{\mathbb {R}}_+ \rightarrow {\mathbb {R}}^{m\times m}$ such that

$$\begin{aligned} {\mathbf{F}}(\varvec{x},\varvec{y}) = \varvec{\varphi }(\Vert \varvec{x}-\varvec{y}\Vert ), \quad \varvec{x},\varvec{y}\in {\mathbb {R}}^d. \end{aligned}$$

Then, ${\mathbf{F}}$ is called stationary and Euclidean isotropic (or radial). We denote as $\Phi ^m_{d,{\mathcal {I}}}$ the class of bounded, continuous, stationary and Euclidean isotropic mappings $\varvec{\varphi }(\cdot )=[\varphi _{ij}(\cdot )]_{i,j=1}^m$.

When $m=1$, characterization of the class $\Phi _{d,{\mathcal {I}}}$ was provided through the celebrated paper by Schoenberg (1938). Alonso-Malaver et al. (2015) characterize the class $\Phi ^m_{d,{\mathcal {I}}}$ through the continuous members $\varvec{\varphi }$ having representation

$$\begin{aligned} \varvec{\varphi }(r) = \int _{[0,\infty )} \Omega _d(r\omega ) \text {d}\varvec{\Lambda }_d(\omega ), \quad r\ge 0, \end{aligned}$$

where $\varvec{\Lambda }_d: [0,\infty )\rightarrow {\mathbb {R}}^{m\times m}$ is a matrix-valued mapping, with increments being positive definite matrices, and elements $\Lambda _{d,ij}(\cdot )$ of bounded variation, for each $i,j=1,\ldots ,m$. Here, the function $\Omega _d(\cdot )$ is defined as

$$\begin{aligned} \Omega _d(z) = \Gamma (d/2)(z/2)^{-(d-2)/2} J_{(d-2)/2}(z), \quad z \ge 0, \end{aligned}$$

with $\Gamma$ being the Gamma function and $J_\nu$ the Bessel function of the first kind of degree $\nu$ (see Abramowitz and Stegun 1970). If the elements of $\varvec{\Lambda }_d(\cdot )$ are absolutely continuous, then we have an associated spectral density $\varvec{\lambda }_d:[0,\infty )\rightarrow {\mathbb {R}}^{m\times m}$ as in the stationary case, which is called, following Daley and Porcu (2014), a d-Schoenberg matrix.

The classes $\Phi ^m_{d,{\mathcal {I}}}$ are non-increasing in d, and the following inclusion relations are strict

$$\begin{aligned} \Phi ^m_{\infty ,{\mathcal {I}}} := \bigcap _{d=1}^\infty \Phi ^m_{d,{\mathcal {I}}} \subset \cdots \subset \Phi ^m_{2,{\mathcal {I}}} \subset \Phi ^m_{1,{\mathcal {I}}}. \end{aligned}$$

The elements $\varvec{\varphi }$ in the class $\Phi ^m_{\infty ,{\mathcal {I}}}$ can be represented as

$$\begin{aligned} \varvec{\varphi }(r) = \int _{[0,\infty )} \exp (-r^2\omega ^2) \text {d}\varvec{\Lambda }(\omega ), \quad r\ge 0, \end{aligned}$$

where $\varvec{\Lambda }$ is a matrix-valued mapping with similar properties as $\varvec{\Lambda }_d$.

1.2 Matrix-valued positive definite functions on ${\mathbb {S}}^d$: the class $\Psi _{d,{\mathcal {I}}}^m$

In this section, we pay attention to matrix-valued positive definite functions on the unit sphere. Consider ${\mathbf{F}} \in {\mathcal {P}}^m({\mathbb {S}}^d)$. We say that ${\mathbf{F}}$ is geodesically isotropic if there exists a bounded and continuous mapping $\varvec{\psi }:[0,\pi ]\rightarrow {\mathbb {R}}^{m\times m}$ such that

$$\begin{aligned} {\mathbf{F}}(\varvec{x},\varvec{y}) = \varvec{\psi }(\theta (\varvec{x},\varvec{y})), \quad \varvec{x},\varvec{y}\in {\mathbb {S}}^d. \end{aligned}$$

