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Cross-Covariance Functions for Multivariate Geostatistics

Marc G. Genton, William Kleiber

arXiv:1507.08017v1stat.ME

TL;DR

Multivariate geostatistics needs cross-covariance functions that remain compatible with marginal covariances and yield valid covariance matrices. This review surveys constructions and extensions, evaluates selected models on climate and temperature data, and identifies unresolved theoretical and practical questions.

  • Problem

    Cross-covariance functions must describe dependence among multiple spatial variables while producing a nonnegative definite covariance structure.

  • Method

    The paper reviews coregionalization, convolution, Matérn, nonstationary, space-time, and specialized cross-covariance constructions, then compares selected models using likelihood and cross-validation co-kriging.

  • Results

    The nonstationary extension of the parsimonious Matérn had the largest climate-output log likelihood, improving the next best model by over 1000, while co-kriging improved predictive RMSE and CRPS by 6–7% for observational temperature data.

  • Takeaways & Limitations

    Flexible cross-covariance models support co-kriging and are necessary for simulating multivariate random fields with nontrivial dependencies.

  • Takeaways & Limitations

    Characterizing valid parameter classes, developing multivariate estimation theory, and determining when nontrivial cross-covariances benefit co-kriging remain open problems.

Abstract

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Continuously indexed datasets with multiple variables have become ubiquitous in the geophysical, ecological, environmental and climate sciences, and pose substantial analysis challenges to scientists and statisticians. For many years, scientists developed models that aimed at capturing the spatial behavior for an individual process; only within the last few decades has it become commonplace to model multiple processes jointly. The key difficulty is in specifying the cross-covariance function, that is, the function responsible for the relationship between distinct variables. Indeed, these cross-covariance functions must be chosen to be consistent with marginal covariance functions in such a way that the second-order structure always yields a nonnegative definite covariance matrix. We review the main approaches to building cross-covariance models, including the linear model of coregionalization, convolution methods, the multivariate Matérn and nonstationary and space-time extensions of these among others. We additionally cover specialized constructions, including those designed for asymmetry, compact support and spherical domains, with a review of physics-constrained models. We illustrate select models on a bivariate regional climate model output example for temperature and pressure, along with a bivariate minimum and maximum temperature observational dataset; we compare models by likelihood value as well as via cross-validation co-kriging studies. The article closes with a discussion of unsolved problems.

1. INTRODUCTION

Multivariate geostatistics requires valid cross-covariance functions that jointly describe dependence among spatial variables while preserving nonnegative definiteness. The section introduces stationarity, isotropy, estimation, and structural properties relevant to co-kriging and simulation.

  • A multivariate random field is fully specified, in the Gaussian case, by its mean vector and cross-covariance matrix function.
  • Valid cross-covariance models must produce a nonnegative definite covariance matrix for every collection of locations and variables.
  • Stationarity means covariance functions depend only on the separation vector, whereas isotropy additionally requires invariance under rotations and reflections.
  • Testing structures such as symmetry and separability can guide the choice of cross-covariance functions, which are used for co-kriging and simulation.
  • Cross-covariance matrix functions satisfy transpose symmetry across locations, but cross-covariances themselves need not be symmetric or maximized at zero separation.
  • Separable structures factor spatial correlation from nonspatial variable covariance, while empirical estimators support least-squares, likelihood-based, or Bayesian model fitting.

2. CROSS-COVARIANCES BUILT FROM UNIVARIATE MODELS

The review builds multivariate cross-covariances by combining valid univariate covariance models through coregionalization, convolution, or latent dimensions. These constructions trade interpretability, flexibility, computational cost, and parameter complexity.

  • The main univariate-based constructions are the linear model of coregionalization, convolution methods, and latent-dimensional models.
  • Linear Model of Coregionalization: The LMC represents a multivariate field as a linear combination of independent univariate fields, requiring only valid univariate correlations to be specified.
  • Linear Model of Coregionalization: With many processes, the LMC can become parameter-heavy and difficult to estimate, while its smoothness is limited by the roughest latent process.
  • Convolution Methods: Kernel convolution uses shared latent dependence and can impose strong dependence among processes, with difficult interpretation and frequent numerical integration.
  • Convolution Methods: Covariance convolution often requires numerical integration, although Matérn correlations with common scales yield a closed-form special case of the multivariate Matérn model.
  • Latent Dimensions: Latent-dimensional models assign each variable a point in an augmented space, guaranteeing nonnegative definiteness through a valid univariate covariance there.

3. MAT´ERN CROSS-COVARIANCE FUNCTIONS

The multivariate Matérn extends the widely used univariate Matérn family so that multiple processes can have distinct marginal smoothness and covariance behavior while retaining valid cross-dependence.

  • The Matérn smoothness parameter controls correlation near the origin and mean-square differentiability, while the length scale controls decay at larger distances.
  • The multivariate Matérn permits marginal and cross-correlation structures from the Matérn family, with collocated coefficients measuring same-location dependence.
  • Validity requires parameter conditions linking marginal and cross-covariance smoothnesses, scales, and collocated correlations.
  • Parsimonious and Full Matérn Models: The parsimonious Matérn uses common scales and cross-smoothnesses equal to marginal averages, reducing complexity while retaining distinct marginal smoothnesses.
  • Parsimonious and Full Matérn Models: The full bivariate Matérn allows distinct smoothness and scale parameters, increasing flexibility beyond the parsimonious specification.
  • Experiments reported substantial improvement over independence when models retained distinct marginal covariance structures alongside cross-process correlation.

