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The Schoenberg kernel and more flexible multivariate covariance models in Euclidean spaces

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Abstract

The J-Bessel univariate kernel \(\Omega _d\) introduced by Schoenberg plays a central role in the characterization of stationary isotropic covariance models defined in a d-dimensional Euclidean space. In the multivariate setting, a matrix-valued isotropic covariance is a scale mixture of the kernel \(\Omega _d\) against a matrix-valued measure that is nondecreasing with respect to matrix inequality. We prove that constructions based on a p-variate kernel \([\Omega _{d_{ij}}]_{i,j=1}^p\) are feasible for different dimensions \(d_{ij},\) at the expense of some parametric restrictions. We illustrate how multivariate covariance models inherit such restrictions and provide new classes of hypergeometric, Matérn, Cauchy and compactly-supported models to illustrate our findings.

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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 AFB220002] (X. Emery) and by the Khalifa University of Science and Technology under Award No. FSU-2021-016 (E. Porcu).

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Correspondence to Emilio Porcu.

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Appendices

Appendix A: Definitions and lemmas

Definition 1

A real symmetric matrix \({\varvec{a}}=[a_{ij}]_{i,j=1}^p\) is conditionally negative semidefinite if \( \sum _{i,j=1}^p \lambda _i \lambda _j {a}_{ij} \le 0\) for all \(\lambda _1,\ldots , \lambda _p \in {\mathbb {R}}\) such that \( \sum _{i=1}^p \lambda _i = 0.\)

Definition 2

A real symmetric matrix \({\varvec{a}}=[a_{ij}]_{i,j=1}^p\) is conditionally null semidefinite if both \({\varvec{a}}\) and \(-{\varvec{a}}\) are conditionally negative semidefinite.

Lemma 1

Let \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) be symmetric conditionally negative semidefinite matrices of size \(p \times p.\) Then,  for any \(t \in (0,1),\) the matrix \(t^{{\varvec{\alpha }}-1} (1-t)^{{\varvec{\beta }}-1}\) is positive semidefinite.

Proof

The claim follows from the Schur product theorem and the fact that, under the mentioned conditions, both \(e^{({\varvec{\alpha }}-1) \ln (t)}\) and \(e^{({\varvec{\beta }}-1) \ln (1-t)}\) are positive semidefinite (Berg et al. 1984, Chapter 3, Theorem 2.2). \(\square \)

Lemma 2

Let \({\varvec{\alpha }}\) be a conditionally null semidefinite real matrix of size \(p \times p.\) Then,  for any \(t \in (0,\infty ),\) the matrix \(t^{{\varvec{\alpha }}-1}\) is positive semidefinite.

Proof

The claim follows from the fact that \({\varvec{\alpha }}\) is addition-separable, i.e., the (ij)-th entry is the arithmetic average of the (ii)-th and (jj)-th entries (Allard et al. 2022). Accordingly, for any \(t \in (0,\infty ),\) \(t^{{\varvec{\alpha }}-1}\) has positive entries and is product-separable, i.e., the (ij)-th entry is the geometric average of the (ii)-th and (jj)-th entries, which entails that it is positive semidefinite (Berg et al. 1984, Chapter 3, Property 1.9). \(\square \)

Lemma 3

Let \({\varvec{a}}=[a_{ij}]_{i,j=1}^p\) be a symmetric conditionally negative semidefinite matrix with nonnegative entries. Then : 

  1. (1)

    for any \(t \le 1,\) \(t^{{\varvec{a}}}\) is positive semidefinite; 

  2. (2)

    for any \(t \ge 1,\) \(t^{-{\varvec{a}}}\) is positive semidefinite; 

  3. (3)

    for any \(t \ge 0,\) \(\left( \frac{1}{1+{\varvec{a}}} \right) ^{t}\) is positive semidefinite.