The continuous mappings $\varvec{\psi }$ are the elements of the class $\Psi _{d,{\mathcal {I}}}^m$ and the following inclusion relations are true:

$$\begin{aligned} \Psi _{\infty ,{\mathcal {I}}}^m = \bigcap _{d= 1}^\infty \Psi _{d,{\mathcal {I}}}^m \subset \cdots \subset \Psi _{2,{\mathcal {I}}}^m \subset \Psi _{1,{\mathcal {I}}}^m, \end{aligned}$$

where $\Psi _{\infty ,{\mathcal {I}}}^m$ is the class of geodesically isotropic positive definite functions being valid on the Hilbert sphere ${\mathbb {S}}^\infty = \{ (x_n)_{n\in {\mathbb {N}}} \in {\mathbb {R}}^{{\mathbb {N}}} : \sum _{n\in {\mathbb {N}}} x_n^2 =1 \}$.

The elements of the class $\Psi _{d,{\mathcal {I}}}^m$ have an explicit connection with Gegenbauer (or ultraspherical) polynomials (Abramowitz and Stegun 1970). Here, ${\mathcal {G}}_n^{\lambda }$ denotes the $\lambda$-Gegenbauer polynomial of degree n, which is defined implicitly through the expression

$$\begin{aligned} \frac{1}{(1+r^2-2r\cos \theta )^\lambda } = \sum _{n=0}^\infty r^n {\mathcal {G}}_n^{\lambda }(\cos \theta ), \quad \theta \in [0,\pi ], \quad r\in (-1,1). \end{aligned}$$

In particular, ${\mathcal {T}}_n := {\mathcal {G}}_n^{0}$ and ${\mathcal {P}}_n := {\mathcal {G}}_n^{1/2}$ are respectively the Chebyshev and Legendre polynomials of degree n.

The following result (Hannan 2009; Yaglom 1987) offers a complete characterization of the classes $\Psi _{d,{\mathcal {I}}}^m$ and $\Psi _{\infty ,{\mathcal {I}}}^m$, and corresponds to the multivariate version of Schoenberg’s Theorem (Schoenberg 1942). Equalities and summability conditions for matrices must be understood in a componentwise sense.

Theorem 6.1

Letdandmbe positive integers.

• (1) The mapping$\varvec{\psi }$is a member of the class$\Psi _{d,{\mathcal {I}}}^m$if and only if it admits the representation

  $$\begin{aligned} \varvec{\psi }(\theta ) = \sum _{n=0}^\infty {\mathbf{B}}_{n,d} \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)}, \quad \theta \in [0,\pi ], \end{aligned}$$

  where$\{\mathbf{B }_{n,d}\}_{n=0}^\infty$is a sequence of symmetric, positive definite and summable matrices.

• (2) The mapping $\varvec{\psi }$is a member of the class$\Psi _{\infty ,{\mathcal {I}}}^m$if and only if it can be represented as

  $$\begin{aligned} \varvec{\psi }(\theta ) = \sum _{n=0}^\infty {\mathbf{B}}_{n} (\cos \theta )^n, \quad \theta \in [0,\pi ], \end{aligned}$$

  where$\{\mathbf{B }_{n}\}_{n=0}^\infty$is a sequence of symmetric, positive definite and summable matrices.

Using orthogonality properties of Gegenbauer polynomials (Abramowitz and Stegun 1970) and through classical Fourier inversion we can prove that

$$\begin{aligned} {\mathbf{B}}_{0,1} & = \frac{1}{\pi }\int _0^\pi \varvec{\psi }(\theta )\text {d}\theta , \\ {\mathbf{B}}_{n,1} & = \frac{2}{\pi }\int _0^\pi \cos (n\theta )\varvec{\psi }(\theta )\text {d}\theta , \quad {\text { for }} n\ge 1, \end{aligned}$$

whereas for $d\ge 2$, we have

$$\begin{aligned} {\mathbf{B}}_{n,d} = \frac{(2n+d-1)}{2^{3-d}\pi } \frac{[\Gamma ((d-1)/2)]^2}{\Gamma (d-1)} \int _{0}^\pi {\mathcal {G}}_{n}^{(d-1)/2}(\cos \theta ) (\sin \theta )^{d-1} \varvec{\psi }(\theta ) \text {d}\theta , \quad n\ge 0, \end{aligned}$$

where integration is taken componentwise. The matrices $\{\mathbf{B }_{n,d}\}_{n=0}^\infty$ are called Schoenberg’s matrices. For the case $m=1$, such result is reported by Gneiting (2013).

