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Isotropic Variogram Matrix Functions on Spheres

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Abstract

This paper is concerned with vector random fields on spheres with second-order increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ 2, log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization.

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Acknowledgements

The authors wish to thank the Editor-in-Chief, an associate editor, two reviewers and Dr. James Stapleton for their valuable comments and suggestions which helped to improve the presentation of this paper. Ma’s work is supported in part by US Department of Energy under Grant DE-SC0005359.

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Appendices

Appendix A: Proof of Theorem 1

Suppose that γ(ϑ) is the variogram matrix function of an m-variate intrinsically stationary and isotropic elliptically contoured random field on \(\mathbb{S}^{d}\). Its direct variogram γ kk (ϑ) is the variogram of kth component \(\{ Z_{k}(\mathbf{x}), \mathbf{x}\in\mathbb{S}^{d} \}\), k∈{1,…,m}, and, by Eq. (7), there exists a summable sequence of nonnegative numbers {b kk;n , n=1,2,…} such that

$$ \gamma_{kk}(\vartheta) = \sum _{n=0}^\infty b_{kk;n} \bigl\{ 1-p^{ ( \frac{d-1}{2} ) }_n (\cos\vartheta) \bigr\}, \quad \vartheta\in[0, \pi]. $$
(18)

Similarly, we can apply Eq. (7) to any linear combination of component random fields, say \(\{ \sum_{k=1}^{m} a_{k}Z_{k}(\mathbf{x}), \ \mathbf{x}\in \mathbb{S}^{d}\}\), which is a univariate random field on \(\mathbb{S}^{d}\) with the variogram

(19)

Specially, for any ij, we take a i =a j =1, a k =0, ki,j. The corresponding variogram possesses an expansion

$$ \gamma_{ii}(\vartheta)+\gamma_{jj}( \vartheta)+ 2\gamma_{ij}(\vartheta)= \sum_{n=0}^\infty c^+_{ij;n} \bigl\{ 1-p^{ ( \frac{d-1}{2} ) }_n (\cos\vartheta) \bigr \}. $$
(20)

By taking a i =−a j =1, a k =0, ki,j, we have the following expansion

$$ \gamma_{ii}(\vartheta)+\gamma_{jj}( \vartheta)- 2\gamma_{ij}(\vartheta)= \sum_{n=0}^\infty c^-_{ij;n} \bigl\{ 1-p^{ ( \frac{d-1}{2} ) }_n (\cos\vartheta) \bigr \}. $$
(21)

This subtracted from Eq. (20) yields

$$ \gamma_{ij;n}= \sum_{n=0}^\infty b_{ij;n} \bigl\{ 1-p^{ ( \frac{d-1}{2} ) }_n (\cos\vartheta) \bigr \}, $$
(22)

where \(b_{ij;n}=\frac{1}{4}(c^{+}_{ij;n}-c^{-}_{ij;n})\), n=0,1,…; i,j=1,2,…,m, which is summable. Letting B n =(b ij;n ) m×m , we showed the existence of the expansion of the format in Eq. (12). It remains to show that the matrices B n are positive definite for the necessity part. It can be obtained by combining Eqs. (7) and (19). In fact, based on the variogram expansion, Eq. (19) equals

$$\sum_{n=0}^\infty\sum _{i=1}^m\sum_{j=1}^m a_ia_j b_{ij,n} \bigl\{ 1-p^{ ( \frac{d-1}{2} ) }_n (\cos\vartheta) \bigr\}, $$

and the coefficients \(\sum_{i=1}^{m} \sum_{j=1}^{m} a_{i}a_{j} b_{ij,n}\) must be nonnegative by applying Remark 1 of Schoenberg (1942). Namely, B n (n=0,1,…) are positive definite. Conversely, suppose that {B n , n=0,1,…} is a summable sequence of positive definite matrices that entails Eq. (10). Then the difference between \(\sum_{n=0}^{\infty}\mathbf{B}_{n}\) and γ(ϑ) determines a well-defined m×m matrix function

By Theorem 1 of Ma (2012), C(ϑ) is a covariance matrix function on \(\mathbb{S}^{d}\), and, moreover, its associated variogram matrix function is γ(ϑ), by Eq. (9).

Appendix B: Proof of Theorem 3

For (i), using the Taylor expansion of function g(x) given by Eq. (4) with a=∞, we rewrite Eq. (13) as

$$\gamma_{ij}(\vartheta)=\sum_{n=0}^\infty \frac{g^{(n)} (0+)}{n!} b_{ij}^n \bigl(1-\cos^n( \vartheta) \bigr). $$

Then Theorem 3(i) follows from Corollary 2 with \(b(n)=\frac{g^{(n)} (0+)}{n!}\) and B 0 =g(0)1, where 1 is an m×m matrix with all entries equal to 1. Similarly, we can show Theorem 3(ii) and (iii) using the expansion of Eq. (4) with a=1, noting that the choice of range of b ij needs to be within the domain supporting the identity of Eq. (4).

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Du, J., Ma, C. & Li, Y. Isotropic Variogram Matrix Functions on Spheres. Math Geosci 45, 341–357 (2013). https://doi.org/10.1007/s11004-013-9441-x

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