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.

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Acknowledgements

The Associate Editor and two Referees are gratefully acknowledged for their thorough revisions that allow for a considerably improved version of the manuscript. The research work of Alfredo Alegría was partially supported by Grant FONDECYT 11190686 from the Chilean government.

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Correspondence to Alfredo Alegría.

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Alegría, A., Cuevas-Pacheco, F. Karhunen–Loève expansions for axially symmetric Gaussian processes: modeling strategies and \(L^2\) approximations. Stoch Environ Res Risk Assess 34, 1953–1965 (2020). https://doi.org/10.1007/s00477-020-01839-4

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Keywords

  • Associated Legendre polynomials
  • Covariance functions
  • Isotropy
  • Great-circle distance
  • Longitudinally independent
  • Longitudinally reversible
  • Spherical harmonics