rmpoint: Generate N Random Multitype Points

Description

Generate a random multitype point pattern with a fixed number of points, or a fixed number of points of each type.

Usage

 ```1 2 3 4``` ``` rmpoint(n, f=1, fmax=NULL, win=unit.square(), types, ptypes, ..., giveup=1000, verbose=FALSE, nsim=1, drop=TRUE) ```

Arguments

 `n` Number of marked points to generate. Either a single number specifying the total number of points, or a vector specifying the number of points of each type. `f` The probability density of the multitype points, usually un-normalised. Either a constant, a vector, a function `f(x,y,m, ...)`, a pixel image, a list of functions `f(x,y,...)` or a list of pixel images. `fmax` An upper bound on the values of `f`. If missing, this number will be estimated. `win` Window in which to simulate the pattern. Ignored if `f` is a pixel image or list of pixel images. `types` All the possible types for the multitype pattern. `ptypes` Optional vector of probabilities for each type. `...` Arguments passed to `f` if it is a function. `giveup` Number of attempts in the rejection method after which the algorithm should stop trying to generate new points. `verbose` Flag indicating whether to report details of performance of the simulation algorithm. `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.

Details

This function generates random multitype point patterns consisting of a fixed number of points.

Three different models are available:

I. Random location and type:

If `n` is a single number and the argument `ptypes` is missing, then `n` independent, identically distributed random multitype points are generated. Their locations `(x[i],y[i])` and types `m[i]` have joint probability density proportional to f(x,y,m).

II. Random type, and random location given type:

If `n` is a single number and `ptypes` is given, then `n` independent, identically distributed random multitype points are generated. Their types `m[i]` have probability distribution `ptypes`. Given the types, the locations `(x[i],y[i])` have conditional probability density proportional to f(x,y,m).

III. Fixed types, and random location given type:

If `n` is a vector, then we generate `n[i]` independent, identically distributed random points of type `types[i]`. For points of type m the conditional probability density of location (x,y) is proportional to f(x,y,m).

Note that the density `f` is normalised in different ways in Model I and Models II and III. In Model I the normalised joint density is g(x,y,m)=f(x,y,m)/Z where

Z = sum_[m] integral lambda(x,y,m) dx dy

while in Models II and III the normalised conditional density is g(x,y|m) = f(x,y,m)/Z[m] where

Z[m] = integral lambda(x,y,m) dx dy.

In Model I, the marginal distribution of types is p[m] = Z[m]/Z.

The unnormalised density `f` may be specified in any of the following ways.

single number:

If `f` is a single number, the conditional density of location given type is uniform. That is, the points of each type are uniformly distributed. In Model I, the marginal distribution of types is also uniform (all possible types have equal probability).

vector:

If `f` is a numeric vector, the conditional density of location given type is uniform. That is, the points of each type are uniformly distributed. In Model I, the marginal distribution of types is proportional to the vector `f`. In Model II, the marginal distribution of types is `ptypes`, that is, the values in `f` are ignored. The argument `types` defaults to `names(f)`, or if that is null, `1:length(f)`.

function:

If `f` is a function, it will be called in the form `f(x,y,m,...)` at spatial location `(x,y)` for points of type `m`. In Model I, the joint probability density of location and type is proportional to `f(x,y,m,...)`. In Models II and III, the conditional probability density of location `(x,y)` given type `m` is proportional to `f(x,y,m,...)`. The function `f` must work correctly with vectors `x`, `y` and `m`, returning a vector of function values. (Note that `m` will be a factor with levels `types`.) The value `fmax` must be given and must be an upper bound on the values of `f(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 `f` is a list of functions, then the functions will be called in the form `f[[i]](x,y,...)` at spatial location `(x,y)` for points of type `types[i]`. In Model I, the joint probability density of location and type is proportional to `f[[m]](x,y,...)`. In Models II and III, the conditional probability density of location `(x,y)` given type `m` is proportional to `f[[m]](x,y,...)`. The function `f[[i]]` must work correctly with vectors `x` and `y`, returning a vector of function values. The value `fmax` must be given and must be an upper bound on the values of `f[[i]](x,y,...)` for all locations `(x, y)` inside the window `win`. The argument `types` defaults to `names(f)`, or if that is null, `1:length(f)`.

