Spatio-Temporal Diffusion (ST-Diff)

• Spatio-Temporal Diffusion (ST-Diff) is a family of probabilistic models that integrate spatial and temporal dependencies via SPDE-driven mechanisms.
• The models employ Gaussian Markov Random Fields and finite element methods to achieve efficient and interpretable representations of complex space–time data.
• Empirical applications, such as global temperature modeling, demonstrate that non-separable covariance structures enhance predictive accuracy and capture nuanced diffusion effects.

Spatio-Temporal Diffusion (ST-Diff) denotes a family of probabilistic models that combine spatial and temporal dependencies through diffusion-like or stochastic partial differential equation (SPDE)-driven mechanisms. In the statistical literature, the archetypal example is the diffusion-based extension of Gaussian Matérn fields, where spatial, temporal, and spatio-temporal dependencies are encoded in the covariance structure via solutions to parametric @@@@1@@@@. These models permit flexible and interpretable representations of space–time data, capturing smoothness, range, and non-separability, and admit efficient implementation as Gaussian Markov Random Fields (GMRF) via finite element methods. Their practical relevance is illustrated in geostatistical and climate applications, such as global temperature field modeling, where capturing both spatial and temporal correlation is critical (Lindgren et al., 2020).

1. Stochastic PDE Formulation and Model Family

The canonical ST-Diff model posits a latent spatio-temporal process $u(s,t)$ on a spatial domain $D\subset\mathbb R^d$ and time $t\in\mathbb R$ as the solution to the fractional SPDE: $$\left(-\gamma_t^2\,\partial_t^2+L_s^{\alpha_s}\right)^{\alpha_t/2}u(s,t) = \dot{\mathcal E}_Q(s,t),$$ where:

• $L_s = \gamma_s^2 - \Delta$ is the damped spatial Laplacian,
• $\alpha_t > 0$, $\alpha_s \ge 0$ govern the temporal and spatial operator orders, respectively,
• $\dot{\mathcal E}_Q$ is Gaussian white noise in time with spatial precision operator $Q = \gamma_e^2L_s^{\alpha_e}$, $\alpha_e \ge 0$,
• $\gamma_s, \gamma_t, \gamma_e > 0$ are scale parameters for space, time, and noise.

For integer $\alpha_t$, the model has an iterated (semi-)differential form: $$\left(\gamma_t \partial_t + L_s^{\alpha_s/2}\right)^{\alpha_t} u(s,t) = \dot{\mathcal E}_Q(s,t).$$ This defines the DEMF$(\alpha_t, \alpha_s, \alpha_e)$ (Diffusion-Extended Matérn Field) family (Lindgren et al., 2020). This formulation accommodates spatially stationary, non-stationary, and manifold-supported fields.

2. Spectral Structure and Covariance Properties

On $D = \mathbb R^d$, the model is stationary in both space and time, and its joint spectral density is: $$S_u(\omega_s, \omega_t) = \frac{1}{(2\pi)^{d+1} \gamma_e^2 [\gamma_t^2 \omega_t^2 + (\gamma_s^2 + \|\omega_s\|^2)^{\alpha_s}]^{\alpha_t} (\gamma_s^2 + \|\omega_s\|^2)^{\alpha_e}},$$ where $\omega_s$ and $\omega_t$ are the spatial and temporal frequencies. The covariance function follows by inverse Fourier transform.

Spatial marginals ($h_t=0$) yield Matérn covariances with smoothness

$$\nu_s = \alpha_e + \alpha_s(\alpha_t - \tfrac{1}{2}) - \frac{d}{2}, \quad r_s = \sqrt{8\nu_s}/\gamma_s,$$

and variance

$$\sigma^2 = \frac{C_{R,\alpha_t} C_{R^d,\alpha} }{ \gamma_s^{2\nu_s} \gamma_e^2 \gamma_t },$$

with constants as defined in (Lindgren et al., 2020). Temporal marginals ($h_s=0$) have smoothness $$\nu_t = \min[\alpha_t - \frac{1}{2}, \nu_s/\alpha_s ]$$. For $\alpha_s = 0$ (separable), temporal marginal is again Matérn, with range $r_t = \sqrt{8(\alpha_t-\frac{1}{2})}/\gamma_t$.

The degree of spatio-temporal non-separability is encoded via

$$\beta_s = 1 - \frac{\alpha_e}{\alpha_e + \alpha_s(\alpha_t-\tfrac{1}{2})},$$

where $\beta_s=0$ denotes separable and $\beta_s=1$ fully non-separable coupling.

