Spatial Point Process

Definition

A spatial point process is a random mechanism generating a locally finite set of points (events) in space (typically $\mathbb R^2$) — tree locations, disease cases, galaxies, crime sites. It is characterized by its intensity function $\rho(u)$ (first-order: the expected number of points per unit area near $u$) and its second-order structure (how points co-occur — clustering or inhibition), summarized by the pair-correlation or Ripley's $K$-function. Inference targets a regression model for the intensity (inhomogeneity driven by covariates) and, separately, a model for the residual dependence.

Key Ideas

• Inhomogeneity via intensity regression. $\rho_\beta(u)=\rho(z(u)^\top\beta)$ (often log-linear) makes the local point density a function of spatial covariates $z(u)$; this is the first-order structure — where points tend to fall.
• Homogeneous Poisson process is the benchmark of "complete spatial randomness": constant intensity, points independent. Departures are clustering (aggregation) or inhibition/regularity (repulsion).
• Second-order structure and the $K$-function. Ripley's $K(r)$ (and its inhomogeneous version) gives the expected number of further points within distance $r$ of a typical point, standardized by intensity; $K(r)>\pi r^2$ indicates clustering, $<\pi r^2$ inhibition. The pair correlation function is the density analogue.
• Cluster and Cox models. Overdispersed/clustered patterns are modeled by Cox processes — a Poisson process with a random intensity field, e.g. the log-Gaussian Cox process (log-intensity is a Gaussian random field) — or by Neyman-Scott cluster processes (parent points scatter offspring). These have tractable $K$-functions but intractable likelihoods.
• Two-step / composite estimation (Waagepetersen-Guan 2009): because intensity is first-order and clustering is second-order, estimate them in sequence — (1) the covariate parameters $\beta$ by a Poisson-likelihood score estimating function (consistent even under clustering, since it uses only the intensity), then (2) the clustering parameters by minimum-contrast fitting of the $K$-function with $\hat\beta$ plugged in. Both stages are asymptotically normal under mixing conditions. This sidesteps the intractable full likelihood.

How It Works

Fit the intensity by treating the observed points as if Poisson and solving the (weighted) Poisson score for $\beta$ — equivalent to a Poisson log-linear regression on a fine grid / quadrature scheme. Then compute the empirical inhomogeneous $K$-function from the residual pattern and choose the cluster-model parameters to minimize the integrated squared difference (minimum contrast) between empirical and theoretical $K$. Standard errors come from the estimating-function / mixing asymptotics, or by simulation from the fitted model.

Why It Matters

• Separates "where" from "how clustered." In ecology this distinguishes aggregation due to habitat/niche (intensity covariates) from aggregation due to dispersal (residual clustering) — the coexistence-of-species question.
• Scales to large patterns. Composite/estimating-function methods make huge modern point patterns (all trees in a forest plot; every case in a registry) fittable where full-likelihood MCMC would not.
• Continuous-space complement. It is the point-location counterpart to the areal CAR/disease-mapping models (aggregate counts per region) and the temporal counting process (events in time).

Open Questions

• Efficiency loss of two-step vs. joint estimation, and optimal weighting of the Poisson score under strong clustering.
• Choice of the $K$-function contrast (range, exponents) and its effect on the clustering estimates.
• Distinguishing habitat-driven inhomogeneity from dependence-driven clustering is only partially identified without covariates or replication.

Related

• Counting Process
• Conditional Autoregressive Model
• Waagepetersen-Guan (2009) Two-Step Estimation for Inhomogeneous Spatial Point Processes
• Rasmus Waagepetersen
• Yongtao Guan