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

• Published in: Stochastic environmental research and risk assessment 2022-11, Vol.36 (11), p.3639-3652
• Main Authors: Bonabifard, M. R., Mosammam, A. M., Ghaemi, M. R.
• Format: Article
• Language: English
• Subjects: Aquatic Pollution; Chemistry and Earth Sciences; Computational Intelligence; Computer Science; Earth and Environmental Science; Earth Sciences; Environment; Environmental research; Euclidean space; Hilbert space; Kernel functions; Math. Appl. in Environmental Science; Mathematical functions; Original Paper; Physics; Probability Theory and Stochastic Processes; Risk assessment; Smoothing; Spatial data; Spheres; Spline functions; Statistics for Engineering; Stochastic models; Waste Water Technology; Water Management; Water Pollution Control
• ISSN: 1436-3240; 1436-3259
• DOI: 10.1007/s00477-022-02217-y

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 R d to the sphere, 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.