3. Interpretability and Parameter Roles

Key parameters and their roles:

• $\sigma^2$: Marginal variance.
• $\nu_s$, $\nu_t$: Numbers of mean-square derivatives, i.e., spatial and temporal smoothness.
• $r_s$, $r_t$: Practical spatial and temporal correlation ranges, as above.
• $\beta_s$: Index of non-separability, controlling how quickly higher spatial frequencies decorrelate over time.

Special cases include classical separable models, critical diffusion (e.g., DEMF(1,2,1), relevant for $\nu_s = \nu_t = 1/2$ in $d=2$), and iterated diffusion (DEMF(2,2,0), giving higher smoothness in both space and time) (Lindgren et al., 2020).

4. Sparse FEM/GMRF Representation and Computation

The latent field is represented in a basis expansion: $$u(s,t) = \sum_{i=1}^{n_s} \sum_{j=1}^{n_t} \psi_i(s)\,\phi_j(t)\,u_{ij},$$ with $\psi_i$ spatial (FEM) and $\phi_j$ temporal (e.g., B-spline) basis functions.

The joint precision matrix $\mathbf{Q}$ for the coefficients $\{u_{ij}\}$ is a sparse sum of Kronecker products: $$\mathbf{Q} = \gamma_e^2 \sum_{k=0}^{2\alpha_t} \gamma_t^{\,k} \bm J_{\alpha_t,k/2} \otimes \bm K_{\alpha_s(\alpha_t-\tfrac k2)+\alpha_e},$$ where matrices capture temporal mass/stiffness ($\bm J$) and spatial stiffness ($\bm K$), ensuring computational tractability for large spatio-temporal domains.

Boundary and stationarity conditions can be handled by adjusting the first/last diagonal blocks, mimicking AR(2)-type processes in time.

5. Implementation and Software

A subset of DEMF models is implemented in R-INLA through the INLAspacetime package with cgeneric interface, leveraging the GMRF framework. The Kronecker-sum structure leads to sparse precision matrices, supporting large-scale inference even with irregular observations. The computational burden remains comparable to separable Matérn–AR(1/2) models; non-separability introduces only modest extra sparsity per node.

The implementation current supports integer $\alpha_t$, local FEM basis in space, linear/quadratic B-splines in time. Extension to fractional $\alpha_t$ and alternative bases is possible but requires further development (Lindgren et al., 2020).

6. Empirical Performance: Global Temperature Modeling

Application to global daily temperature data (GHCN-Daily, 2022) demonstrates the modeling flexibility and improved predictive performance. The response is modeled as: $$y_i = \mu + \alpha E(s_i) + b(s_i, t_i) + v(s_i, t_i) + u(s_i, t_i) + \epsilon_i,$$ where:

• $\mu, \alpha, E(s)$ capture mean and elevation,
• $b$ encodes low-frequency seasonal trends,
• $v$ are seasonal spatial Matérn fields,
• $u$ is the short-term spatio-temporal field (various DEMF variants tested).

Spatial mesh and temporal B-spline bases deliver a discretized latent space $(\sim\!4.5\times10^5)$ in dimension. Penalized complexity (PC) priors and efficient inference with R-INLA allow full posterior analysis.

Results illustrate (i) non-separable models (DEMF(1,2,1), DEMF(2,2,0)) more accurately capture spatial diffusion effects in both in-sample and multi-horizon out-of-sample forecasts; (ii) estimated spatial correlation ranges ($\sim\!2200$ km), temporal ranges ($\sim\!50$ days), and standard deviations ($\sim\!4^\circ{}$C), with stability and accuracy advantages for non-separable models at multi-day horizons (Lindgren et al., 2020).

7. Extensions, Limitations, and Outlook

The DEMF family generalizes to Whittle-Matérn fields and supports curved manifolds (e.g., global geostatistics) and non-stationary fields (by varying coefficients). Parameters separately control spatial and temporal smoothness/range and degree/type of non-separability, with explicit regimes of separability.

A finite element approach endowed with sparse GMRF structure renders the models practical for Bayesian or penalized inference on large networks and grids. The framework is broadly extensible to other spatial–temporal domains given appropriate basis and precision matrix specification.

A limitation is current support for only integer-order $\alpha_t$ and local FEM/temporal bases; fractional differentiation and more general non-stationary or non-Euclidean domains require custom development. The interpretability and flexibility of DEMF-type ST-Diff models make them a principled tool for geostatistical, environmental, and other spatio-temporal data analyses demanding explicit, physics-inspired correlation structure (Lindgren et al., 2020).