Proof

Assertions (1) and (2) are a consequence of Theorem 2.2 in Chapter 3 of Berg et al. (1984). Assertion (3) holds for \(t=0;\) for positive t,  one has

$$\begin{aligned} \left( \frac{1}{1+{\varvec{a}}} \right) ^{t} = \left( \frac{1}{(1+{\varvec{a}})^{\frac{t}{\lceil t \rceil }}} \right) ^{\lceil t \rceil }. \end{aligned}$$

The result follows from the application of Corollary 2.10 and Exercise 2.21 in Chapter 3 of Berg et al. (1984) and of the Schur product theorem. \(\square \)

Appendix B: Proofs

Proof of Proposition 1

We start by proving the first set of conditions. Under conditions (1.b) and (1.c), for any \(r>0,\) \([B_{ij}(r)]_{i,j=1}^p\) is the sum of a nonnegative diagonal matrix and a min matrix with nonnegative entries, hence it is positive semidefinite (Horn and Johnson 2013, problem 7.1.P18). So is \([\rho _{ij} \, \kappa _{ij,d}]_{i,j=1}^p\) due to conditions (1.a) and (1.c). The positive semidefiniteness of \({\varvec{A}}(r)\) then stems from the Schur product theorem.

To prove the sufficiency of the second set of conditions, let us use Newton’s generalized binomial theorem (Olver et al. 2010, formula 4.6.7) to rewrite \(A_{ij}(r)\) as

$$\begin{aligned} A_{ij}(r)= & {} \sum _{k=0}^{\infty } \rho _{ij} \, \kappa _{ij,d} \, {\mathbb {I}}_{(b_{ij},\infty )}(r^{-1}) \, (\gamma _{ij})^{\underline{k}} \, \frac{(-b_{ij}^2 \, r^2)^k}{k!} \\= & {} \sum _{k=0}^{\infty } \rho _{ij} \, \kappa _{ij,d} \, {\mathbb {I}}_{(b_{ij},\infty )}(r^{-1}) (\gamma _{ij})^{\underline{2k}} \, (b_{ij}^2 \, r^2)^{2k} \left[ \frac{1}{(2k)!}+\frac{2k-\gamma _{ij}}{(2k+1)!} b_{ij}^2 \, r^2 \right] \\= & {} \sum _{k=0}^{\infty } \rho _{ij} \, \kappa _{ij,d} \, {\mathbb {I}}_{(b_{ij},\infty )}(r^{-1}) \prod _{q=0}^{2k-1} \left( q-\gamma _{ij}\right) (b_{ij}^2 \, r^2)^{2k} \left[ \frac{1}{(2k)!}+\frac{2k-\gamma _{ij}}{(2k+1)!} b_{ij}^2 \, r^2 \right] . \end{aligned}$$

Under conditions (2.a), (2.b) and (2.c), the claim follows from the fact that the set of symmetric positive semidefinite matrices is closed under Schur products, sums and limits.

The third set of conditions is obtained by following the reasoning of Daley et al. (2015, Lemma 2); it ensures that, for all \(r>0,\) \({\varvec{A}}(r)\) is a diagonally dominant matrix with nonnegative diagonal entries, hence positive semidefinite (Horn and Johnson 2013, Theorem 6.1.10). \(\square \)

Proof of Proposition 2

We invoke formula 6.567.1 in Gradshteyn and Ryzhik (2007) to write \(\Omega _{\nu _{ij}}\) as

$$\begin{aligned} \Omega _{\nu _{ij}}(x) = \kappa _{ij,d} \int _{0}^{1} \Omega _{d} (rx) r^{d-1} \big ( 1 - r^2 \big )^{\gamma _{ij}} {\textrm{d}} r, \end{aligned}$$

with \(\kappa _{ij,d}\) defined through (8). Hence, we can write \(\lambda _{ij}\) in (11) as