pixel image:

If `f` is a pixel image object of class `"im"` (see `im.object`), the unnormalised density at a location `(x,y)` for points of any type is equal to the pixel value of `f` for the pixel nearest to `(x,y)`. In Model I, the marginal distribution of types is uniform. 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 `f` is a list of pixel images, then the image `f[[i]]` determines the density values 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(f)`, or if that is null, `1:length(f)`.

The implementation uses the rejection method. For Model I, `rmpoispp` is called repeatedly until `n` points have been generated. It gives up after `giveup` calls if there are still fewer than `n` points. For Model II, the types are first generated according to `ptypes`, then the locations of the points of each type are generated using `rpoint`. For Model III, the locations of the points of each type are generated using `rpoint`.

Value

A point pattern (an object of class `"ppp"`) if `nsim=1`, or a list of point patterns if `nsim > 1`.

Author(s)

and \rolf

`ppp.object`, `owin.object`

Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82``` ```abc <- c("a","b","c") ##### Model I rmpoint(25, types=abc) rmpoint(25, 1, types=abc) # 25 points, equal probability for each type, uniformly distributed locations rmpoint(25, function(x,y,m) {rep(1, length(x))}, types=abc) # same as above rmpoint(25, list(function(x,y){rep(1, length(x))}, function(x,y){rep(1, length(x))}, function(x,y){rep(1, length(x))}), types=abc) # same as above rmpoint(25, function(x,y,m) { x }, types=abc) # 25 points, equal probability for each type, # locations nonuniform with density proportional to x rmpoint(25, function(x,y,m) { ifelse(m == "a", 1, x) }, types=abc) rmpoint(25, list(function(x,y) { rep(1, length(x)) }, function(x,y) { x }, function(x,y) { x }), types=abc) # 25 points, UNEQUAL probabilities for each type, # type "a" points uniformly distributed, # type "b" and "c" points nonuniformly distributed. ##### Model II rmpoint(25, 1, types=abc, ptypes=rep(1,3)/3) rmpoint(25, 1, types=abc, ptypes=rep(1,3)) # 25 points, equal probability for each type, # uniformly distributed locations rmpoint(25, function(x,y,m) {rep(1, length(x))}, types=abc, ptypes=rep(1,3)) # same as above rmpoint(25, list(function(x,y){rep(1, length(x))}, function(x,y){rep(1, length(x))}, function(x,y){rep(1, length(x))}), types=abc, ptypes=rep(1,3)) # same as above rmpoint(25, function(x,y,m) { x }, types=abc, ptypes=rep(1,3)) # 25 points, equal probability for each type, # locations nonuniform with density proportional to x rmpoint(25, function(x,y,m) { ifelse(m == "a", 1, x) }, types=abc, ptypes=rep(1,3)) # 25 points, EQUAL probabilities for each type, # type "a" points uniformly distributed, # type "b" and "c" points nonuniformly distributed. ###### Model III rmpoint(c(12, 8, 4), 1, types=abc) # 12 points of type "a", # 8 points of type "b", # 4 points of type "c", # each uniformly distributed rmpoint(c(12, 8, 4), function(x,y,m) { ifelse(m=="a", 1, x)}, types=abc) rmpoint(c(12, 8, 4), list(function(x,y) { rep(1, length(x)) }, function(x,y) { x }, function(x,y) { x }), types=abc) # 12 points of type "a", uniformly distributed # 8 points of type "b", nonuniform # 4 points of type "c", nonuniform ######### ## Randomising an existing point pattern: # same numbers of points of each type, uniform random locations (Model III) rmpoint(table(marks(demopat)), 1, win=Window(demopat)) # same total number of points, distribution of types estimated from X, # uniform random locations (Model II) rmpoint(npoints(demopat), 1, types=levels(marks(demopat)), win=Window(demopat), ptypes=table(marks(demopat))) ```

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