Hierarchical LGCPs: Multi-Scale Spatial Models

• Hierarchical LGCPs are extended spatial models that incorporate latent segmentation and process dependencies to capture complex heterogeneity.
• They employ advanced Bayesian inference methods such as MCMC, Laplace approximations, and SPDE-based discretizations to ensure computational efficiency.
• These models have practical applications in ecology and forestry, providing improved bias correction and sharper spatial segmentation in multi-scale analyses.

A hierarchical Log-Gaussian Cox Process (LGCP) is an extension of the classical LGCP framework, introducing additional latent structure or dependence between layers of spatial point processes. This framework allows for flexible modeling of spatial or spatiotemporal point patterns exhibiting complex heterogeneity, interactions, or region-specific behavior. Hierarchical LGCPs are prominently utilized for multi-scale or multi-type point patterns, latent spatial segmentation, and conditional dependencies between distinct sets of events. Major recent advances include the Level Set Cox Process (LSCP) for automatic spatial segmentation and models coupling multi-type processes with one-way influence, such as those used for regeneration analysis in forests (Hildeman et al., 2017, Kuronen et al., 2020).

1. Model Construction and Hierarchical Structures

Two principal constructions of hierarchical LGCPs have been advanced. The first class uses a spatial mixture framework to capture fundamentally different behaviors in spatial subdomains. For the LSCP, a latent continuous Gaussian random field $Z(s)\sim GP(m(s),K(s,t))$ is segmented into $K$ classes via ordered thresholds $-\infty = u_0 < u_1 < \ldots < u_K = \infty$: $C(s) = \sum_{k=1}^K k\, 1_{[u_{k-1},u_k)}(Z(s))$ Each region then follows its own independent Gaussian field $Y_k(s)\sim GP(\mu_k(s),\mathrm{Cov}_k(s,t))$, and intensity is defined as $\lambda(s)=\exp\{Y_{C(s)}(s)\}$ (Hildeman et al., 2017).

The second principal class arises in bivariate settings, capturing directed dependencies such as the effect of a “parent” point process $X$ on a “child” process $Y$. Conditionally on $X$, $Y$ is modeled as an LGCP with intensity: $$\Lambda(s;\beta_0,\beta_1,\theta,x,Z) = \exp\left\{ \beta_0 + \beta_1 C(s;\theta,x) + Z(s) \right\}$$ where $C(s;\theta,x)$ is an “influence field” derived from superposed kernels emitted by $X$ (Kuronen et al., 2020). This allows for modeling spatial dependence where one process modifies the environment for another.

2. Covariance Structures and Prior Specification

In both frameworks, the covariance structure of Gaussian fields is typically Matérn, which can flexibly control smoothness, marginal variance, and effective range. The Matérn kernel for class $k$ is: $$\mathrm{Cov}_k(s,t) = \sigma_k^2 \frac{2^{1-\nu_k}}{\Gamma(\nu_k)} (\kappa_k\lVert s-t \rVert)^{\nu_k} K_{\nu_k}(\kappa_k \lVert s-t \rVert)$$ with $\sigma_k > 0$ (variance), $\nu_k > 0$ (smoothness), $\kappa_k > 0$ (scale), and $K_\nu$ the modified Bessel function.

Priors for hyperparameters are hierarchical: thresholds $u_k$ use ordered Gaussians or diffuse uniforms; field variance and range use exponential or gamma priors subject to resolution constraints; smoothness may be fixed or given PC priors penalizing roughness. In the LSCP, mean functions $\mu_k(s)$ are often linear in covariates, with normal priors on coefficients. Kernel parameters for influence fields in regeneration models (e.g., range $\theta$, exponent $\alpha$, or scale $\delta$ for mark dependence) employ weakly-informative gamma or exponential priors guided by domain size and data resolution (Hildeman et al., 2017, Kuronen et al., 2020).

3. Inference Methodologies and Computational Techniques

Efficient Bayesian inference for hierarchical LGCPs leverages discretization strategies, MCMC, and Laplace-type approximations. The LSCP implements a Metropolis-within-Gibbs scheme, introducing a discretized classification field with spatial “nugget” to enable parallel sampling:

• Classification field $C$ updated cell-wise;
• Hyperparameters sampled using MALA or pCN-MALA;
• Gaussian fields $Y_k(s)$ for each class updated in parallel;
• Truncation in spectral (Fourier/eigen) basis enables $O(N\log N)$ updates.

For bivariate LGCPs with influence, the high-dimensional Gaussian field is marginalized by Laplace approximation (following [Tierney & Kadane 1986]) using an SPDE-based GMRF representation for sparse precision. Remaining parameters are sampled with robust adaptive Metropolis (RAM) algorithms.

Both models use finite-dimensional approximations—grid-based for spatial domains—and prove consistency of these approximations as grid size increases. Mesh-invariant proposals (e.g., pCN-MALA) ensure stable acceptance rates under refinement (Hildeman et al., 2017).

4. Edge Correction and Model Diagnostics

In hierarchical models with spatial influence fields, edge effects are significant. Contributions from outside the observation window are modeled analytically or via FFT-based convolution, under assumptions such as Poisson processes for the influencer process $X$. This adjustment prevents bias near boundaries in parameter estimates for influence kernels and interaction effects.

Model diagnostics employ rank envelope tests on spatial summary functions including the $L$-function, pair-correlation $g(r)$, and empty-space $F(r)$; for bivariate processes, cross-summary functions such as $L_{12}(r)$ are used. Posterior-predictive checking via simulation assesses fit across replicates, with global envelopes indicating adequacy (Kuronen et al., 2020).

5. Applications and Empirical Performance

Hierarchical LGCPs have demonstrated substantial improvement over standard LGCPs for spatially heterogeneous data. In the LSCP application to tropical rainforest data (2461 tree locations in Panama), the LSCP with $K=2$ identified a near-zero intensity “swamp class” and a forest class, recovering sharp boundaries and producing unbiased parameter estimates. The LSCP envelopes for spatial summary statistics fully contained the empirical curves, whereas the standard LGCP exhibited bias—inflated estimates of variance and range, and spurious covariate significance (Hildeman et al., 2017).

In forestry regeneration models, hierarchical LGCPs demonstrated the effect of large trees on seedling locations, showing that models with kernels for influence outperformed null models in predicting cross-type spatial statistics.

6. General Implications and Lessons

The introduction of hierarchical structure—either via latent class segmentation (LSCP) or explicit process hierarchy (bivariate LGCPs)—enables the model to accommodate sharp spatial discontinuities, conditional dependencies, and context-specific interactions invisible to standard LGCPs. Parametric “influence kernels” and latent segmentation facilitate ecologically and spatially meaningful inference, while advanced computational techniques (MALA, RAM, Laplace/SPDE) ensure tractable inference for high-dimensional spatial applications.

Replication enhances identifiability of spatially structured effects. Posterior-predictive checks provide principled model assessment. These methodologies generalize to various scientific domains, including ecology, forestry, and epidemiology, wherever multiple spatial processes interact or data exhibit nonstationarity (Hildeman et al., 2017, Kuronen et al., 2020).