INLA-SPDE for Bayesian Spatio-Temporal Models

• The INLA-SPDE framework is a computational method that combines SPDEs and GMRF approximations to enable scalable Bayesian inference in latent Gaussian models.
• It leverages hierarchical modeling, mesh discretization, and PC priors to recover high-resolution spatio-temporal fields from aggregated observations.
• The approach has been effectively applied to environmental data disaggregation, such as Aerosol Optical Depth estimation in India, demonstrating its robustness and efficiency.

The INLA-SPDE framework provides a computationally efficient methodology for Bayesian inference in latent Gaussian models, where the latent field is modeled as a continuous spatio-temporal Gaussian process governed by stochastic partial differential equations (SPDEs). The approach leverages the equivalence between certain classes of Gaussian processes and discretized Markov random fields to facilitate scalability and analytic tractability. It is particularly well-suited for disaggregation tasks, where coarse areal or temporal aggregates are observed, and the goal is to reconstruct high-resolution estimates of an underlying spatio-temporal phenomenon. This framework is exemplified in the modeling of Aerosol Optical Depth (AOD) in India, where satellite-derived measurements at $0.75^\circ$ spatial and 3-hour temporal accuracy are disaggregated to $0.25^\circ$ and 1-hour resolution, respectively, to support environmental and public health analysis (Avellaneda et al., 9 Nov 2025).

1. Hierarchical Model Specification

The core of the INLA-SPDE framework is a hierarchical model:

1. Latent process: The underlying spatio-temporal field $y(\mathbf{s}, t)$ is modeled as

$$y(\mathbf{s}, t) = \beta_0 + \sum_{k=1}^p \beta_k x_k(\mathbf{s}, t) + z(\mathbf{s}, t)$$

where $x_k(\mathbf{s}, t)$ are covariates, $\{\beta_k\}$ are fixed effects, and $z(\mathbf{s}, t)$ is a zero-mean latent Gaussian process.

1. SPDE for the latent field: The process $z(\mathbf{s}, t)$ is defined as the solution to a spatio-temporal SPDE:

$$\left(\gamma_t \frac{\partial}{\partial t} + L_s^{\alpha_s}\right)^{\alpha_t} v(\mathbf{s}, t) = d\mathcal{E}_Q(\mathbf{s}, t)$$

where $L_s = \gamma_s^2 - \Delta$ is the spatial Helmholtz operator, $d\mathcal{E}_Q$ is a spatially correlated, temporally uncorrelated Gaussian noise, and the exponents $\alpha_s$, $\alpha_t$, $\alpha_e$ modulate spatial/temporal/nonseparability properties.

1. Aggregation for observed data: Observations are spatially and temporally aggregated:

$$Y_{ij} = \beta_0 + \sum_{k=1}^p \beta_k X_k(R_i, T_j) + z(R_i, T_j) + \varepsilon_{ij}, \quad \varepsilon_{ij} \sim N(0, \sigma_\varepsilon^2)$$

with

$$z(R_i, T_j) = \frac{1}{|R_i|\,|T_j|} \int_{R_i} \int_{T_j} z(\mathbf{s}, t) \, dt \, d\mathbf{s}$$

The latent process supports both separable ($\alpha_e=0$) and non-separable ($\alpha_e>0$) spatio-temporal covariance structures. This spatial-temporal white-noise–driven construct captures smooth propagation and is robust to irregular observational supports.

2. Hyperparameter Priors and Covariate Structure

• Fixed effects ($\beta_k$): $N(0, 10^6)$, expressing vague prior knowledge.
• Measurement error precision ($\tau_\varepsilon$): Penalized Complexity (PC) prior with $$P(\sigma_\varepsilon > u_\varepsilon) = \alpha_{\varepsilon}$$, e.g., $u_\varepsilon=1$, $\alpha_\varepsilon=0.01$.
• SPDE hyperparameters (marginal standard deviation $\sigma$, spatial range $r_s$, temporal range $r_t$): Each equipped with independent PC priors: $P(\sigma > u_\sigma) = \alpha_\sigma$, $P(r_s < u_r) = \alpha_r$, and analogous for $r_t$.
• Covariates: In the AOD case, only elevation is considered: $$y(\mathbf{s}, t) = \beta_0 + \beta_1\,\mathrm{elevation}(\mathbf{s}) + z(\mathbf{s}, t)$$. More generally, the $x_k$ may capture temporally or spatially varying predictors.

The use of PC priors (Simpson et al. 2017) provides regularization and interpretability in terms of probabilistic deviation from a base model.