4. NONSTATIONARY CROSS-COVARIANCE FUNCTIONS

Spatial dependence may vary with terrain, land use, or dynamical environments, motivating nonstationary cross-covariance models whose covariance depends on location pairs rather than only lags. The review discusses parametric extensions and nonparametric estimation, while noting computational and interpretive challenges.

  • Terrain, land use, and prevailing winds can make spatial dependence evolve across a region, which stationary models do not capture well.
  • Nonstationary cross-covariance functions depend on the spatial pair (s1,s2), rather than only the separation vector.
  • Nonstationary Extensions: A nonstationary LMC can replace latent stationary correlations with univariate nonstationary correlations, but this approach had not been implemented according to the review.
  • Nonstationary Extensions: Nonstationary multivariate Matérn models allow variance, smoothness, and length-scale parameters to vary spatially, accommodating longer-range dependence over ocean than land.
  • Nonstationary Extensions: Spatially varying collocated cross-correlation can represent different relationships between variables, such as weaker minimum–maximum temperature dependence in mountainous regions.
  • Nonparametric Approaches: Nonparametric multivariate covariance estimators avoid model choice and can capture nonstationarity, while convolution-based nonstationary methods may require numerical integration and have ambiguous interpretations.

5. CROSS-COVARIANCE FUNCTIONS WITH SPECIAL FEATURES

The section develops specialized cross-covariance constructions for asymmetry, compact support, and spherical domains, while identifying open challenges in nonstationary extensions and large-scale computation.

  • 5.1 Asymmetric Cross-Covariance Functions: Asymmetric cross-covariance models are needed when empirical tests reject the symmetry assumptions of stationary models.
  • 5.1 Asymmetric Cross-Covariance Functions: Shifting each variable by a distinct vector introduces delays that generate asymmetry, while constraints on the shifts ensure identifiability.
  • 5.1 Asymmetric Cross-Covariance Functions: Asymmetric constructions can be applied to LMC and multivariate Matérn models, and simulations and data examples reported prediction improvements when asymmetry was required.
  • 5.2 Compactly Supported Cross-Covariance Functions: Compactly supported multivariate covariance models use scale mixtures and valid matrix-valued functions to induce sparse covariance structures.
  • 5.2 Compactly Supported Cross-Covariance Functions: Multivariate Askey, Buhmann, B-spline, and Wendland constructions provide compact-support extensions, including tapering tools for multivariate Matérn covariances.
  • 5.3 Cross-Covariance Functions on the Sphere: Spherical-domain models address datasets collected across the Earth, while relaxed axial-symmetry extensions for multivariate fields remain an open problem.

6. DATA EXAMPLES

Two bivariate climate applications compare cross-covariance models using likelihood and pseudo cross-validation. The climate-model output favors a nonstationary Matérn by likelihood, whereas observational temperatures show predictive gains from multivariate co-kriging.

  • Data sets: The examples use gridded regional climate-model output and irregularly located minimum- and maximum-temperature observations.
  • 6.1 Climate Model Output Data: Climate-model residuals show smoother temperature, rougher precipitation, similar correlation length scales, and strong negative empirical correlation of −0.67.
  • 6.1 Climate Model Output Data: Six multivariate models are compared with independence, including parsimonious, full, lagged, nonstationary Matérn, and latent-dimensional formulations estimated primarily by maximum likelihood.
  • 6.1 Climate Model Output Data: Temperature and precipitation have estimated smoothness values near 1.3 and 0.55, respectively, with similar length scales and strongly negative cross-correlation in the bivariate Matérn fits.
  • 6.1 Climate Model Output Data: The nonstationary parsimonious Matérn achieves the largest log likelihood, improving the next best model by over 1000.
  • 6.2 Observational Temperature Data: For Colorado temperature observations, pseudo cross-validation repeatedly withholds 25% of locations and evaluates co-kriging with RMSE and CRPS.
  • 6.2 Observational Temperature Data: Predictive RMSE and CRPS improve by 6–7% with the parsimonious lagged Matérn compared with marginal kriging.

7. DISCUSSION

The discussion surveys specialized cross-covariance constructions for long-range dependence, spatio-temporal structure, asymmetry, physical constraints, and computational efficiency. It closes by identifying unresolved theoretical, modeling, and application questions.

  • Specialized constructions: Specialized models address variables with mixed short- and long-range dependence, potentially substantial cross-correlation, and opposing dependence behaviors.These constructions extend multivariate covariance classes such as the Cauchy family.
  • Specialized constructions: SPDE systems provide valid matrix covariances while approximating Gaussian random fields by Gaussian Markov random fields for computationally efficient analysis.Hu et al. applied this approach jointly to temperature and humidity.
  • Spatio-temporal models: Spatio-temporal extensions support stationary cross-covariances, latent-dimension constructions, nonseparability, and asymmetric time delays between variables.The latent-dimension framework can control dependence among space, time, and variables; separate parameters introduce temporal asymmetry and spatial anisotropy.
  • Physics-constrained models: Physics-constrained covariance models incorporate relationships such as geostrophic flow and divergence-free or curl-free vector-field structure.These constructions use physical restrictions or derivatives of specified variograms to preserve specialized field properties.
  • Physics-constrained models: Extending climate-model covariance results from temperature fields to variables such as pressure and wind, including multivariate Matérn cross-covariances, remains open.The cited temperature results include Matérn spatial correlation with smoothness parameter ν = 1 and derivations on a uniform sphere.
  • Open problems: Open problems include characterizing valid cross-covariance classes, determining when nontrivial cross-covariances benefit co-kriging, and developing models for simulation, lattices, and non-Gaussian fields.Further needs include validity conditions for multivariate power exponential models, estimation theory, and connections among splines, co-kriging, and multivariate numerical analysis.
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