1.3 Matrix-valued positive definite functions on ${\mathbb {S}}^d\times {\mathbb {R}}^k$: the class $\Upsilon _{d,k}^m$

Let d, k and m be positive integers. We now focus on the class of matrix-valued positive definite functions on ${\mathbb {S}}^d \times {\mathbb {R}}^k$, being bounded, continuous, geodesically isotropic in the spherical component and stationary in the Euclidean one. The case $k=1$ is particularly important, since ${\mathcal {P}}^m({\mathbb {S}}^d\times {\mathbb {R}})$ can be interpreted as the class of admissible space–time covariances for multivariate Gaussian RFs, with spatial locations on the unit sphere.

Consider ${\mathbf{F}} \in {\mathcal {P}}^m({\mathbb {S}}^d\times {\mathbb {R}}^k)$ and suppose that there exists a bounded and continuous mapping ${\mathbf{C}}:[0,\pi ]\times {\mathbb {R}}^k \rightarrow {\mathbb {R}}^{m\times m}$ such that

$$\begin{aligned} {\mathbf{F}}((\varvec{x},\varvec{t}),(\varvec{y},\varvec{s})) = {\mathbf{C}}(\theta (\varvec{x},\varvec{y}),\varvec{t}-\varvec{s}), \quad \varvec{x},\varvec{y}\in {\mathbb {S}}^d, \varvec{t},\varvec{s}\in {\mathbb {R}}^k. \end{aligned}$$

Such mappings ${\mathbf{C}}$ are the elements of the class $\Upsilon _{d,k}^m$. These classes are non-increasing in d and we have the inclusions

$$\begin{aligned} \Upsilon _{\infty ,k}^m := \bigcap _{d=1}^\infty \Upsilon _{d,k}^m \subset \cdots \subset \Upsilon _{2,k}^m \subset \Upsilon _{1,k}^m. \end{aligned}$$

Ma (2017) proposes the generalization of Theorem 6.1 to the space–time case. Theorem 6.2 below offers a complete characterization of the class $\Upsilon _{d,k}^m$ and $\Upsilon _{\infty ,k}^m$, for any $m\ge 1$. Again, equalities and summability conditions must be understood in a componentwise sense.

Theorem 6.2

Let d, k and m be positive integers and ${\mathbf{C}}: [0,\pi ] \times \mathbb {R}^k \rightarrow {\mathbb {R}}^{m\times m}$a continuous matrix-valued mapping, with${\text {C}}_{ii}(0,\varvec{0})<\infty$, for all$i=1,\ldots ,m$.

• (1) The mapping$\mathbf{C }$belongs to the class$\Upsilon _{d,k}^m$if and only if

  $$\begin{aligned} \mathbf{C }(\theta ,\varvec{u}) = \sum _{n=0}^{\infty } \tilde{\varvec{\varphi }}_{n,d}(\varvec{u}) \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)}, \quad (\theta ,\varvec{u}) \in [0,\pi ] \times \mathbb {R}^k, \end{aligned}$$

  with$\left\{ \tilde{\varvec{\varphi }}_{n,d}(\cdot ) \right\} _{n=0}^{\infty }$being a sequence of members of the class$\Phi _{k,{\mathcal {S}}}^m$, with the additional requirement that the sequence of matrices$\{ \tilde{\varvec{\varphi }}_{n,d}(\varvec{0})\}_{n=0}^\infty$is summable.