$$\begin{aligned} \lambda _{ij}(x)= & {} \rho _{ij} \frac{\kappa _{ij,d}}{b_{ij}^{d}} \int _0^1 \Omega _{d} \Big ( r \frac{x}{b_{ij} }\Big ) r^{d-1} \big ( 1 - r^2 \big )^{\gamma _{ij}} \textrm{d } r \nonumber \\= & {} \rho _{ij} \, \kappa _{ij,d} \int _{0}^{b_{ij}^{-1}} \Omega _{d} \big ( v x \big ) v^{d-1} \big ( 1 - b_{ij}^2 v^2 \big )^{\gamma _{ij}} \textrm{d } v \nonumber \\= & {} \rho _{ij} \, \kappa _{ij,d} \int _{0}^{\infty } \Omega _{d} \big ( v x \big ) v^{d-1} \big ( 1 - b_{ij}^2 v^2 \big )_+^{\gamma _{ij}} \textrm{d } v \nonumber \\= & {} \int _{0}^{\infty } \Omega _{d} \big ( v x \big ) v^{d-1} A_{ij} (v) {\textrm{d}} v. \end{aligned}$$
(22)

We note that \(x \mapsto \Omega _{d}(vx)\) belongs to \(\Phi _{d}\) for any \(v > 0.\) Also, by assumption, \({\varvec{A}}(v)\) is symmetric positive semidefinite for each \(v>0.\) Hence, we can invoke Theorem 1 in Porcu and Zastavnyi (2011) to claim that the matrix-valued function

$$\begin{aligned} x \mapsto \Omega _{d} ( v x ) {\varvec{A}} (v), \quad x \ge 0, \end{aligned}$$

belongs to \(\Phi _d^p\) for all \(v > 0.\) The proof is completed by invoking again Theorem 1 in Porcu and Zastavnyi (2011) in concert with the fact that the integral above is well-defined because \(0 \le x \mapsto |\Omega _{d}(x) |\) is uniformly bounded by 1 and the function \(A_{ij}\) is compactly supported, strictly decreasing, and bounded at zero, which ensures integrability. \(\square \)

Proof of Proposition 3

We provide a constructive proof. First, note that the integral in (12) is well-defined because \(v \mapsto A_{ij}(v)\) is compactly supported and the integrand is a continuous function of v. By assumption \(\varphi \) is a member of the class \(\Phi _d.\) Hence, we can invoke Schoenberg’s theorem (Schoenberg 1938) to claim that \(\varphi \) admits a uniquely determined expansion of the type

$$\begin{aligned} \varphi (x)= \int _{0}^{\infty } \Omega _{d}(rx ) F_d({\textrm{d}}r), \quad x \ge 0, \end{aligned}$$
(23)

where \(F_d\) is a d-Schoenberg measure on \([0,\infty ).\) Hence, we have \({\varvec{\psi }}=[\psi _{ij}]_{i,j=1}^p\) with

$$\begin{aligned} \psi _{ij}(x)= & {} \int _{0}^{\infty } \varphi (vx ) v^{d-1} A_{ij}(v) {\textrm{d}} v \nonumber \\= & {} \int _{0}^{\infty } \int _{0}^{\infty } \Omega _{d}(rvx ) F_d({\textrm{d}}r) v^{d-1} A_{ij}(v) {\textrm{d}} v \nonumber \\= & {} \int _{0}^{\infty } \left( \int _{0}^{\infty } \Omega _{d}(rvx ) v^{d-1} A_{ij}(v) {\textrm{d}} v \right) F_d({\textrm{d}}r) \\= & {} \int _{0}^{\infty } \lambda _{{ij}} (rx) F_d({\textrm{d}}r),\nonumber \end{aligned}$$
(24)

which completes the proof because the class \(\Phi _d^p\) is closed in the topology of finite measures, so that scale mixtures provide elements within the same class. The interchange of the integrals in (24) is justified by Fubini’s theorem, insofar as \(|\Omega _{d} |\) is uniformly bounded by 1 and \(A_{ij}\) is continuous and compactly supported, so that

$$\begin{aligned} \int _0^\infty \int _0^\infty |\Omega _{d}(rvx) v^{d-1} A_{ij}(v) |F_d({\textrm{d}}r) {\textrm{d}}v \le \int _0^\infty F_d({\textrm{d}}r) \int _0^{b_{ij}^{-1}} v^{d-1} A_{ij}(v) {\textrm{d}}v < \infty . \end{aligned}$$

