Asymmetric Space-Time Covariance Functions

• Asymmetric space-time covariance functions are statistical models that capture nonseparable, directional dependencies in spatial and temporal data, especially under transport phenomena like advection.
• They are constructed using approaches such as Lagrangian frameworks, hierarchical mixtures, and coregionalization with temporal shifts, ensuring valid and flexible covariance modeling.
• These models enhance empirical performance in fields like atmospheric science and climate modeling by addressing issues such as the dimple problem and nonstationarity.

Asymmetric space-time covariance functions constitute a fundamental class of models in spatial and spatiotemporal statistics, enabling representation of physically realistic dependence in systems where the assumption of space-time symmetry or stationarity is inappropriate. These constructions model statistical dependence structures in multivariate or univariate random fields indexed over space and time, accounting for nonseparability, directionality, and transport phenomena. Asymmetry arises naturally in many domains such as atmospheric and oceanographic sciences, geophysics, and climate modeling, often due to transport (advection), propagation, or causality constraints. Contemporary research has yielded a diverse array of principled approaches—ranging from hierarchical mixtures and coregionalization, to Lagrangian frameworks and designer kernels—that rigorously address validity, flexibility, and interpretability.

1. Definition and General Formulation

Let $Z(\mathbf{x}, t)$ be a (possibly multivariate) mean-zero random field defined on $\mathbb{R}^d \times T$ with $T \subseteq \mathbb{R}$. The space-time covariance function is $$C\bigl( (\mathbf{x}, t), (\mathbf{x}', s) \bigr) = \mathbb{E}\bigl[ Z(\mathbf{x}, t) Z(\mathbf{x}', s) \bigr]$$. In classic stationary/isotropic models, the covariance is a function of spatial lag $\mathbf{h} = \mathbf{x} - \mathbf{x}'$ and temporal lag $u = t - s$, and exhibits the symmetry $C(\mathbf{h}, u) = C(\mathbf{h}, -u)$. Asymmetric space-time covariances, by contrast, encode directionality with respect to time (or space), so that $C(\mathbf{h}, u) \neq C(\mathbf{h}, -u)$ in general (Alegria et al., 2016).

For a valid (positive-definite) covariance $C$, any finite collection of points produces a nonnegative-definite covariance matrix. For the multivariate case, the cross-covariances

$$C_{ij}(\mathbf{h}, u) = \operatorname{Cov}\big\{ Z_i(\mathbf{x} + \mathbf{h}, t + u),\; Z_j(\mathbf{x}, t) \big\}$$

must ensure nonnegative definiteness when assembled into a block matrix (Genton et al., 2015).

2. Principal Constructions for Asymmetric Space-Time Covariance

Several major modeling strategies have been developed to explicitly construct asymmetric space-time covariances, each providing specific interpretability and flexibility.

2.1 Lagrangian (Transport) Models

Models motivated by the Lagrangian framework incorporate physically-derived mechanisms such as advection or transport by prevailing flows. Define a stationary spatial field $Y(\mathbf{x})$ and a random velocity vector $\mathbf{V}$; the Lagrangian field is $Z(\mathbf{x}, t) = Y(\mathbf{x} - t\mathbf{V})$. The resulting space-time covariance is

$$C(\mathbf{h}, u) = \mathbb{E} \big[ C_S(\mathbf{h} - u\mathbf{V}) \big],$$

with $C_S$ the spatial covariance (Alegria et al., 2016, Ma, 11 Nov 2025). Asymmetry in time results whenever the law of $\mathbf{V}$ is not symmetric.

2.2 Hierarchical Mixture Models

The hierarchical mixture framework introduces asymmetry by (i) randomizing over velocity vectors (location mixing) and (ii) integrating over hyperparameters such as range or smoothness (scale mixing). The general construction is: $$K(\mathbf{h}, u) \propto \int_0^\infty \bigg[ \int_{\mathbb{R}^d} C_S(\mathbf{h} - \mathbf{v}u;\rho) f(\mathbf{v}|\rho) d\mathbf{v} \bigg] g(\rho) d\rho,$$ where $f$ and $g$ are suitable mixing densities (Ma, 11 Nov 2025). Explicit closed forms are available for many choices, including Matérn and Cauchy class kernels.

2.3 Coregionalization with Temporal Shifts

In the multivariate setting, the linear model of coregionalization (LMC) is extended with variable-specific temporal delays: $$C_{ij}(\mathbf{h}, u) = \sum_{k=1}^K A_{ik} A_{jk} C_k\big(\mathbf{h},\; u+\tau_{ik}-\tau_{jk}\big),$$ where $\tau_{ik}$ encodes component- and factor-specific time lags, shifting the cross-covariances asymmetrically in time (Genton et al., 2015).

2.4 Nonstationary Designer Kernels on the Half-Line

Nonstationary "designer" kernels restricted to $[0, \infty)$ eliminate time-translation invariance at the kernel level. Using Laguerre polynomial expansions and Mercer’s theorem, a temporal covariance of the form

$$K_\ell(t,s) = \frac{\Gamma(\alpha+1)}{(1 - 2\delta)^{\alpha+1}}(ts\omega)^{-\alpha/2}\exp\!\left[-\left(\delta + \frac{\omega}{1-\omega}\right)(t + s)\right] I_\alpha\!\left(\frac{2\sqrt{ts\omega}}{1 - \omega}\right)$$

is obtained, where asymmetry in $t$ and $s$ arises inherently from the half-line domain and parameter configuration (McCourt et al., 2018).

