Karhunen-Loève Expansions for Axially Symmetric Gaussian Processes: Modeling Strategies and $L^2$ Approximations

Alfredo Alegría; Francisco Cuevas-Pacheco

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arXiv · 2007.01961

Karhunen-Loève Expansions for Axially Symmetric Gaussian Processes: Modeling Strategies and $L^2$ Approximations

Alfredo Alegría·Francisco Cuevas-Pacheco

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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.

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Alfredo Alegría, Francisco Cuevas-Pacheco. 2020-07-03. Karhunen-Loève Expansions for Axially Symmetric Gaussian Processes: Modeling Strategies and $L^2$ Approximations. https://arxiv.org/abs/2007.01961

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