Connecting reflective asymmetries in multivariate spatial and spatio-temporal covariances

TL;DR

This work introduces reflective asymmetric covariances to construct asymmetric multivariate spatial and univariate space-time covariances via spectral-domain multipliers, yielding a two-term decomposition that splits a symmetric base from an asymmetric component. The authors develop closed-form and semi-closed forms for squared-exponential, Cauchy, and Matérn cross-covariances (in various dims) and extend these ideas to univariate space-time via $f(\mathbf{x},\eta) = \sigma f^{*}(\|\mathbf{x}\|,|\eta|) [1 + \xi \operatorname{sign}(\langle \mathbf{x}, \tilde{\mathbf{x}}\rangle)\operatorname{sign}(\eta)]$, with separable-type and fully nonseparable Gneiting-style variants. They demonstrate improved model fit and prediction on simulated data and the Irish wind data, and provide practical tools including efficient spectral simulation, nonstationary/anisotropic extensions, and Vecchia-based inference for large datasets. The approach offers a parsimonious alternative to Lagrangian models and enables straightforward likelihood-based symmetry testing. Limitations include incomplete closed-form Matérn cross-covariances in higher dimensions and opportunities to further integrate Lagrangian and reflective frameworks for even greater flexibility.

Abstract

In the analysis of multivariate spatial and univariate spatio-temporal data, it is commonly recognized that asymmetric dependence may exist, which can be addressed using an asymmetric (matrix or space-time, respectively) covariance function within a Gaussian process framework. This paper introduces a new paradigm for constructing asymmetric space-time covariances, which we refer to as "reflective asymmetric," by leveraging recently-introduced models for multivariate spatial data. We first provide new results for reflective asymmetric multivariate spatial models that extends their applicability. We then propose their asymmetric space-time extension, which come from a substantially different perspective than Lagrangian asymmetric space-time covariances. There are fewer parameters in the new models, one controls both the spatial and temporal marginal covariances, and the standard separable model is a special case. In simulation studies and analysis of the frequently-studied Irish wind data, these new models also improve model fit and prediction performance, and they can be easier to estimate. These features indicate broad applicability for improved analysis in environmental and other space-time data.

Figures (12)

• Figure 1: Squared-exponential, Cauchy, and Matérn cross-covariances $\Re(\sigma_{jk})C_{jk}^{\Re}({\boldsymbol{h}}) + \Im(\sigma_{jk})C_{jk}^{\Im}({\boldsymbol{h}})$ in $d=2$. In all cases, we take $a=1$, $\tilde{{\boldsymbol{x}}} = (1,1)^\top / \sqrt{2}$. On the top row, we plot $C_{jk}^{\Im}({\boldsymbol{h}})$, while the bottom row takes $\Re(\sigma_{jk}) = \Im(\sigma_{jk}) = 1/2$.
• Figure 2: Separable-type covariances for $d=1$ and plots based on squared-exponential (spatial) and Cauchy (temporal) model with $a_s = a_t = 1$, $\alpha = 1/2$, and $\sigma = 1$. (A) The symmetric case $\xi = 0$, (B) the asymmetric part only when taking $\xi = 1$ for illustration, (C) $\xi = 0.4$, (D) $\xi = -0.9$.
• Figure 3: Cross-sections of the separable-type covariances for $d=2$ using the Cauchy (spatial) and Matérn (temporal) model with $a_{{\boldsymbol{s}}} = a_t = 1$, $\alpha = 1$, $\nu = 1$, $\sigma = 1$, $\tilde{{\boldsymbol{x}}} = (1,1)^\top / \sqrt{2}$ and $\xi = -0.9$. (A) Covariance as a function of $u$ for varying ${\boldsymbol{h}} = (h_1, h_2)^\top$, with the symmetric model $\xi = 0$ also plotted in dashed red; (B) covariance as a function of ${\boldsymbol{h}}$ for varying $u$.
• Figure 4: Gneiting-type covariances as in Example \ref{['ex:sq_exp_gneiting_separability']} with separability parameter $b$ and asymmetry parameter $\xi$ for $d=1$ with $a_s = a_t = 1$, $\sigma = 1$, and $\delta = 1$.
• Figure 5: Simulation at different $({\boldsymbol{s}} \in \mathbb{R}^2,t)$ on a grid of $301 \times 301 \times 12$. The spatial covariance is Matérn with $\nu_{{\boldsymbol{s}}} = 1$ and $a_{{\boldsymbol{s}}} = 1$. The temporal covariance is Matérn with $\nu_t = 2$ and $a_t = 1$. The other parameters are $\sigma = 1$, $\xi = 0.9$, $\tilde{{\boldsymbol{x}}} = (1,0)^\top$. The proposals were Matérn spectral densities with $\nu = 0.5$ and $a = 1$, and $L = 30{,}000$.

Theorems & Definitions (33)

• Theorem 1: Simplified multivariate Bochner
• Definition 1: Multivariate symmetry
• Theorem 2: Simplified Bochner, space-time covariances
• Definition 2: Separability
• Definition 3: Full symmetry
• Definition 4: Axial symmetry
• Proposition 1
• Proposition 2
• Proposition 3
• Example 1: Special case $d=1$