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<p>Modern spatial data modeling extends beyond Euclidean domains, accounting for curvature and domain-specific constraints. Motivated by these challenges, this paper explores multivariate random fields on the paraboloid, constructed using an infinite series expansion of Gegenbauer polynomials, which exploits the one-to-one correspondence between the paraboloid and the upper hypersphere. This expansion yields matrix-valued covariance functions that depend on the metric derived from this correspondence. We further investigate how truncating these series leads to practical approximations, quantify the resulting accuracy using appropriate metrics, and demonstrate how these approximations suggest an efficient simulation algorithm, validated through numerical studies.</p>

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Series expansion and fast simulations of multivariate isotropic random fields on the paraboloid

  • 期刊论文

  • 发表日期：

    2025年12月18日

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  • Alfredo Alegría,
  • Galatia Cleanthous,
  • Athanasios G. Georgiadis

摘要

Modern spatial data modeling extends beyond Euclidean domains, accounting for curvature and domain-specific constraints. Motivated by these challenges, this paper explores multivariate random fields on the paraboloid, constructed using an infinite series expansion of Gegenbauer polynomials, which exploits the one-to-one correspondence between the paraboloid and the upper hypersphere. This expansion yields matrix-valued covariance functions that depend on the metric derived from this correspondence. We further investigate how truncating these series leads to practical approximations, quantify the resulting accuracy using appropriate metrics, and demonstrate how these approximations suggest an efficient simulation algorithm, validated through numerical studies.