Point Process Models (WebR)

PPOL 6805 / DSAN 6750: GIS for Spatial Data Science
Fall 2025

Jeff Jacobs

jj1088@georgetown.edu

Wednesday, October 15, 2025

A Menagerie of Models

Poisson Point Processes (CSR)

spatstat.random::rpoispp(lambda, win)
  • \(N \sim \text{Pois}(\lambda)\)
  • For \(i \in \{1, \ldots, N\}\): Generate \(X_i, Y_i \sim \mathcal{U}(\texttt{win})\)

Simple Sequential Inhibition (SSI)

spatstat.random::rSSI(r, n=Inf, giveup=1000, win)
  • Initialize \(\mathbf{S} = \varnothing\). While not done:
    • Generate \(\mathbf{E} = (X, Y) \sim \mathcal{U}(\texttt{win})\)
    • Check if \(\mathbf{E}\) within r units of any existing point in \(\mathbf{S}\)
      • If it is, throw \(\mathbf{E}\) away. Otherwise, add \(\mathbf{E}\) to \(\mathbf{S}\)
    • done=TRUE if \(\mathbf{S}\) has n points OR has been the same for giveup steps

Matérn Cluster Process

spatstat.random::rMatClust(kappa, scale, mu, win)
  • Generate \(K(\kappa)\) parent points via Poisson Point Process with intensity \(\lambda = \kappa\)
  • For each parent point \(\mathbf{s}_i \in \left\{\mathbf{s}_1, \ldots, \mathbf{s}_{K(\kappa)}\right\}\):
    • Generate \(N(\mu)\) offspring points via Poisson Point Process with intensity \(\lambda = \mu\), distributed uniformly within a circle of radius scale centered at \(\mathbf{s}_i\)
  • Offspring points form the outcome (parent points are thrown away)

Matérn Inhibition Process (I and II)

spatstat.random::rMaternI(kappa, r, win)
spatstat.random::rMaternII(kappa, r, win)
rMaternI() [Docs] rMaternII() [Docs]
  • Generate events \(\mathbf{S} = \{\mathbf{s}_1, \ldots, \mathbf{s}_{N(\lambda)}\}\) via Poisson point process with \(\lambda = \kappa\)
  • Generate events \(\mathbf{S} = \{\mathbf{s}_1, \ldots, \mathbf{s}_{N(\lambda)}\}\) via Poisson point process with \(\lambda = \kappa\), plus timestamp \(t_i \sim \mathcal{U}(0,1)\) for each \(\mathbf{s}_i\)
  • Delete all pairs of points \(\mathbf{s}_i\), \(\mathbf{s}_j\) for which \(\textsf{dist}(\mathbf{s}_i, \mathbf{s}_j) < \texttt{r}\)
  • For each pair of points \(\mathbf{s}_i\), \(\mathbf{s}_j\): if \(\textsf{dist}(\mathbf{s}_i, \mathbf{s}_j) < \texttt{r}\), delete the later point

Cox Processes: Random Intensity → Random Events

spatstat.random::rLGCP(model, mu, param, win)
models=c("exponential", "gauss", "stable", "gencauchy", "matern")

Docs