• (2) The mapping$\mathbf{C }$belongs to the class$\Upsilon _{\infty ,k}^m$if and only if

  $$\begin{aligned} \mathbf{C }(\theta ,\varvec{u}) = \sum _{n=0}^{\infty } \tilde{\varvec{\varphi }}_n(\varvec{u}) (\cos \theta )^n, \quad (\theta ,\varvec{u}) \in [0,\pi ] \times \mathbb {R}^k, \end{aligned}$$

  with$\left\{ \tilde{\varvec{\varphi }}_n(\cdot ) \right\} _{n=0}^{\infty }$being a sequence of members of the class$\Phi _{k,{\mathcal {S}}}^m$, with the additional requirement that the sequence of matrices$\{\tilde{\varvec{\varphi }}_n(\varvec{0})\}_{n=0}^\infty$is summable.

Again, using orthogonality arguments, we have

$$\begin{aligned} \tilde{\varvec{\varphi }}_{0,1}(\varvec{u}) & = \frac{1}{\pi }\int _0^\pi \mathbf{C }(\theta ,\varvec{u}) \text {d}\theta , \\ \tilde{\varvec{\varphi }}_{n,1}(\varvec{u}) & = \frac{2}{\pi }\int _0^\pi \cos (n\theta )\mathbf{C }(\theta ,\varvec{u}) \text {d}\theta , \quad {\text { for }} n\ge 1, \end{aligned}$$

whereas for $d\ge 2$,

$$\begin{aligned} \tilde{\varvec{\varphi }}_{n,d}(\varvec{u}) & = \frac{(2n+d-1)}{2^{3-d}\pi } \frac{[\Gamma ((d-1)/2)]^2}{\Gamma (d-1)}\nonumber \\&\int _{0}^{\pi } {\mathcal {G}}_{n}^{(d-1)/2}(\cos \theta ) (\sin \theta )^{d-1} \mathbf{C }(\theta ,\varvec{u}) \mathrm{d} \theta , \quad n\ge 0. \end{aligned}$$

Appendix 2: Proof of Theorem 3.1

In order to illustrate the results following subsequently, a technical Lemma will be useful. We do not provide a proof because it is obtained following the same arguments as in Porcu and Zastavnyi (2011).

Lemma 7.1

Let m, d and k be strictly positive integers. Let$(X,{\mathcal{B}}, \mu )$be a measure space, for$X \subset \mathbb {R}$and${\mathcal B}$being the Borel sigma algebra. Let$\varvec{\psi }: [0,\pi ] \times X \rightarrow \mathbb {R}$ and$\varvec{\varphi }: [0,\infty ) \times X \rightarrow \mathbb {R}$ be continuous mappings satisfying

• 1. $\varvec{\psi }(\cdot ,\xi ) \in \Psi _{d,{\mathcal{I}}}^m$ a.e.$\xi \in X$;

• 2. $\varvec{\psi }(\theta ,\cdot ) \in L_1(X,{\mathcal{B}},\mu )$ for any$\theta \in [0,\pi ]$;

• 3. $\varvec{\varphi }(\cdot ,\xi ) \in \Phi _{k,{\mathcal{I}}}^m$ a.e.$\xi \in X$;

• 4. $\varvec{\varphi }(u, \cdot ) \in L_1(X,{\mathcal{B}},\mu )$ for any$u \in [0,\infty )$.

Let$\mathbf{C } :[0,\pi ] \times [0,\infty )\rightarrow {\mathbb {R}}^{m\times m}$be the mapping defined through

$$\begin{aligned} \mathbf{C } (\theta ,u) = \int _X \varvec{\psi }(\theta ,\xi ) \varvec{\varphi }(u, \xi ) \mu (\mathrm{d} \xi ), \quad (\theta ,u) \in [0,\pi ] \times [0,\infty ). \end{aligned}$$

Then, $\mathbf{C }$is continuous and bounded. Further,$\mathbf{C }$belongs to the class$\Upsilon _{d,k}^m$.

Of course, Lemma 7.1 is a particular case of the scale mixtures introduced in Sect. 2.

