Karhunen–Loève expansions for axially symmetric Gaussian processes: modeling strategies and $$L^2$$ approximations

Abstract

Axially symmetric processes on spheres, for which the second-order dependency structure may substantially vary with shifts in latitude, are a prominent alternative to model the spatial uncertainty of natural variables located over large portions of the Earth. In this paper, we focus on Karhunen-Loève expansions of axially symmetric Gaussian processes. First, we investigate a parametric family of Karhunen-Loève coefficients that allows for versatile spatial covariance functions. The isotropy as well as the longitudinal independence can be obtained as limit cases of our proposal. Second, we introduce a strategy to render any longitudinally reversible process irreversible, which means that its covariance function could admit certain types of asymmetries along longitudes. Then, finitely truncated Karhunen-Loève expansions are used to approximate axially symmetric processes. For such approximations, bounds for the L 2-error are provided. Numerical experiments are conducted to illustrate our findings.

Strict positive definiteness of convolutional and axially symmetric kernels on d-dimensional spheres

“…The discussion in Section 2 allows to construct kernels with different geometric properties. Even though these kernels could only be constructed approximately as truncated series we believe that they can nevertheless be used in applications as discussed in [2] for axially symmetric kernels. We believe that taking into account a wider range of geometric properties of the kernel will allow the improvement of approximation results for many areas of application.…”

Point pattern analysis and classification on compact two–point homogeneous spaces evolving time

“…Assume that {N t (•), t ∈ T } defines a spatiotemporal Cox process with random log-intensity log(X t ), whose infinite-dimensional marginals have characteristic functional (1). The n-dimensional micro-scale behavior of the random point pattern is then characterizes by its n-order product density ρ…”

“…The interest of the extended family of Cox processes analyzed here relies on well-known examples of compact two-point homogeneous spaces like the sphere S d ⊂ R d+1 , and the projective spaces over different algebras (see Section 2 in [22] for more details). Recent advances on modeling, analysis and simulation of Gaussian spherical isotropic random fields, including random fields obeying a fractional stochastic partial differential equation on the sphere, can be exploited in our more general L 2 (M d , dν)-valued Gaussian log-risk process framework (see [1]; [3]; [14]; [21], among others). Particularly, [3] and [21] focalize on Cosmic Microwave Background (CMB) evolution modeling and data analysis.…”

Point pattern analysis and classification on compact two-point homogeneous spaces evolving time

“…2 in Ma and Malyarenko ( 2020 ), for more details). Recent advances on modeling, analysis and simulation of Gaussian spherical isotropic random fields, including random fields obeying a fractional stochastic partial differential equation on the sphere, can be exploited in our more general -valued Gaussian log-risk process framework (see Alegría and Cuevas-Pacheco 2020 ; Anh et al. 2018 ; Emery and Porcu 2019 ; Cleanthous et al.…”