rstpoispp: Simulate an inhomogeneous spatio-temporal Poisson point...

In SepTest: Tests for First-Order Separability in Spatio-Temporal Point Processes

Simulate an inhomogeneous spatio-temporal Poisson point process

Description

Generates a realization of an inhomogeneous Poisson point process (STPP) in space and time using the standard thinning method. The user provides an intensity function \lambda(u, t) and an upper bound L_{\max} on its value over the observation window. The algorithm first samples candidate events uniformly over space and time and then retains each candidate with probability proportional to its normalized intensity \lambda(u,t)/L_{\max}.

Usage

rstpoispp(
  lambda,
  Lmax,
  s.region = splancs::as.points(c(0, 1, 1, 0), c(0, 0, 1, 1)),
  t.region = c(0, 1)
)

Arguments

lambda	A function of the form lambda(u, t) that returns the intensity value at coordinates (x, y, t).
Lmax	A numeric value giving the known or estimated maximum of the intensity function lambda over the spatial and temporal window. Used for thinning.
s.region	A matrix with two columns giving the polygonal spatial window. Each row is a vertex of the polygon. Default is the unit square.
t.region	A numeric vector of length 2 giving the temporal observation window. Default is c(0,1).

Details

The method implements the classical thinning algorithm for simulating inhomogeneous Poisson processes:

1. Draw N^* \sim \mathrm{Poisson}(L_{\max} \, |W| \, |T|), where |W| and |T| denote the spatial and temporal window measures.

2. Generate N^* candidate points uniformly over W \times T.

3. Retain each point (u_i, t_i) independently with probability p_i = \lambda(u_i, t_i) / L_{\max}.

The result is a realization of an inhomogeneous STPP with intensity function \lambda(u, t).

This simulator underpins the spatio-temporal framework introduced in Ghorbani et al. (2021, 2025) for studying first-order separability. By selecting appropriate intensity functions (see get.lambda.function), users can generate fully separable, partially separable, or non-separable spatio-temporal patterns, enabling direct evaluation of separability tests such as chi2.test, global.envelope.test, or dHS.test.

Value

A numeric matrix with three columns (x, y, t) representing the retained points from the inhomogeneous Poisson process.

Note

The intensity function \lambda(u, t) should return non-negative numeric values and be bounded above by Lmax across the observation domain.

Author(s)

Mohammad Ghorbani [e-mail]
Nafiseh Vafaei [e-mail]

References

Ghorbani M., Vafaei N., Dvořák J., Myllymäki M. (2021). Testing the first-order separability hypothesis for spatio-temporal point patterns. Computational Statistics & Data Analysis, 161, 107245.

Ghorbani, M., Vafaei, N. and Myllymäki, M. (2025). A kernel-based test for the first-order separability of spatio-temporal point processes, TEST .

See Also

get.lambda.function to construct spatio-temporal intensity models; get.lambda.max to compute intensity maxima; estimate.st.intensity for intensity estimation; plot_stpp for visualization.

Examples

# Example 1: Simulate a separable spatio-temporal Poisson process
lambda <- get.lambda.function(N = 200, g = 50, model = 1)
Lmax   <- get.lambda.max(N = 200, g = 50, model = 1)
X <- rstpoispp(lambda, Lmax)
head(X)

# Example 2: Non-separable model (Model 4)
lambda <- get.lambda.function(N = 200, g = 50, model = 4)
Lmax   <- get.lambda.max(N = 200, g = 50, model = 4)
sim_data <- rstpoispp(lambda, Lmax)

# Spatial projection of simulated events
plot(sim_data[, 1:2], asp = 1, main = "Spatial Projection of Simulated stPP")
# Example 3: 3D visualization using plot_ST_pp()
plot_stpp(X, type = "3D", title="Realisation of a stPP")

SepTest documentation built on Feb. 3, 2026, 5:07 p.m.