\(\square \)

Proof of Proposition 4

We start by evaluating the integral

$$\begin{aligned} \int _{0}^x u \Upsilon _d (\varphi )(u) {\textrm{d}} u= & {} \int _{0}^x u \int _{0}^{\infty } \left( \int _{0}^{\infty } \Omega _{d}(vur) v^{d-1} {\varvec{A}}(v) {\textrm{d}} v \right) F_d({\textrm{d}}r) {\textrm{d}} u \\= & {} \int _{0}^{\infty } v^{d-1} {\varvec{A}}(v) \int _{0}^{\infty } F_d({\textrm{d}} r) \int _{0}^x u \, \Omega _{d} (vur) {\textrm{d}} u \, {\textrm{d}} v \\= & {} \int _{0}^{\infty } v^{d-3} {\varvec{A}}(v) \int _{0}^{\infty } \frac{d-2}{r^2} \Big (\Omega _{d-2}(0)- \Omega _{d-2}(vxr) \Big ) F_{d}({\textrm{d}}r) {\textrm{d}} v, \end{aligned}$$

where the last identity comes from Equation (2.3) in Daley and Porcu (2014). The last inner integrand is everywhere positive for \(x>0.\) When \(F_{d-2} ({\textrm{d}} r) = r^{-2}F_d({\textrm{d}}r)\) is a finite measure on \({\mathbb {R}}_+,\) we can bound the absolute value of the difference in the last inner integral by 2 and use dominated convergence to justify taking the limit for \(x \rightarrow \infty \) there. This proves that \(\int _{0}^{\infty } u \Upsilon _d (\varphi )(u) {\textrm{d}} u \) is well-defined. Also, we notice that \(F_{d-2}\) is the \((d-2)\)-Schoenberg measure associated with the montée of order 2 (sensu Matheron 1965) of \(\varphi ,\) say \(\hat{\varphi }.\) We can now use the previous chain of equalities to write

$$\begin{aligned} {\mathcal {I}} (\Upsilon _d (\varphi ))(x)= & {} \int _{0}^{\infty } v^{d-3} {\varvec{A}}(v) \int _{0}^{\infty } \Omega _{d-2}(vxr) F_{d-2}(\textrm{d } r) {\textrm{d}} v \nonumber \\= & {} \Upsilon _{d-2} ({\hat{\varphi }(x)} ), \end{aligned}$$
(25)

which provides an element of \(\Phi _{d-2}^p\) thanks to Proposition 3. \(\square \)

Proof of Proposition 5

The proof can be made by recursivity on account of (14), (15), (16) and Lemmas 1 and 2 in Appendix A, based on the fact that a mixture of functions belonging to \(\Phi _d^p\) weighted by positive semidefinite matrices still belongs to \(\Phi _d^p,\) insofar as \(\Phi _d^p\) is closed under Schur products, sums and limits. \(\square \)

Proof of Proposition 6

The proposition results from Proposition 2 and the fact that \(\Phi _d^p\) is closed under Schur products, sums and limits. \(\square \)

Proof of Proposition 7

Let \(\nu \) be a positive integer. From the spectral representation of the Matérn covariance in \({\mathbb {R}}^{\nu }\) (Lantuéjoul 2002; Arroyo and Emery 2021), one has

$$\begin{aligned} {{\mathcal {M}}(x;b,\mu ) = \frac{2}{B(\mu ,\frac{\nu }{2})} \int _0^{\infty } \Omega _{\nu } \left( \frac{rx}{b}\right) \frac{r^{\nu -1}}{(1+ r^2)^{\mu +\frac{\nu }{2}}} {\textrm{d}}r, \quad x \ge 0,} \end{aligned}$$

a formula that is actually valid for any \(\nu >0\) (not necessarily an integer) and \(\mu >0\) (Erdélyi 1954, formula 8.5.20). Accordingly, for \({\varvec{\mu }}\) and \({\varvec{\nu }}-d\) with positive entries:

