[Paper Review] Probability, Statistics and Planet Earth, I: Geotemporal covariances

Ν. H. Bingham, Tasmin L. Symons|arXiv (Cornell University)|Jun 9, 2017

Statistical and numerical algorithms4 references3 citations

TL;DR

This paper establishes the theoretical foundation for geotemporal covariance functions on the sphere cross time (S^d × ℝ), extending classical Bochner-Schoenberg theory to spatio-temporal processes. It proves that valid isotropic geotemporal covariances are characterized by mixtures of products of Gegenbauer (or Legendre in 3D) polynomials in space and positive definite functions in time, enabling rigorous modeling of climate and weather data with proper statistical structure and numerical tractability.

ABSTRACT

The study of covariances (or positive definite functions) on the sphere (the Earth, in our motivation) goes back to Bochner and Schoenberg (1940--42) and to the first author (1969, 1973), among others. Extending to the geotemporal case (sphere cross line, for position and time) was for a long time an obstacle to geostatistical modelling. The characterisation question here was raised by the authors and Mijatović in 2016, and answered by Berg and Porcu in 2017. Extensions to multiple products (of spheres and lines) follows similarly (Guella, Menegatto and Peron, 2016). We survey results of this type, and related applications e.g. in numerical weather prediction.

Motivation & Objective

• To extend classical positive definite function theory on spheres to the geotemporal setting (sphere × time), crucial for modeling atmospheric and oceanic processes.
• To resolve the long-standing challenge of characterizing valid spatio-temporal covariance functions on the Earth’s surface plus time.
• To provide a mathematically rigorous framework for constructing isotropic and stationary geotemporal covariance functions using harmonic analysis on symmetric spaces.
• To support practical applications in numerical weather prediction and climate modeling by enabling the use of finite mixture models with separable structure.
• To unify and generalize existing results on positive definite functions on product spaces, particularly through the Berg-Porcu theorem for S^d × ℝ.

Proposed method

• Adapts the Bochner-Schoenberg theorem to the geotemporal case by combining isotropic spherical harmonics (Gegenbauer polynomials) with positive definite functions on the real line.
• Applies the Schur product theorem to ensure that pointwise products of positive definite functions on S^d and ℝ remain positive definite on the product space.
• Uses the Berg-Porcu theorem (2017) as a key result to characterize all isotropic and stationary geotemporal covariance functions as mixtures of separable components.
• Employs finite mixture models with non-negative weights on Gegenbauer polynomials and time-dependent covariance functions to enable statistical estimation and numerical implementation.
• Leverages block-diagonal structure of large covariance matrices to simplify inversion and reduce computational cost in high-dimensional spatio-temporal data.
• Integrates statistical tools such as the EM algorithm and matrix norms for estimating mixture components, adapting methods from finite mixture models to covariance estimation.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a function on S^d × ℝ to be a valid geotemporal covariance function?
• RQ2 How can isotropic and stationary spatio-temporal covariance structures be constructed on the spherical Earth with time?
• RQ3 What is the role of Gegenbauer polynomials and their connection to spherical harmonics in characterizing geotemporal covariances?
• RQ4 How can the resulting covariance models be practically implemented for large-scale climate and weather data?
• RQ5 In what ways do the separable structures in space and time simplify statistical inference and numerical computation?

Key findings

• The Berg-Porcu theorem provides a complete characterization of isotropic and stationary geotemporal covariance functions on S^d × ℝ as mixtures of products of Gegenbauer polynomials (in space) and positive definite functions on ℝ (in time).
• Valid geotemporal covariances are shown to be expressible as convex combinations of separable kernels, ensuring positive definiteness and enabling statistical modeling.
• The use of Legendre polynomials (for d=2) or Gegenbauer polynomials (for general d) as spatial components ensures compatibility with spherical harmonic decomposition and spectral theory.
• Finite mixture approximations of these infinite mixtures are feasible and computationally tractable, with block-diagonal covariance matrices allowing efficient inversion.
• The framework supports practical applications in numerical weather prediction and climate modeling by providing a mathematically sound basis for spatio-temporal random fields.
• The theory extends naturally to multiple product spaces (e.g., S^d × S^1 × ℝ), allowing modeling of cyclical components like seasonal effects through additional compact factors.