Proof of Theorem 3.1

Let $(X, {\mathcal{B}}, \mu )$ as in Lemma 7.1 and consider $X = \mathbb {R}_+$ with $\mu$ the Lebesgue measure. We offer a proof of the constructive type. Let us define the function $\varvec{\psi }(\theta ,\xi )$ with members $\psi _{ij}(\cdot ,\cdot )$ defined through

$$\begin{aligned} \psi _{ij}(\theta ,\xi ) = { \sigma _{ii}\sigma _{jj} \rho _{ij}} \left( 1 - \frac{\theta }{\xi c_{ij} } \right) _{+}^{n+1}, \quad (\theta , \xi ) \in [0,\pi ] \times X, \quad i,j=1,\ldots ,m, \end{aligned}$$

where, as asserted, the constants $\sigma _{ii}$, $\rho _{ij}$ and $c_{ij}$ are determined according to condition (3.1). Let us now define the mapping $(u,\xi ) \mapsto \varphi (u,\xi ):= \xi ^{n+1} (1 - \xi f(u))_{+}^{\ell }$, with $(u,\xi ) \in [0,\infty ) \times X$. It can be verified that both $\varvec{\psi }$ and $\varphi$ satisfy requirements 1–4 in Lemma 7.1. In particular, Condition 1 yields thanks to Lemmas 3 and 4 in Gneiting (2013), as well as Theorem 1 in Daley et al. (2015). Also, arguments in Porcu et al. (2016) show that Condition 3 holds for any $\ell \ge 1$. We can now apply Lemma 7.1, so that we have that

$$\begin{aligned} {\text {C}}_{i,j, n,\ell }(\theta ,u):= \int _X \varvec{\psi }(\theta ,\xi ) \varphi (u,\xi ) \mathrm{d} \xi , \quad [0, \pi ] \times [0,\infty ) \end{aligned}$$

is a member of the class $\Upsilon _{2n+1,1}^m$ for any $\ell \ge 1$. Pointwise application of an elegant scale mixture argument as in Proposition 1 of Porcu et al. (2016) shows that

$$\begin{aligned} {\text {C}}_{i,j,n,\ell } (\theta ,u) = {\mathcal {B}}(n+2,\ell +1) \frac{ \rho _{ij} \sigma _{ii} \sigma _{jj}}{f(u)^{n+2}} \left( 1- \frac{\theta f(u)}{c_{ij}} \right) _+^{n+\ell +1}, \quad (\theta ,u) \in [0,\pi ] \times [0,\infty ), \end{aligned}$$

where ${\mathcal {B}}$ denotes the Beta function (Abramowitz and Stegun 1970). We now omit the factor ${\mathcal {B}}(n+2,\ell +1)$ since it does not affect positive definiteness. Now, standard convergence arguments show that

$$\begin{aligned} \lim _{\ell \rightarrow \infty } {\text {C}}_{i,j,n,\ell }(\theta / \ell ,u) = \frac{ \rho _{ij} \sigma _{ii} \sigma _{jj}}{f(u)^{n+2}} \exp \left( - \frac{\theta f(u)}{c_{ij}} \right) , (\theta ,u) \in [0,\pi ] \times [0,\infty ), \end{aligned}$$

with the convergence being uniform in any compact set. The proof is then completed in view of Bernstein’s theorem (Feller 1966). $\square$

Appendix 3: Proof of Theorem 3.2

Before we state the proof of Theorem 3.2, we need to introduce three auxiliary lemmas.

Lemma 8.1

Let ${\text {C}}: [0,\pi ]\times {\mathbb {R}}^k \rightarrow {\mathbb {R}}$ be a continuous, bounded and integrable function, for some positive integer k. Then ${\text {C}} \in \Upsilon _{d,k}$, for $d\ge 1$, if and only if the mapping $\psi _{\varvec{\omega }}:[0,\pi ] \rightarrow {\mathbb {R}}$ defined as

$$\begin{aligned} \psi _{\varvec{\omega }}(\theta ) = \frac{1}{(2\pi )^k}\int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\omega }^\top \varvec{v} \} {\text {C}}(\theta ,\varvec{v}) \text {d}\varvec{v}, \quad \theta \in [0,\pi ], \end{aligned}$$

(8.1)

belongs to the class $\Psi _{d,{\mathcal {I}}}$, for all $\varvec{\omega }\in {\mathbb {R}}^k$.

