Spatially-Dependent Modelling Framework

• Spatially-dependent modelling frameworks are explicit methods that capture location-indexed dependence by integrating mathematical and algorithmic approaches.
• They incorporate techniques such as random scale mixtures, spatial deformations, and copula-based models to flexibly manage both asymptotic dependence and sub-asymptotic decay in extremes.
• These frameworks enable precise quantification and simulation of complex spatial correlations, proving essential in environmental, neurobiological, and high-dimensional spatio-temporal analyses.

A spatially-dependent modelling framework refers to any mathematical, statistical, or algorithmic methodology in which model components, parameters, or latent variables are explicitly characterized by their dependence on spatial location or, more generally, on space-time coordinates. Such frameworks are foundational in the analysis of multivariate and high-dimensional spatial or spatio-temporal data, enabling the rigorous quantification, simulation, and inference for complex spatial dependence, heterogeneity, and (increasingly) extremes. Spatial dependence is essential to capture spatial correlation, propagate uncertainty, and explain processes ranging from environmental extremes to neurobiological signals.

1. Core Principles and Model Classes

At its core, a spatially-dependent modelling framework is built upon the explicit recognition and mathematical modeling of location-indexed dependencies. The design and classification of such frameworks vary depending on application and phenomena:

• Random scale mixtures: Processes are modeled as mixtures in which spatial or spatio-temporal random fields are multiplied by latent scales, which may also evolve over space or time. For example, the space-time scale mixture

$$X(s,t) = R(t)^{\delta} W(s,t)^{1-\delta}, \quad (s,t) \in S \times T, \quad \delta \in [0,1]$$

represents a flexible mechanism for capturing both asymptotic dependence and independence, with $R(t)$ a temporal global scale (e.g., Pareto-distributed), and $W(s,t)$ an AI process (Dell'Oro et al., 2024). The single parameter $\delta$ controls the transition between dependence classes.

• Spatial deformations: Nonstationary dependencies are handled by warping the spatial domain into a latent space (the D-plane) where the dependence structure becomes stationary and isotropic. A thin-plate spline mapping $f: \mathbb{R}^2 \to \mathbb{R}^2$ is optimized so that the empirical and theoretical extremal-dependence measures (e.g., the $\chi$-statistic) agree as closely as possible in the transformed coordinates (Richards et al., 2021).
• Gaussian process mixtures and copula-based models: By combining Gaussian processes with spatially-dependent random scales, one can flexibly capture local, long-range, and non-stationary tail dependence. Notably, spatial scale-aware models allow both the random scale $R(s)$ and the Pareto tail index $\phi(s)$ to vary smoothly over space, modulated by compactly supported kernels (Shi et al., 2024).
• Hierarchical and factor models: High-dimensional and multivariate data are decomposed into lower-rank spatial factors, which may themselves be spatially indexed Gaussian processes or latent feature models (e.g., the spatially-dependent Indian Buffet Process (Sugasawa et al., 2024)), facilitating scalability and interpretability.

These frameworks capture a wide spectrum of dependence structures—encompassing both asymptotic dependence (AD) and independence (AI), non-stationarity, nonlinearity, and sub-asymptotic decay rates crucial for extremal analysis.

2. Quantification of Spatial and Spatio-Temporal Extreme Dependence

Dependence in tails is a central theme in the analysis of extremes. The upper-tail dependence coefficient

$$\chi = \lim_{u \to 1^-} \frac{P\{X_1 > F^{-1}_{X_1}(u), X_2 > F^{-1}_{X_2}(u)\}}{1-u}$$

discriminates between asymptotic dependence ($\chi>0$) and independence ($\chi=0$), with the Ledford–Tawn residual coefficient $\eta \in (0,1]$ quantifying sub-asymptotic decay

$$\chi(u) \sim \ell\left((1-u)^{-1}\right) (1-u)^{1/\eta - 1}, \quad u \to 1^-.$$

In frameworks such as the spatio-temporal random scale mixture, $\delta$ determines whether the process inherits its extremal class from the global scale process $R(t)$ or the local field $W(s,t)$ (Dell'Oro et al., 2024). If $\delta > 1/2$, extremes are dominated by the temporal scale and the field exhibits asymptotic dependence as inherited from $R(t)$; for $\delta < 1/2$, the spatial field $W(s,t)$ controls the tail regime. The mixture formulation allows regimes with AD in space but AI in time, or vice versa.

For models with spatially varying random scales and tail indices (Shi et al., 2024), dependence classes can be local: a pair of sites $s_i, s_j$ is AD if both have $\phi(s) > \alpha$ and their Wendland supports overlap; otherwise, conditional independence emerges at longer spatial ranges.

Extremal dependence is thus parameterized, interpreted, and inferred within these spatially-dependent frameworks, often via the estimation and interpolation of empirical $\chi(u)$ statistics over spatial lags.

