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

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Abstract

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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Data availability

The data are available in the Global Historical Climatology Network (GHCN) in http://www.ncdc.noaa.gov.

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Acknowledgements

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

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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). https://doi.org/10.1007/s00477-022-02217-y

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