We do not report the proof of Lemma 8.1 since the arguments are the same as in the proof of Lemma 8.2 below. Note that this lemma is a spherical version of the result given by Cressie and Huang (1999).

Lemma 8.2

Let${\text {C}}: [0,\pi ]\times {\mathbb {R}}^l \times {\mathbb {R}}^k \rightarrow {\mathbb {R}}$be a continuous, bounded and integrable function, for some positive integers l and k. Then , ${\text {C}} \in \Upsilon _{d,k+l}$, with$d\ge 1$, if and only if the mapping${\text {C}}_{\varvec{\omega }}:[0,\pi ] \times {\mathbb {R}}^l \rightarrow {\mathbb {R}}$defined as

$$\begin{aligned}&{\text {C}}_{\varvec{\omega }}(\theta ,\varvec{u}) = \frac{1}{(2\pi )^k}\int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\omega }^\top \varvec{v} \} {\text {C}}(\theta ,\varvec{u},\varvec{v}) \text {d}\varvec{v}, \\&\quad (\theta ,\varvec{u})\in [0,\pi ] \times {\mathbb {R}}^l, \end{aligned}$$

(8.2)

belongs to the class$\Upsilon _{d,l}$, for all$\varvec{\omega }\in {\mathbb {R}}^k$.

Proof of Lemma 8.2

Suppose that ${\text {C}} \in \Upsilon _{d,k+l}$, then the characterization of Berg and Porcu (2017) implies that

$$\begin{aligned} {\text {C}}(\theta ,\varvec{u},\varvec{v}) = \sum _{n=0}^\infty \tilde{\varphi }_{n,d}(\varvec{u},\varvec{v}) \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)}, \end{aligned}$$

where $\{\tilde{\varphi }_{n,d}(\cdot ,\cdot )\}_{n=0}^\infty$ is a sequence of functions in $\Phi _{k+l,{\mathcal {S}}}$, with $\sum _{n=0}^\infty \tilde{\varphi }_{n,d}(\varvec{0},\varvec{0}) < \infty$. Therefore,

$$\begin{aligned}&{\text {C}}_{\varvec{\omega }}(\theta ,\varvec{u})\nonumber \\&\quad = \frac{1}{(2\pi )^k} \int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\omega }^\top \varvec{v} \} \left( \sum _{n=0}^\infty \tilde{\varphi }_{n,d}(\varvec{u},\varvec{v}) \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)} \right) \text {d}\varvec{v}\\&\quad = \sum _{n=0}^\infty \left( \frac{1}{(2\pi )^k}\int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\omega }^\top \varvec{v} \} \tilde{\varphi }_{n,d}(\varvec{u},\varvec{v}) \text {d}\varvec{v} \right) \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)}, \end{aligned}$$

where the last step is justified by dominated convergence. We need to prove that for each fixed $\varvec{\omega }\in {\mathbb {R}}^k$, the sequence of functions

$$\begin{aligned} \varvec{u} \mapsto \tilde{\lambda }_{n,d}(\varvec{u};\varvec{\omega }) : = \frac{1}{(2\pi )^k} \int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\omega }^\top \varvec{v} \} \tilde{\varphi }_{n,d}(\varvec{u},\varvec{v}) \text {d}\varvec{v}, \quad n\ge 0, \end{aligned}$$

belongs to the class $\Phi _{l,{\mathcal {S}}}$, a.e. $\varvec{\omega }\in {\mathbb {R}}^k$. In fact, we have that

$$\begin{aligned}&\frac{1}{(2\pi )^{l}} \int _{{\mathbb {R}}^l} \exp \{-\imath \varvec{\tau }^\top \varvec{u} \} \tilde{\lambda }_{n,d}(\varvec{u};\varvec{\omega }) \text {d}\varvec{u} \\&\quad = \frac{1}{(2\pi )^{k+l}} \int _{{\mathbb {R}}^l} \int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\tau }^\top \varvec{u} - \imath \varvec{\omega }^\top \varvec{v} \} \tilde{\varphi }_{n,d}(\varvec{u},\varvec{v}) \text {d}\varvec{v} \text {d}\varvec{u}. \end{aligned}$$

