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pattern.sim {voronoi}

Simulate an inhomogeneous Poisson process
Package: 
voronoi
Version: 
1.1

Description

Draw simulations from one of five pre-defined Poisson point process models. Parameters for thre of the models - 'homogeneous', 'linear' and 'quadratic' - can be adjusted by the user. For the two other models - 'moon', and 'ridge' - the user may specify the maximum intensity, but cannot change any other features of the model. All simulations are on the unit square.

Usage

pattern.sim(r=NULL, a=NULL, b=NULL, type=c('homogeneous', 'linear', 
   'quadratic', 'moon', 'ridge'), pts=NULL)

Arguments

r
The maximum intensity of the process, specified on the log scale.
a
Parameter used to control the increasing intensity from left to right when type is 'linear'. When type='quadratic', controls the how strongly peaked the intensity is from left to right.
b
Parameter used to control the increasing intensity from bottom to top when type is 'linear'. When type='quadratic', controls the how strongly peaked the intensity is from bottom to top.
type
One of five Poisson point process models: "homogeneous", "linear", "quadratic", "moon", or "ridge".
pts
An m-by-2 matrix of points, where column 1 represents the horizontal location of the points and column 2 the vertical location. If pts is specified, pattern.sim returns an m-by-3 matrix; the first two columns are identical to the input pts object and the third column is the intensity at those locations for the model.

Details

Point patterns are simulated by first drawing a sample from a homogeneous Poisson point process with intensity equal to the user-specified maximum, then thinning according to the desired model. The models for type='linear' and 'quadratic' are actually log-linear and log-quadratic. The 'ridge' model includes a tall ridge along the western side of the unit square, similar to that used by Barr and Schoenberg (2011). The 'moon' type model includes a ridge, peak and crescent, similar to that used by Heikkinen and Arjas (1998) and Barr and Schoenberg (2011).

Values

An n-by-2 matrix. Each row represents a point, and each column represents a dimension. If pts is specified, then an m-by-3 matrix; the first two columns are identical to the input pts object and the third column is the intensity at those locations for the model.

References

Barr CD and Schoenberg FP (2011). On the Voronoi estimator for the intensity of an inhomogeneous planar Poisson process. Biometrika, 97(4), 977-984.

Heikkinen J and Arjas E (1998). Non-parametric Bayesian estimation of a spatial Poisson intensity. Scandinavian Journal of Statistics, 25, 435-50.

See Also

ve, centroidal, bw

Examples

par(mfrow = c(2,2))
plot(pattern.sim(type = "linear"), main = "linear")
plot(pattern.sim(type = "quadratic"), main = "quadratic")
plot(pattern.sim(type = "moon"), main = "moon")
plot(pattern.sim(type = "ridge"), main = "ridge")
 
for(i in 1:4) {
   plot(pattern.sim(a = 2*i - 1, b = 11 - 2*i, type = "linear"), 
     main = paste("a = ", 2*i - 1, ", ", "b = ", 11 - 2*i),
     pch = 20, cex = 0.2)
}
 
 
for(i in 1:4) {
   plot(pattern.sim(a = 2*i - 1, b = 11 - 2*i, type = "quadratic"), 
     main = paste("a = ", 2*i - 1, ", ", "b = ", 11 - 2*i),
     pch = 20, cex = 0.2)
}
 
for(i in 1:4) {
   plot(pattern.sim(i + 6, type = "moon"), 
     main = paste("r = ", i + 6), pch = 20, cex = 0.2)
}
 
for(i in 1:4) {
   plot(pattern.sim(i + 6, type = "ridge"), 
     main = paste("r = ", i + 6), pch = 20, cex = 0.2)
}
 
n      <- 101
x      <- seq(0, 1, length = n)
y      <- seq(0, 1, length = n)
xy     <- expand.grid(x, y)
r      <- 7
a1     <- pattern.sim(r, type = "linear", pts = xy)
a2     <- pattern.sim(r, type = "quadratic", pts = xy)
a3     <- pattern.sim(r, type = "moon", pts = xy)
a4     <- pattern.sim(r, type = "ridge", pts = xy)
cols   <- gray(seq(1, 0, len = 1500))
 
image(matrix(a1[,3], n, n), col = cols, main = "linear")
image(matrix(a2[,3], n, n), col = cols, main = "quadratic")
image(matrix(a3[,3], n, n), col = cols, main = "moon")
image(matrix(a4[,3], n, n), col = cols, main = "ridge")

Author(s)

Christopher D. Barr

Documentation reproduced from package voronoi, version 1.1. License: GPL-3