Spatial Point Process

The fundamental null hypothesis in spatial point pattern analysis. Under CSR, points are distributed independently and uniformly across the study region, following a homogeneous Poisson process. The intensity λ (expected number of points per unit area) is constant. Rejection of CSR indicates the presence of biological structure—either clustering (aggregation) or regularity (inhibition). CSR is tested using quadrat counts, nearest-neighbor distances, and the empirical Ripley's K function.

The first-order property describing the expected density of points at any location. In spatial transcriptomics, intensity maps reveal tissue regions with high cell-type or transcript abundance. Kernel density estimation smooths discrete point locations into a continuous surface, with bandwidth selection controlling the scale of detected features. Log-Gaussian Cox processes model the intensity as a latent random field, capturing overdispersion and spatial heterogeneity in molecular distributions.

Second-order summary statistics that characterize spatial dependence between points. Ripley's K function counts the expected number of additional points within distance r of a typical point, normalized by overall intensity. Its derivative, the pair correlation function g(r), reveals clustering at specific length scales. In tissue analysis, a peak in g(r) at 20-50μm may indicate paracrine signaling between neighboring cells, while a flat profile supports spatial independence.

An extension where each point carries a mark—a categorical or continuous attribute such as cell type, gene expression level, or transcript identity. Marked models test whether marks are spatially correlated beyond chance. Mark connection functions quantify the probability that two points separated by distance r share a specific mark. This enables formal testing of cell-type co-localization and ligand-receptor spatial coupling in tumor microenvironments.

Real tissues exhibit spatially varying intensity due to anatomical structures. Inhomogeneous point process models separate true inter-point interaction from background density gradients. Replicated point patterns—multiple tissue sections or regions of interest—are analyzed using hierarchical models with shared parameters, enabling population-level inference about spatial organization while accounting for biological and technical variability across samples.

Explicit parametric models for point interactions. Strauss processes penalize point pairs within a critical distance, modeling cellular exclusion or territoriality. Geyer saturation processes allow clustering without the degeneracy issues of simpler attraction models. These models are fit via maximum pseudolikelihood or Bayesian MCMC, yielding interpretable parameters—interaction strength γ and range R—that quantify the nature and scale of spatial organization in tissue architecture.

The simplest spatial point process model, where points are distributed independently and uniformly across a study region. It serves as the null model of complete spatial randomness (CSR) against which observed patterns are tested.

Key properties:

• The number of points in any subregion follows a Poisson distribution
• Point locations are independent of each other
• Intensity λ is constant across space (homogeneous)

In practice, biological point patterns rarely follow CSR. Deviations from the Poisson model provide evidence for cell-cell interactions, tissue architecture, or disease-associated spatial reorganization.

A data structure where each cell or spot is represented as a node, and edges connect neighboring locations based on spatial proximity. Construction methods include:

• k-nearest neighbors (k-NN): Connect each point to its k closest neighbors
• Distance threshold: Connect all point pairs within radius r
• Delaunay triangulation: Connect points forming a mesh of non-overlapping triangles

These graphs enable graph neural networks and spatial domain detection algorithms to propagate information across tissue topology, capturing local microenvironment relationships that point process intensities alone cannot describe.

A non-parametric statistical method for assessing the significance of observed spatial patterns by generating a null distribution through random shuffling of spatial labels.

Procedure:

1. Compute an observed spatial statistic (e.g., Moran's I, Ripley's K)
2. Randomly permute the spatial coordinates or cell-type labels n times
3. Recalculate the statistic for each permutation
4. Compare the observed value against the empirical null distribution

This approach avoids parametric assumptions about the underlying spatial process and is widely used to identify spatially variable genes and test for cell-type co-localization in tissue sections.

A computational process that estimates the cell-type proportions within each spatial transcriptomics spot by separating the mixed gene expression signal into its constituent parts.

Approaches include:

• Regression-based methods: Use reference single-cell signatures to fit proportions
• Probabilistic models: Apply Bayesian frameworks with spatial priors
• Deep learning: Train neural networks to map mixture profiles to cell-type compositions

Spatial deconvolution transforms coarse spot-level data into cell-type-resolved maps, enabling point process analysis of specific cell populations even when single-cell resolution is not experimentally achievable.