Strictly and non-strictly positive definite functions on spheres

By Tilmann Gneiting. 2011

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Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

Subfields: Environmental Engineering, Global and Planetary Change, Statistics and Probability, Signal Processing

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Citations in the corpus

Neural Tangent Kernel: Convergence and Generalization in Neural Networks (2018, background): “The other key ingredient for proving Proposition 2 is the following theorem, which is a slight reformulation of Theorem 1(b) in [8], which itself is a generalization of a classical result of Schönberg: Theorem 3.”
On Sampling Spatially-Correlated Random Fields for Complex Geometries (2019, background): “There are examples of correlation functions, e.g. the square-exponential, which are not positive deﬁnite even with exact geodesic distance: for instance, the sphere with great circle distance [8].”
A class of Matérn-like covariance functions for smooth processes on a sphere (2015, methodology): “Nonetheless, as Gneiting (2013) argues, great circle distance is a physically more natural distance metric on the surface of a sphere, and it is worth developing valid covariance models with great circle distance.”
The Hipparcos–Gaia Catalog of Accelerations (2018, background): “This function is positive definite on the sphere for c > 0 and 0 ν ≤ 0.5 (Gneiting 2013).”
Spatial regression models over two-dimensional manifolds (2016, background): “…e.g., the spherical splines introduced by Wahba (1981) and Baramidze et al. (2006), or the models described by Jun & Stein (2007), Jun (2011) and Gneiting (2013) and methods that work on more general sphere-like domains, such as those of Alfeld et al. (1996) and Lindgren et al. (2011).”
From Schoenberg Coefficients to Schoenberg Functions (2015, methodology): “Statistical and probabilistic communities are especially interested in this class, because the functions C as in Equation (1) are the autocorrelation functions of isotropic Gaussian fields in Sd. Gneiting finishes his essay [12] with a list of open problems contained in [13].”
Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere (2016, methodology): “The names given to each of them (first column) are either parenthetical to the spatial margin, according to Gneiting (2013) (second and fourth entries), or to the coefficients bk in the series expansion in Equation (6).”
Excursion probability of Gaussian random fields on sphere (2014, methodology): “See Huang, Zhang and Robeson [16] and Gneiting [13] for further comments on (2.”