get.lambda.function: Construct spatio-temporal intensity functions with controlled separability

Description

Returns an intensity function $\lambda(x,y,t)$ corresponding to one of four models used for simulation experiments in Ghorbani et al. (2021). Each model is a mixture of a separable "background" component and a structured (generally non-separable) spatio-temporal Gaussian bump. The models provide different degrees of space–time separability, allowing for controlled experiments on separability testing.

Usage

get.lambda.function(
  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)
)

Value

A function of the form function(x, y, t) representing the selected intensity surface.

Arguments

N: Numeric scalar (> 0). Baseline intensity level (interpreted as an expected total count after scaling in the calling simulator; see details below).
g: Numeric scalar (>= 0). Weight of the structured (non-separable) component.
model: Integer in 1:4 indicating the structure of the intensity 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 (models 2 and 4).
mu2: Numeric vector of length 2. Mean of the spatial 2D Gaussian background term (models 3 and 4).
sd2: Numeric vector of length 2 with positive entries. Standard deviations of the spatial 2D Gaussian background term (models 3 and 4).
mu3: Numeric vector of length 3. Mean of the structured (non-separable) 3D Gaussian component.
sd3: Numeric vector of length 3 with positive entries. Standard deviations of the structured 3D Gaussian component.

Author

Mohammad Ghorbani mohammad.ghorbani@slu.se
Nafiseh Vafaei nafiseh.vafaei@slu.se

Details

The returned function is intended for use in simulation (e.g., for generating spatio-temporal Poisson point patterns under varying degrees of separability).

The intensity is constructed as: $$\lambda(x,y,t) = \lambda_{\mathrm{bg}}(x,y,t) + g\, f_{st}(x,y,t),$$ where $f_{st}$ is a nonnegative 3D Gaussian density (via norm3d) and the background term $\lambda_{\mathrm{bg}}$ depends on model:

1: Homogeneous background: $\lambda_{\mathrm{bg}}(x,y,t) = (N-g)$
2: Temporal inhomogeneity only: $\lambda_{\mathrm{bg}}(x,y,t) = (N-g)\, f_t(t)$, where $f_t$ is a 1D Gaussian density (dnorm).
3: Spatial inhomogeneity only: $\lambda_{\mathrm{bg}}(x,y,t) = (N-g)\, f_s(x,y)$, where $f_s$ is a 2D Gaussian density (norm2d).
4: Separable spatial-temporal inhomogeneity: $\lambda_{\mathrm{bg}}(x,y,t) = (N-g)\, f_s(x,y)\, f_t(t)$.

Note: since Gaussian densities can exceed 1 for small standard deviations, N is best interpreted as a scaling parameter used by the calling simulator. Ensure $\lambda(x,y,t)$ is nonnegative over the intended domain.

See more details in Ghorbani et al. (2021), Section 6.1.

References

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

Examples

Run this code


# Choose model 4: non-separable spatio-temporal intensity
lambda <- get.lambda.function(N = 210, g = 50, model = 4)
lambda(0.5, 0.5, 0.5)  # Evaluate intensity at center of space-time domain

# Visualize spatial intensity at fixed time for model 2
lambda2 <- get.lambda.function(N = 200, g = 50, model = 2)
x <- y <- seq(0, 1, length.out = 100)
z <- outer(x, y, function(x, y) lambda2(x, y, t = 0.5))

fields::image.plot(
x, y, z,
main = "Intensity at t = 0.5 (Model 2)",
col = topo.colors(50))

Run the code above in your browser using DataLab