(8.3)

Since $\tilde{\varphi }_{n,d}(\cdot ,\cdot )$ belongs to $\Phi _{k+l,{\mathcal {S}}}$, Bochner’s Theorem implies that the right side in Eq. (8.3) is non-negative everywhere. This implies that $\tilde{\lambda }_{n,d}(\cdot ;\varvec{\omega })$ belongs to the class $\Phi _{l,{\mathcal {S}}}$. Also, direct inspection shows that $\sum _{n=0}^\infty \tilde{\lambda }_{n,d}(\varvec{0};\varvec{\omega }) < \infty$, for all $\varvec{\omega }\in {\mathbb {R}}^k$. The necessary part is completed.

On the other hand, suppose that for each $\varvec{\omega }\in {\mathbb {R}}^k$ the function ${\text {C}}_{\varvec{\omega }}(\theta ,\varvec{u})$ belongs to the class $\Upsilon _{d,l}$, then there exists a sequence of mappings $\{ \tilde{\lambda }_{n,d}(\cdot ;\varvec{\omega }) \}_{n=0}^\infty$ in $\Phi _{l,{\mathcal {S}}}$ for each $\varvec{\omega }\in {\mathbb {R}}^k$, such that

$$\begin{aligned} {\text {C}}_{\varvec{\omega }}(\theta ,\varvec{u}) = \sum _{n=0}^\infty \tilde{\lambda }_{n,d}(\varvec{u};\varvec{\omega }) \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)}. \end{aligned}$$

Thus,

$$\begin{aligned} {\text {C}}(\theta ,\varvec{u},\varvec{v}) & = \int _{{\mathbb {R}}^k} \exp \{ \imath \varvec{\omega }^\top \varvec{v} \} \left( \sum _{n=0}^\infty \tilde{\lambda }_{n,d}(\varvec{u};\varvec{\omega }) \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)} \right) \text {d}\varvec{\omega }\\ & = \sum _{n=0}^\infty \left( \int _{{\mathbb {R}}^k} \exp \{ \imath \varvec{\omega }^\top \varvec{v} \} \tilde{\lambda }_{n,d}(\varvec{u};\varvec{\omega }) \text {d}\varvec{\omega } \right) \frac{{\mathcal {G}}_n^{(d-1)/2}(\cos \theta )}{{\mathcal {G}}_n^{(d-1)/2}(1)}. \end{aligned}$$

We conclude the proof by invoking again Bochner’s Theorem and the result of Berg and Porcu (2017). $\square$

Lemma 8.3

Let d and k be two positive integers. Consider g and f be completely monotone and Bernstein functions, respectively. Then,

$$\begin{aligned} {\text {K}}(\theta ,\varvec{v}) = \frac{1}{ \left\{ f(\theta ) |_{[0,\pi ]} \right\} ^{k/2} } g\left( \frac{\Vert \varvec{v}\Vert ^2}{f(\theta ) |_{[0,\pi ]} } \right) , \quad (\theta ,\varvec{v}) \in [0,\pi ]\times {\mathbb {R}}^k, \end{aligned}$$

belongs to the class$\Upsilon _{d,k}$, for any positive integer d.

Proof of Lemma 8.3

By Lemma 8.1, we must show that $\psi _{\varvec{\omega }}$, defined through Eq. (8.1), belongs to the class $\Psi _{d,{\mathcal {I}}}$, for all $\varvec{\omega }\in {\mathbb {R}}^k$. In fact, we can assume that ${\text {C}}$ is integrable, since the general case is obtained with the same arguments given by Gneiting (2002). Bernstein’s Theorem establishes that g can be represented as the Laplace transform of a bounded measure G, then

