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Semi-reproducing kernel Hilbert spaces, splines and increment kriging on the sphere

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The concept of reproducing kernel Hilbert space does not capture the key features of the spherical smoothing problem. A semi- reproducing kernel Hilbert space (SRKHS), provides a more natural setting for the smoothing spline solution. In this paper, we carry over the concept of the SRKHS from the \({\mathbb {R}}^d\) to the sphere, \({\mathbb {S}}^{d-1}\). In addition, a systematic study is made of the properties of an spherical SRKHS. Next, we present the one to one correspondence between increment-reproducing kernels and conditionally positive definite functions and its consequences on spherical optimal smoothing. The smoothing and interpolation issues on the sphere are considered in the proposed SRKHS setting. Finally, a simulation study is done to illustrate the proposed methodology and an analysis of world average temperature from 1963 to 1967 and 1993–1997 is done using the proposed methods.

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We thank the Editor in Chief, the Associate Editor and anonymous reviewers for insightful comments that helped clarify and improve the paper significantly.


The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to A. M. Mosammam.

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Bonabifard, M.R., Mosammam, A.M. & Ghaemi, M.R. Semi-reproducing kernel Hilbert spaces, splines and increment kriging on the sphere. Stoch Environ Res Risk Assess 36, 3639–3652 (2022).

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