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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The data are available in the Global Historical Climatology Network (GHCN) in http://www.ncdc.noaa.gov.
References
Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol 55. US Government Printing Office, Washington
Aronszajn N (1950) Theory of reproducing kernels. Trans Am Math Soc 68(3):337–404
Atteia M (2014) Hilbertian kernels and spline functions. Elsevier, Amsterdam
Banerjee S (2005) On geodetic distance computations in spatial modeling. Biometrics 61(2):617–625
Beatson RK, Zu Castell W et al (2018) Thinplate splines on the sphere. SIGMA 14:1083
Bussberg NW, Shields J, Huang C (2021) Non-homogeneity estimation and universal kriging on the sphere. arXiv preprint arXiv:2107.02871
Chiles J-P, Delfiner P (2009) Geostatistics: modeling spatial uncertainty, vol 497. Wiley, Hoboken
Craven P, Wahba G (1978) Smoothing noisy data with spline functions. Numer Math 31(4):377–403
Cressie N (1993) Statistics for spatial data. Wiley, Hoboken
Freeden W (1984) Spherical spline interpolation—basic theory and computational aspects. J Comput Appl Math 11(3):367–375
Freeden W (1990) Spherical spline approximation and its application in physical geodesy. In: Vogel A (ed) Geophysical data inversion methods and applications. Springer, Berlin, pp 79–104
Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere with applications to geomathematics. Clarendon Press, New York
Gneiting T et al (2013) Strictly and non-strictly positive definite functions on spheres. Bernoulli 19(4):1327–1349
Huang C, Zhang H, Robeson SM (2016) Intrinsic random functions and universal kriging on the circle. Stat Probab Lett 108:33–39
Huang C, Zhang H, Robeson SM, Shields J (2019) Intrinsic random functions on the sphere. Stat Probab Lett 146:7–14
Hubbert S, Lê Gia QT, Morton TM (2015) Spherical radial basis functions, theory and applications. Springer, Berlin
Jeong J, Castruccio S, Crippa P, Genton MG (2017) Statistics-based compression of global wind fields.
Keller W, Borkowski A (2019) Thin plate spline interpolation. J Geod 93(9):1251–1269
Kennedy RA, Sadeghi P, Khalid Z, McEwen JD (2013) Classification and construction of closed-form kernels for signal representation on the 2-sphere. In: Wavelets and sparsity XV, vol 8858. International Society for Optics and Photonics, p 88580M
Kent J, Mardia K (1994) The link between kriging and thin plates probability, statistics and optimization ed fp keller. Biometrika 93:989–995
Levesley J, Light W, Ragozin D, Sun X (1999) A simple approach to the variational theory for interpolation on spheres. In: Muller M (ed) New developments in approximation theory. Springer, Berlin, pp 117–143
Mardia KV, Kent JT, Goodall CR, Little JA (1996) Kriging and splines with derivative information. Biometrika 83(1):207–221
Martinez-Morales JL (2005) Generalized legendre series and the fundamental solution of the laplacian on the n-sphere. Anal Math 31(2):131–150
Matheron G (1973) The intrinsic random functions and their applications. Adv Appl Probab 5:439–468
Meinguet J (1979) Multivariate interpolation at arbitrary points made simple. Z angew Math Phys ZAMP 30(2):292–304
Mosamam A, Kent J (2010) Semi-reproducing kernel hilbert spaces, splines and increment kriging. J Nonparametric Stat 22(6):711–722
Mosammam AM, Mateu J (2018) A penalized likelihood method for nonseparable space-time generalized additive models. AStA Adv Stat Anal 102(3):333–357
Müller C (1966) Lecture notes in mathematics, Spherical harmonics, vol 17. Springer, Berlin
Parker RL (1994) Geophysical inverse theory, vol 1. Princeton University Press, Princeton
Porcu E, Alegria A, Furrer R (2018) Modeling temporally evolving and spatially globally dependent data. Int Stat Rev 86(2):344–377
Porcu E, Bevilacqua M, Genton MG (2016) Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J Am Stat Assoc 111(514):888–898
Shure L, Parker RL, Backus GE (1982) Harmonic splines for geomagnetic modelling. Phys Earth Planetary Interiors 28(3):215–229
Taijeron H, Gibson AG, Chandler C (1994) Spline interpolation and smoothing on hyperspheres. SIAM J Sci Comput 15(5):1111–1125
Thomas-Agnan C et al (1991) Spline functions and stochastic filtering. Ann Stat 19(3):1512–1527
Wahba G (1981) Spline interpolation and smoothing on the sphere. SIAM J Sci Stat Comput 2(1):5–16
Wahba G (1982) Erratum: spline interpolation and smoothing on the sphere. SIAM J Sci Stat Comput 3(3):385–386
Wahba G (1990) Spline models for observational data. In: SIAM
Wang H, Sloan IH (2017) On filtered polynomial approximation on the sphere. J Fourier Anal Appl 23(4):863–876
Wendelberger JG (1981). The computation of laplacian smoothing splines with examples. Technical report, Wisconsin Univ-Madison Dept Of Statistics
Whaler K, Gubbins D (1981) Spherical harmonic analysis of the geomagnetic field: an example of a linear inverse problem. Geophys J Int 65(3):645–693
Xu Y, Cheney EW (1992) Strictly positive definite functions on spheres. Proc Am Math Soc 116:977–981
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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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DOI: https://doi.org/10.1007/s00477-022-02217-y