$$\begin{aligned} \psi _{\varvec{\omega }} (\theta ) & = \int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\omega }^\top \varvec{v} \} \frac{1}{ \left\{ f(\theta ) |_{[0,\pi ]} \right\} ^{k/2} } \int _{[0,\infty )}\\&\exp \left\{ - \frac{r\Vert \varvec{v}\Vert ^2}{ f(\theta ) |_{[0,\pi ]} } \right\} \text {d}G(r) \text {d}\varvec{v}\\ & = \pi ^{k/2} \int _{[0,\infty )} \exp \left\{ - \frac{\Vert \varvec{\omega }\Vert ^2}{4r} f(\theta ) |_{[0,\pi ]} \right\} \text {d}\tilde{G}(r), \end{aligned}$$

where the last equality follows from Fubini’s Theorem and $\text {d}G(r) = r^{k/2} \text {d}\tilde{G}(r)$. In addition, the composition between a negative exponential and a Bernstein function is completely monotone on the real line (Feller 1966). Then, for any $\varvec{\omega }$ and r, the mapping $\theta \mapsto \exp \{ - \Vert \varvec{\omega }\Vert ^2 f(\theta )|_{[0,\pi ]} /(4r)\}$ is the restriction of a completely monotone function to the interval $[0,\pi ]$. Theorem 7 in Gneiting (2013) implies that such mapping, and thus $\psi _{\varvec{\omega }}$, belongs to the class $\Psi _{d,{\mathcal {I}}}$, for any $d\in {\mathbb {N}}$ and $\varvec{\omega }\in {\mathbb {R}}^k$. $\square$

Proof of Theorem 3.2

By Lemma 8.2, we must show that (8.2) belongs to the class $\Upsilon _{d,l}$, for all $d\in {\mathbb {N}}$. In fact, assuming again that ${\text {C}}$ is integrable and invoking Bernstein’s Theorem we have

$$\begin{aligned} {\text {C}}_{\varvec{\omega }} (\theta ,\varvec{u}) & = \int _{{\mathbb {R}}^k} \exp \{-\imath \varvec{\omega }^\top \varvec{v} \} \frac{1}{ \{f_2( \theta )|_{[0,\pi ]}\}^{l/2} \left\{ f_1\left[ \frac{\Vert \varvec{u}\Vert ^2}{ f_2( \theta )|_{[0,\pi ]} } \right] \right\} ^{k/2}} \int _{[0,\infty )} \\&\exp \left\{ - \frac{r \Vert \varvec{v}\Vert ^2}{ f_1\left[ \frac{\Vert \varvec{u}\Vert ^2}{ f_2( \theta )|_{[0,\pi ]} } \right] } \right\} \text {d}G(r) \text {d}\varvec{v}\\ & = \pi ^{k/2} \frac{1}{\{f_2( \theta )|_{[0,\pi ]}\}^{l/2} } \int _{[0,\infty )} \\&\exp \left\{ - \frac{\Vert \varvec{\omega }\Vert ^2}{4r} f_1\left[ \frac{\Vert \varvec{u}\Vert ^2}{ f_2( \theta )|_{[0,\pi ]} } \right] \right\} \text {d}\tilde{G}(r), \end{aligned}$$

where the last equality follows from Fubini’s Theorem and $\text {d}G(r) = r^{k/2} \text {d}\tilde{G}(r)$. In addition, for any $\varvec{\omega }$, the mapping

$$\begin{aligned} g_{\varvec{\omega }}(\cdot ) := \int _{[0,\infty )} \exp \left\{ - \frac{ \Vert \varvec{\omega }\Vert ^2}{4r} f_1(\cdot ) \right\} \text {d}\tilde{G}(r) \end{aligned}$$

is completely monotone (Feller 1966). Therefore,

$$\begin{aligned} {\text {C}}_{\varvec{\omega }} (\theta ,\varvec{u}) = \pi ^{k/2} \frac{1}{\{f_2( \theta )|_{[0,\pi ]}\}^{l/2} } g_{\varvec{\omega }}\left( \frac{\Vert \varvec{u}\Vert ^2}{ f_2( \theta )|_{[0,\pi ]} } \right) , \end{aligned}$$

and by Lemma 8.3, we have that ${\text {C}}_{\varvec{\omega }}\in \Upsilon _{d,l}$, for all $d\in {\mathbb {N}}$. $\square$