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

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

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.

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).

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
\(p(x,y,m) = \lambda(x,y,m)/\mbox{lmax}\).

`rpoispp`

for unmarked Poisson point process;
`rmpoint`

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

,
`owin.object`

.

# 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 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)) # }