3. Estimation and Inference Approaches

The complex, high-dimensional, and often intractable likelihoods of spatially-dependent extreme value models necessitate advanced inference strategies:

• Simulation-based and likelihood-free methods: For models like the spatial random scale mixture, exhaustive simulation of the generative process under a grid of parameter values is used to generate “images” of summary statistics (empirical $\chi(u)$ over spatial/temporal lags). These serve as input to convolutional neural networks trained to perform point and interval parameter estimation (Dell'Oro et al., 2024). This approach circumvents the computational intractability of full likelihood-based inference in high dimensions.
• Composite and pairwise likelihood: Spatial deformation models optimize the agreement between theoretical and empirical extremal dependence via weighted least squares, often eschewing full likelihood evaluation (Richards et al., 2021). Candidate deformations are selected by comparing composite likelihood information criteria (e.g., CLAIC).
• Bayesian hierarchical and copula frameworks: Joint inference on marginals and dependence is achieved using copula-based Bayesian hierarchies, as in the spatial scale-aware tail model (Shi et al., 2024). Adaptive MCMC algorithms (e.g., random-walk Metropolis within Gibbs) are essential for tractable sampling when latent fields, scale variables, and spatial surfaces are high-dimensional and non-Gaussian.
• Neural network-based estimation: Model-specific summary statistics are mapped to parameter space via neural networks (e.g., CNNs), achieving both computational scalability and data-driven adaptivity to complex dependence structures (Dell'Oro et al., 2024).

Collectively, these strategies facilitate not only parameter recovery in simulation but also uncertainty quantification and model selection for observed spatio-temporal data.

4. Characterization of Nonstationarity and Sub-Asymptotic Behavior

Spatially-dependent modelling frameworks address the prevalence of nonstationarity and locally varying tail behavior, which are unresolved by traditional max-stable or stationary extremes models:

• Spatial deformations remove nonstationarity: By optimizing a bijective map to a latent space where simple, stationary models fit the empirical extremal dependence, the deformation approach provides a practical and interpretable solution to nonstationary extremes (Richards et al., 2021).
• Local mixture/scale-aware extremes: Allowing both the random scale and the tail index to vary over space simultaneously yields models where local patches can be AD or AI, but AI is enforced at long spatial ranges—the observed “patchy” extremal dependence phenotype in weather and environmental data (Shi et al., 2024).
• Flexible copula constructions: Hierarchical mixture models combining global, spatially-constant, and spatially-dependent components enable the modeling of strong tail dependence, sub-asymptotic decay, and identifiability of latent mechanisms (Yadav et al., 2021).

Model validation is typically based on the reproduction of observed $\chi(u)$ decay rates, higher-order extremal statistics, or through held-out predictive performance.

5. Applications and Empirical Performance

Spatially-dependent frameworks have demonstrated utility in various substantive domains:

• Environmental spatio-temporal extremes: Analysis of spring rainfall in the Netherlands finds AD in space but AI in time, recoverable by the spatial random scale mixture model (Dell'Oro et al., 2024).
• Extreme precipitation over continental domains: Spatial scale-aware models applied to US summer precipitation reveal local patches of stronger tail dependence embedded within an overall AI regime—features missed by max-stable or simpler mixture constructions (Shi et al., 2024).
• Spatially non-stationary temperature and precipitation: Spatial deformation models re-map data to achieve stationarity, improving model fit and predictive reliability in applications over topographically complex or heterogeneous regions (Richards et al., 2021).
• Simulation studies: All frameworks confirm, via simulation, their ability to recover key parameters (e.g., $\delta$ in scale mixture; tail index $\phi(s)$ in scale-aware models), correctly discriminate regime boundaries, and yield frequentist coverage close to nominal for inferential summaries.

Empirical model diagnostics are based on graphical comparison of empirical and model-based extremal dependence summaries, higher-order exceedance probabilities, and predictive log-likelihoods on held-out datasets.

6. Comparison, Strengths, and Limitations

The spatially-dependent modelling framework unifies and extends existing methodologies for dependent extremes:

• Strengths: Simultaneous modeling of both AD and AI in space and time, closed-form marginals, accommodation of nonstationarity, flexible inference via neural or simulation-based methods, and compatibility with a variety of empirical summaries derived from observed data.
• Limitations: The computational complexity of simulation- or neural-based inference; intractability of closed-form likelihoods in high dimensions; interpretability may depend on the choice and parametrization of latent models; and reliance on high thresholds for the validity of asymptotic tail approximations.
• Alternatives and extensions: Max-stable and inverted max-stable models remain adequate for stationary and homogeneous extremes but struggle with abrupt or heterogeneous shifts in dependence regime. Conditional extremes frameworks are an alternative for modeling event-based dependence structures but are not, in their standard form, designed for simultaneous, smoothly varying spatial AI/AD transitions.

Spatially-dependent frameworks thus provide the technical and inferential machinery required for the modern analysis of high-dimensional, heterogeneous, and non-stationary spatial extremes and are adaptable to rapidly increasing volumes of environmental and spatio-temporal data (Dell'Oro et al., 2024, Shi et al., 2024, Richards et al., 2021).