$$\begin{aligned} {\mathcal {M}}(x;{\varvec{b}},{\varvec{\mu }},{\varvec{\sigma }}) = 2 \frac{{\varvec{\sigma }}}{B({\varvec{\mu }},\frac{{\varvec{\nu }}}{2})} \int _0^{\infty } \Omega _{{\varvec{\nu }}} \left( \frac{rx}{{\varvec{b}}}\right) \frac{r^{{\varvec{\nu }}-1}}{(1+ r^2)^{{\varvec{\mu }}+\frac{{\varvec{\nu }}}{2}}} {\textrm{d}}r, \quad x \ge 0. \end{aligned}$$

Under condition (3), \(x \mapsto \frac{{\varvec{\rho }}}{{\varvec{b}}^d} \Omega _{{\varvec{\nu }}} \left( \frac{rx}{{\varvec{b}}}\right) \) belongs to \(\Phi _d^p\) (Proposition 2). Furthermore, owing to Lemma 3 in Appendix A, \((r^2/(1+r^2))^{\frac{{\varvec{\nu }}}{2}}\) and \(({1+r^2})^{-{\varvec{\mu }}}\) are positive semidefinite for all \(r \ge 0\) under conditions (1) and (2). The claim follows from Proposition 6. \(\square \)

Proof of Proposition 8

Let \(\nu \) be a positive integer less than \(2\alpha \). From the spectral representation of the Gauss hypergeometric covariance in \({\mathbb {R}}^{\nu }\) (Emery and Alegría 2022), one has

$$\begin{aligned} {\mathcal {H}}(x;b,\alpha ,\beta ,\gamma ,\nu )= & {} \frac{2^{1-\nu } \Gamma (\alpha ) \Gamma (\beta - \frac{\nu }{2}) \Gamma (\gamma - \frac{\nu }{2})}{\Gamma ( \frac{\nu }{2}) \Gamma (\alpha - \frac{\nu }{2}) \Gamma (\beta ) \Gamma (\gamma )} \\{} & {} \times \int _0^{\infty } \Omega _{\nu } \left( \frac{rx}{b}\right) {r^{\nu -1}} {}_1 F_2\left( \alpha ;\beta ,\gamma ; -\frac{r^2}{4} \right) {\textrm{d}}r, \quad x \ge 0, \end{aligned}$$

a formula that is actually valid for any real value (not necessarily an integer) \(\nu \in (0,2\alpha )\) owing to formulae 3–10 in Emery and Alegría (2022).

We now prove (1). Under the assumption that \({\varvec{\nu }}\) has entries in \((0,2\alpha ),\) one can write:

$$\begin{aligned} {\mathcal {H}}(x;{\varvec{b}},\alpha ,{\beta },{\gamma },{\varvec{\nu }},{\varvec{\sigma }}) = \frac{2 \Gamma (\alpha ) {\varvec{\rho }}}{\Gamma ({\beta }) \, \Gamma ({\gamma }) {\varvec{b}}^{d}} \int _0^{\infty } \Omega _{{\varvec{\nu }}} \left( \frac{rx}{{\varvec{b}}}\right) {r^{{\varvec{\nu }}-1}} {}_1 F_2\left( \alpha ;{\beta },{\gamma }; -\frac{r^2}{4} \right) {\textrm{d}}r, \end{aligned}$$

with \({\varvec{\rho }}\) defined as in (19). The claim then follows from Proposition 6, Lemma 2 and from the fact that, under the specified conditions on \((\alpha ,{\beta },{\gamma }),\) the mapping \(r \mapsto {}_1 F_2( \alpha ;{\beta },{\gamma }; -\frac{r^2}{4})\) is nonnegative on \([0,\infty )\) (Cho et al. 2020).

