norm3d: Trivariate normal density with independent components in...

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

Trivariate normal density with independent components in space-time

Description

Evaluates a trivariate normal density on (x,y,t) with independent components (diagonal covariance). The density is the product of three univariate normal densities:

f(x,y,t) = \phi(x;\mu_x,\sigma_x)\,\phi(y;\mu_y,\sigma_y)\,\phi(t;\mu_t,\sigma_t).

Usage

norm3d(x, y, t, mu = c(0.3, 0.3, 0.2), sd = c(0.05, 0.05, 0.05), log = FALSE)

Arguments

x	Numeric vector of x-coordinate(s).
y	Numeric vector of y-coordinate(s).
t	Numeric vector of time coordinate(s).
mu	Numeric vector of length 3 giving c(mu_x, mu_y, mu_t).
sd	Numeric vector of length 3 giving positive standard deviations c(sd_x, sd_y, sd_t).
log	Logical; if TRUE, return the log-density.

Value

Numeric vector of densities (or log-densities) with length determined by standard recycling rules for x, y, and t.

Author(s)

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 & Data Analysis, 161, 107245.

See Also

norm2d, get.lambda.function, estimate.st.intensity

Examples

norm3d(0.3, 0.3, 0.2)  # peak value at the mean (with default parameters)
norm3d(c(0.2, 0.3), 0.3, 0.2)

x <- y <- seq(0, 1, length.out = 100)
z <- outer(x, y, function(x, y) norm3d(x, y, t = 0.2))
image(x, y, z, col = heat.colors(50), main = "Spatial slice of norm3d at t = 0.2")

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