# rmpoispp

##### Generate Multitype Poisson Point Pattern

Generate a random point pattern, a realisation of the (homogeneous or inhomogeneous) multitype Poisson process.

##### Usage

```
rmpoispp(lambda, lmax=NULL, win, types, …,
nsim=1, drop=TRUE, warnwin=!missing(win))
```

##### Arguments

- lambda
Intensity of the multitype Poisson process. Either a single positive number, a vector, a

`function(x,y,m, …)`

, a pixel image, a list of functions`function(x,y, …)`

, or a list of pixel images.- lmax
An upper bound for the value of

`lambda`

. May be omitted- win
Window in which to simulate the pattern. An object of class

`"owin"`

or something acceptable to`as.owin`

. Ignored if`lambda`

is a pixel image or list of images.- types
All the possible types for the multitype pattern.

- …
Arguments passed to

`lambda`

if it is a function.- nsim
Number of simulated realisations to be generated.

- drop
Logical. If

`nsim=1`

and`drop=TRUE`

(the default), the result will be a point pattern, rather than a list containing a point pattern.- warnwin
Logical value specifying whether to issue a warning when

`win`

is ignored.

##### Details

This function generates a realisation of the marked Poisson
point process with intensity `lambda`

.

Note that the intensity function
\(\lambda(x,y,m)\) is the
average number of points **of type m** per unit area
near the location \((x,y)\).
Thus a marked point process with a constant intensity of 10
and three possible types will have an average of 30 points per unit
area, with 10 points of each type on average.

The intensity function may be specified in any of the following ways.

- single number:
If

`lambda`

is a single number, then this algorithm generates a realisation of the uniform marked Poisson process inside the window`win`

with intensity`lambda`

for each type. The total intensity of points of all types is`lambda * length(types)`

. The argument`types`

must be given and determines the possible types in the multitype pattern.- vector:
If

`lambda`

is a numeric vector, then this algorithm generates a realisation of the stationary marked Poisson process inside the window`win`

with intensity`lambda[i]`

for points of type`types[i]`

. The total intensity of points of all types is`sum(lambda)`

. The argument`types`

defaults to`names(lambda)`

, or if that is null,`1:length(lambda)`

.- function:
If

`lambda`

is a function, the process has intensity`lambda(x,y,m,…)`

at spatial location`(x,y)`

for points of type`m`

. The function`lambda`

must work correctly with vectors`x`

,`y`

and`m`

, returning a vector of function values. (Note that`m`

will be a factor with levels equal to`types`

.) The value`lmax`

, if present, must be an upper bound on the values of`lambda(x,y,m,…)`

for all locations`(x, y)`

inside the window`win`

and all types`m`

. The argument`types`

must be given.- list of functions:
If

`lambda`

is a list of functions, the process has intensity`lambda[[i]](x,y,…)`

at spatial location`(x,y)`

for points of type`types[i]`

. The function`lambda[[i]]`

must work correctly with vectors`x`

and`y`

, returning a vector of function values. The value`lmax`

, if given, must be an upper bound on the values of`lambda(x,y,…)`

for all locations`(x, y)`

inside the window`win`

. The argument`types`

defaults to`names(lambda)`

, or if that is null,`1:length(lambda)`

.- pixel image:
If

`lambda`

is a pixel image object of class`"im"`

(see`im.object`

), the intensity at a location`(x,y)`

for points of any type is equal to the pixel value of`lambda`

for the pixel nearest to`(x,y)`

. The argument`win`

is ignored; the window of the pixel image is used instead. The argument`types`

must be given.- list of pixel images:
If

`lambda`

is a list of pixel images, then the image`lambda[[i]]`

determines the intensity of points of type`types[i]`

. The argument`win`

is ignored; the window of the pixel image is used instead. The argument`types`

defaults to`names(lambda)`

, or if that is null,`1:length(lambda)`

.

If `lmax`

is missing, an approximate upper bound will be calculated.

To generate an inhomogeneous Poisson process
the algorithm uses ``thinning'': it first generates a uniform
Poisson process of intensity `lmax`

for points of each type `m`

,
then randomly deletes or retains each point independently,
with retention probability
\(p(x,y,m) = \lambda(x,y,m)/\mbox{lmax}\).

##### Value

A point pattern (an object of class `"ppp"`

) if `nsim=1`

,
or a list of point patterns if `nsim > 1`

.
Each point pattern is multitype (it carries a vector of marks
which is a factor).

##### See Also

`rpoispp`

for unmarked Poisson point process;
`rmpoint`

for a fixed number of random marked points;
`ppp.object`

,
`owin.object`

.

##### Examples

```
# NOT RUN {
# uniform bivariate Poisson process with total intensity 100 in unit square
pp <- rmpoispp(50, types=c("a","b"))
# stationary bivariate Poisson process with intensity A = 30, B = 70
pp <- rmpoispp(c(30,70), types=c("A","B"))
pp <- rmpoispp(c(30,70))
# works in any window
data(letterR)
pp <- rmpoispp(c(30,70), win=letterR, types=c("A","B"))
# inhomogeneous lambda(x,y,m)
# note argument 'm' is a factor
lam <- function(x,y,m) { 50 * (x^2 + y^3) * ifelse(m=="A", 2, 1)}
pp <- rmpoispp(lam, win=letterR, types=c("A","B"))
# extra arguments
lam <- function(x,y,m,scal) { scal * (x^2 + y^3) * ifelse(m=="A", 2, 1)}
pp <- rmpoispp(lam, win=letterR, types=c("A","B"), scal=50)
# list of functions lambda[[i]](x,y)
lams <- list(function(x,y){50 * x^2}, function(x,y){20 * abs(y)})
pp <- rmpoispp(lams, win=letterR, types=c("A","B"))
pp <- rmpoispp(lams, win=letterR)
# functions with extra arguments
lams <- list(function(x,y,scal){5 * scal * x^2},
function(x,y, scal){2 * scal * abs(y)})
pp <- rmpoispp(lams, win=letterR, types=c("A","B"), scal=10)
pp <- rmpoispp(lams, win=letterR, scal=10)
# florid example
lams <- list(function(x,y){
100*exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6)
}
# log quadratic trend
,
function(x,y){
100*exp(-0.6*x+0.5*y)
}
# log linear trend
)
X <- rmpoispp(lams, win=unit.square(), types=c("on", "off"))
# pixel image
Z <- as.im(function(x,y){30 * (x^2 + y^3)}, letterR)
pp <- rmpoispp(Z, types=c("A","B"))
# list of pixel images
ZZ <- list(
as.im(function(x,y){20 * (x^2 + y^3)}, letterR),
as.im(function(x,y){40 * (x^3 + y^2)}, letterR))
pp <- rmpoispp(ZZ, types=c("A","B"))
pp <- rmpoispp(ZZ)
# randomising an existing point pattern
rmpoispp(intensity(amacrine), win=Window(amacrine))
# }
```

*Documentation reproduced from package spatstat, version 1.55-1, License: GPL (>= 2)*