# pattern.sim {voronoi}

### 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")

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