3. Analytical Properties and Positive-Definiteness

The validity (positive-definiteness) of these constructions is critical.

• In Lagrangian and convolution models, positive-definiteness follows from the mixture of positive-definite spatial kernels along shifted paths, under integrability of mixing distributions (Alegria et al., 2016, Ma, 11 Nov 2025).
• For coregionalization with temporal shifts, positive-definiteness is guaranteed if the underlying latent processes and weight matrices are chosen to ensure block matrix positivity (Genton et al., 2015).
• The designer kernel construction leverages orthonormal bases (Laguerre polynomials) with strictly positive, summable Mercer eigenvalues to ensure valid kernels on $[0, \infty)$ (McCourt et al., 2018).

Parameter identifiability and constraints—such as zero mean constraints on temporal lags or bounds on correlation matrices—are necessary to obtain interpretable and valid models.

4. Characterization of Asymmetry and Phenomena: The Dimple Problem

Asymmetric space-time covariances often display phenomena absent in symmetric models. Of central interest is the "dimple problem": for some fixed spatial lags, the covariance, as a function of temporal lag, has a local minimum at lag zero—reflecting that features may be more correlated at a temporally shifted position than at the same instant, consistent with advection-dominated regimes (Alegria et al., 2016).

Analytic criteria for the emergence of dimples involve second-order derivatives of the spatial generator and the distribution of the velocity field. For instance, under isotropy,

$$\Delta C_S(\mathbf{h}) = \phi''(r) + \frac{d-1}{r}\phi'(r)$$

determines the occurrence of dimples via the sign change for subsets of spatial lags. Similar tests are derived for spherical domains.

5. Flexible Model Classes and Closed-Form Examples

The hierarchical mixture and Lagrangian frameworks admit extremely flexible and explicit closed-form covariances. The Lagrangian Matérn and confluent hypergeometric (CH) families, constructed by specific choices of base spatial kernel, velocity mixing, and range mixing, yield covariances such as

$$K(\mathbf{h}, u) = \sigma^2 |\mathbf{I}_d + u^2\Sigma|^{-1/2} \mathcal{M}(h_u;\nu, \phi )$$

with

$$h_u := \sqrt{ (\mathbf{h} - u\boldsymbol\mu)^\top (\mathbf{I}_d + u^2\Sigma)^{-1} (\mathbf{h} - u\boldsymbol\mu) }$$

and $\mathcal{M}$ the Matérn correlation. When further mixing over the scale parameter, one obtains heavy-tailed (long-range dependent) classes (Ma, 11 Nov 2025).

Nonstationary kernels on the half-line, such as those built using Laguerre polynomials, allow modeling of scenarios where time begins at a special origin (e.g., initial condition problems), with covariance behavior sharply distinct near $t=0$ and fully asymmetric globally (McCourt et al., 2018).

6. Practical Applications and Empirical Performance

Asymmetric space-time covariance functions have demonstrated empirical superiority in applications involving directional transport and temporally structured dynamics. Illustrations include:

• Irish wind-speed data: Lagrangian Matérn and CH models outperformed symmetric Gaussian models by >30,000 log-likelihood units, capturing smoother yet long-tailed behavior (Ma, 11 Nov 2025).
• U.S. daily temperature: empirical semivariograms showed strong asymmetry not captured by symmetric models; Lagrangian Matérn and CH yielded improved fit and predictive accuracy.
• Regional climate model residuals (bivariate temperature and precipitation): LMCs with lagged cross-terms and transport convolution models produced marked improvements in log-likelihood and cross-validation RMSE/CRPS over symmetric alternatives (Genton et al., 2015).
• Observational temperature (minimum/maximum): asymmetric lagged Matérn models improved RMSE and CRPS by 6–7% in cross-validation.

7. Extensions and Theoretical Developments

Contemporary work has characterized general conditions for positive-definiteness under arbitrary isotropic base kernels (Theorem 3, (Ma, 11 Nov 2025)), enabling essentially any standard spatial covariance (Matérn, Cauchy, powered-exponential, etc.) to enter this hierarchy with parametrically tunable asymmetry. This advances and unifies earlier Lagrangian models that were previously limited to Gaussian cases, and extends to nonstationary, nonseparable, and physically-constrained variants.

These general model classes, together with analytic tools for characterizing phenomena such as the dimple effect and rigorous identifiability conditions, now provide a systematic foundation for constructing, fitting, and interpreting asymmetric space-time dependence in geostatistics, environmental sciences, and related multivariate fields.

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Key references:

• "Cross-Covariance Functions for Multivariate Geostatistics" (Genton et al., 2015)
• "A Nonstationary Designer Space-Time Kernel" (McCourt et al., 2018)
• "Asymmetric Space-Time Covariance Functions via Hierarchical Mixtures" (Ma, 11 Nov 2025)
• "The dimple problem related to space-time modeling under the Lagrangian framework" (Alegria et al., 2016)