3. Discretization: Mesh and Finite Element Basis

The SPDE is discretized over space and time using a triangulated mesh for the spatial domain ($G$ vertices $\{\mathbf{s}_k\}$) and a regular grid for the temporal domain ($D$ points $\{t_p\}$). A piecewise-linear finite element basis $\psi_{k,p}(\mathbf{s},t)$ is constructed such that

$$z(\mathbf{s}, t) \approx \sum_{k=1}^G \sum_{p=1}^D \psi_{k,p}(\mathbf{s}, t) \mathcal{Z}_{k,p}$$

where the collection $\boldsymbol{\mathcal{Z}} = \{\mathcal{Z}_{k,p}\}$ forms a high-dimensional GMRF with a sparse precision matrix induced by the SPDE operator.

Observation-level aggregation is encoded via the $A$-matrix:

$$z(R_i, T_j) \approx \sum_{k,p} A_{kp}^{ij} \mathcal{Z}_{k,p}$$

where $A_{kp}^{ij}$ reflects the proportion of the support $\left(R_i, T_j\right)$ associated with mesh node $(k, p)$. The resulting mapping from latent nodes to observations is highly sparse and preserves the physical meaning of integration over arbitrary areal supports.

4. Bayesian Inference with INLA-SPDE

The model is cast into the INLA framework:

• Latent vector: $$x = (\beta_0, ..., \beta_p, \boldsymbol{\mathcal{Z}})$$
• Hyperparameters: $\theta = (\tau_\varepsilon, \sigma, r_s, r_t)$

The INLA algorithm proceeds via:

1. Marginal likelihood approximation: $$\pi(\theta \mid Y) \propto \pi(\theta) \pi(Y \mid \theta)$$ via Laplace approximation.
2. Posterior for latent vector: For given $\theta$, approximate $\pi(x \mid Y, \theta)$ Gaussian around its mode.
3. Marginals: Calculate $\pi(\theta_i \mid Y)$, $\pi(x_j \mid Y)$.

The central computational advantage is the exploitation of GMRF sparsity: the precision matrix for $\boldsymbol{\mathcal{Z}}$ is band-sparse, resulting in linear scaling in the number of mesh nodes and enabling practical resolution for large spatio-temporal domains.

5. Simulation Studies and Model Performance

Simulation experiments are conducted on $24 \times 24$ spatial grids and 24 temporal points, subject to varying aggregation coarseness. Both separable ($(\alpha_s, \alpha_e, \alpha_t) = (0,2,1)$) and non-separable ($2,1,1$) SPDE parameter settings are examined. Metrics include RMSE, empirical coverage probability (ECP) for nominal 95% credible intervals, and posterior interval width.

Key findings:

• Error behavior: RMSE increases as aggregation resolution becomes coarser in either space or time.
• Model comparison: The continuous disaggregation (SPDE-INLA) model outperforms areal Besag–AR(1) formulations, especially when the latent process is temporally persistent.
• Uncertainty calibration: Interval coverage and width reflect the known bias–variance tradeoff; high coverage at coarser resolutions is achieved via wider intervals in the posterior.

This suggests that detailed spatio-temporal fields can be more faithfully recovered—even under strong aggregation—when adopting the continuous SPDE construction and mesh discretization.

6. Application: AOD Nowcasting in India

A real-world application involves disaggregating ECMWF AOD$_{550}$ fields from $0.75^\circ \times 0.75^\circ$ grids (1258 cells) every 3 hours to $0.25^\circ$ and hourly resolutions. The procedure employs:

• A mesh with $\sim$1500 spatial nodes and hourly temporal nodes for the 5-day period,
• Non-separable SPDE parameters $(\alpha_t, \alpha_s, \alpha_e) = (1,2,1)$, reparameterized via $(\sigma, r_s, r_t)$,
• A single covariate (elevation) for physical interpretability.

Posterior summaries demonstrate:

• Strong negative effect of elevation on AOD: $\beta_1 \approx -0.08$ per km,
• Spatial range: $r_s \approx 15^\circ$,
• Temporal range: $r_t \approx 6108$ hours.

Hourly, high-resolution posterior means and credible intervals for AOD are obtained, with posterior mean maps exhibiting smooth and physically plausible interpolation across space and time. Exceedance probabilities (e.g., for AOD $> 3.0$) are readily computed, supporting threshold-based environmental monitoring.

7. Significance and Generalization

The INLA-SPDE framework provides a unifying solution for spatio-temporal disaggregation in settings where observations are indirect, aggregated, and noisy. Its computational scalability, flexibility in specifying spatial and temporal dependence, and capacity to handle nonseparable interactions recommend it for use in environmental science, epidemiology, and other domains requiring fine-scale inference from coarse or heterogeneous data (Avellaneda et al., 9 Nov 2025).

A plausible implication is that, as computational infrastructure and mesh generation tools advance, the INLA-SPDE approach will increasingly dominate latent field inference tasks with complex or irregular supports. This framework is robustly justified both theoretically, through its SPDE-GMRF foundations, and empirically, via demonstration in large-scale simulation and real data settings.