Concerning (2), one has (Emery and Alegría 2022, Equation 25)

$$\begin{aligned}{} & {} {}_1F_2\left( \alpha ;{\varvec{\beta }},\varvec{\gamma };-\frac{r^2}{4}\right) = \frac{\Gamma ({\varvec{\beta }}) \Gamma (\varvec{\gamma })}{\Gamma (\beta ) \Gamma ({\varvec{\beta }}-\beta ) \Gamma (\gamma ) \Gamma (\varvec{\gamma }-\gamma )}\\{} & {} \quad \times \int _0^1 \int _0^1 {}_1F_2 \left( \alpha ;\beta ,\gamma ;-t_1 t_2 \frac{r^2}{4}\right) t_1^{\beta -1}(1-t_1)^{{\varvec{\beta }}-\beta -1} t_2^{\gamma -1}(1-t_2)^{\varvec{\gamma }-\gamma -1} \text {d}t_1 \text {d}t_2, \end{aligned}$$

with the integrand being a positive semidefinite matrix for any \(t_1, t_2 \in [0,1]\) and \(r \in [0,\infty )\) under conditions (a), (b) and (c) (Lemma 3). The claim follows from Proposition 6 and Lemma 2, which apply under conditions (d) and (e). \(\square \)

Proof of Proposition 9

For \(b>0\) and \(\nu> \mu >0,\) one has (Gradshteyn and Ryzhik 2007, formula 6.576.7)

$$\begin{aligned} {\mathcal {C}}(x;{b},{\mu }) = \frac{2}{\Gamma ({\mu }) \Gamma \left( \frac{\nu }{2}\right) } \int _0^{\infty } \Omega _{{\nu }} \left( \frac{rx}{{b}}\right) \left( \frac{r}{2}\right) ^{{\mu }+\frac{\nu }{2}-1} K_{{\mu }-\frac{\nu }{2}}(r) {\textrm{d}}r, \quad x \ge 0. \end{aligned}$$

Accordingly, under the assumption that \({\varvec{b}},\) \({\varvec{\mu }}\) and \({\varvec{\nu }}-{\varvec{\mu }}\) have positive entries, one has:

$$\begin{aligned} {\mathcal {C}}(x;{\varvec{b}},{\varvec{\mu }},{\varvec{\sigma }}) = \frac{2{\varvec{\sigma }}}{\Gamma ({\varvec{\mu }}) \Gamma \left( \frac{{\varvec{\nu }}}{2}\right) } \int _0^{\infty } \Omega _{{\varvec{\nu }}} \left( \frac{rx}{{\varvec{b}}}\right) \left( \frac{r}{2}\right) ^{{\varvec{\mu }}+\frac{{\varvec{\nu }}}{2}-1} K_{{\varvec{\mu }}-\frac{{\varvec{\nu }}}{2}}(r) {\textrm{d}}r, \quad x \ge 0, \end{aligned}$$

where (Gradshteyn and Ryzhik 2007, formula 3.471.9)

$$\begin{aligned} 2\left( \frac{r}{2}\right) ^{{\varvec{\mu }}+\frac{{\varvec{\nu }}}{2}-1} K_{{\varvec{\mu }}-\frac{{\varvec{\nu }}}{2}}(r) = \left( \frac{r}{2}\right) ^{{\varvec{\nu }}-1} \int _0^{\infty } \exp \left( -\frac{r^2}{4 t} \right) \exp (-t) t^{{\varvec{\mu }}-\frac{{\varvec{\nu }}}{2}-1} {\textrm{d}}t \end{aligned}$$

is, under condition (1), positive semidefinite for all \(r \in [0,\infty )\) owing to Lemma 2 and the fact that positive semidefinite matrices are closed under Schur products, sums and limits. The claim follows from Proposition 6, considering condition (2) and the fact that, from condition (1), the entries of \({\varvec{\nu }}\) are greater than d. \(\square \)

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Emery, X., Porcu, E. The Schoenberg kernel and more flexible multivariate covariance models in Euclidean spaces. Comp. Appl. Math. 42, 148 (2023). https://doi.org/10.1007/s40314-023-02275-0

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