get.lambda.max: Upper bound for spatio-temporal intensity models

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

Upper bound for spatio-temporal intensity models

Description

Computes a practical upper bound for the spatio-temporal intensity models used in get.lambda.function. The bound is intended for thinning/rejection sampling in simulation routines such as rstpoispp.

Usage

get.lambda.max(
  N,
  g,
  model = 1L,
  mu1 = 0.5,
  sd1 = 0.2,
  mu2 = c(0.5, 0.5),
  sd2 = c(0.2, 0.2),
  mu3 = c(0.3, 0.3, 0.2),
  sd3 = c(0.05, 0.05, 0.05)
)

Arguments

N	Numeric scalar (> 0). Total expected number of events (baseline intensity level).
g	Numeric scalar (>= 0). Weight of the structured (non-separable) component.
model	Integer in 1:4. See get.lambda.function.
mu1	Numeric scalar. Mean of the temporal Gaussian background term (models 2 and 4).
sd1	Numeric scalar (> 0). Standard deviation of the temporal Gaussian background term.
mu2	Numeric vector of length 2. Mean of the spatial Gaussian background term (models 3 and 4).
sd2	Numeric vector of length 2 with positive entries. Standard deviations of the spatial background term.
mu3	Numeric vector of length 3. Mean of the structured spatio-temporal Gaussian component.
sd3	Numeric vector of length 3 with positive entries. Standard deviations of the structured component.

Details

The bound is computed using analytic maxima of Gaussian density components (at their modes), which yields a conservative and fast-to-evaluate upper bound when the component functions are Gaussian product densities.

The intensity models are mixtures of a background term and a structured spatio-temporal Gaussian bump. This function returns an upper bound obtained by evaluating each Gaussian density component at its mode. This upper bound is typically sufficient for rejection sampling when generating realizations of inhomogeneous Poisson or Cox point processes.

If norm2d and norm3d in this package are Gaussian product densities (independent components), then the maxima are available in closed form:

• \max_t \phi(t;\mu,\sigma) = 1/(\sigma\sqrt{2\pi})

• \max_{x,y} \phi(x;\mu_x,\sigma_x)\phi(y;\mu_y,\sigma_y) = 1/(2\pi\sigma_x\sigma_y)

• \max_{x,y,t} \prod_{d=1}^3 \phi(\cdot;\mu_d,\sigma_d) = 1/((2\pi)^{3/2}\sigma_x\sigma_y\sigma_t)

If your norm2d/norm3d use a different parameterization, this bound should be updated accordingly.

Value

Numeric scalar. A conservative upper bound for the selected intensity model.

Author(s)

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

References

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

See Also

get.lambda.function, rstpoispp

Examples

# Example 1: Homogeneous model (Model 1)
get.lambda.max(N = 200, g = 50, model = 1)

# Example 2: Non-separable spatio-temporal model (Model 4)
get.lambda.max(N = 200, g = 50, model = 4)

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