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```
#' @title Simulate Boolean Model with Grains Scaled According to a Truncated
#' Pareto Distribution
#' @export rbpto bpto.coverageprob bpto.covar bpto.germintensity
#' @description Functions for simulation and computing theoretical values of a
#' Boolean model with identically shaped grains with size given by a
#' truncated Pareto distribution.
#'
#' @param lambda Intensity of the germ process (which is a Poisson point
#' process)
#' @param grain A single \code{owin} object that gives the shape and size of the grain
#' at scale 1
#' @param xm A parameter governing the shape of the Pareto distribution used -
#' see details
#' @param alpha A parameter governing the shape of the Pareto distribution used
#' - see details
#' @param lengthscales A list of scales of the \code{grain} for which to
#' approximate the Pareto distribution: The grain for a germ is chosen by
#' selecting a scaled version of \code{grain} where \code{lengthscales}
#' specifies the possible scales and the Pareto distribution is used to
#' specify the probability of selection of each scale.
#' @param coverp Coverage probability of the Boolean model.
#' @param win The window to simulate in (an \code{owin} object)
#' @param seed Optional input (default in NULL). Is an integer passed to
#' \code{\link[base]{set.seed}}. Used to reproduce patterns exactly.
#' @param xy A raster object that specifies pixel coordinates of the final
#' simulated binary map. It is used the same way as \code{xy} is
#' \code{\link[spatstat]{as.mask}} in \pkg{spatstat}. If non-null then the
#' computations will be performed using rasters. Otherwise if \code{grain} and
#' \code{win} are polygonal then computations may be all polygonal.
#' @details
#' The parameters \code{xm} and \code{alpha} are such that the CDF of the Pareto distribution is \eqn{P(s <= x) = 1 - (xm / x)^{alpha}}.
#' The distribution of grains scales is a step-function approximation to the CDF with steps at \code{lengthscales}.
#'
#'
#'
#' @return
#' An \code{owin} object.
#' @keywords spatial datagen
#' @examples
#' lambda <- 0.2
#' win <- square(r = 10)
#' grain <- disc(r = 0.2)
#' xm <- 0.01
#' alpha <- 2
#' lengthscales <- seq(1, 5, by = 0.1)
#' xi <- rbpto(lambda, grain, win, xm, alpha, lengthscales = lengthscales)
#'
#' # Compute properties of the Boolean model from parameters
#' bpto.coverageprob(lambda, grain, xm, alpha, lengthscales = lengthscales)
#' covar <- bpto.covar(lambda, grain, xm, alpha, lengthscales = lengthscales,
#' xy = as.mask(win, eps = 2))
#' @describeIn rbpto Simulate Boolean model with grain size distributed according to a truncated Pareto distribution.
rbpto <- function(lambda, grain, win, xm, alpha, lengthscales,
seed = NULL, xy = NULL){
#check that smallest scale is larger than xm
stopifnot(lengthscales[1] >= xm)
#get scaled versions of grain
grainlib <- mapply(scalardilate, X = list(grain), f = lengthscales, SIMPLIFY = FALSE)
grainlib <- as.solist(grainlib)
#get weights of these grains from pmf
weights <- alpha * xm ^ alpha / (lengthscales ^ (alpha + 1) )
weights <- weights / sum(weights) #standardise
#now simulate Boolean model!
bufferdist <- diameter.owin(grain) * max(lengthscales)
set.seed(seed)
pp <- rpoispp(lambda, win = dilation.owin(win, bufferdist), nsim = 1, drop = TRUE)
#plot(pp)
#plot(add = TRUE, w)
#place grains
set.seed(seed) #this is not best way, must be a way to continue using sequence of pseudo-independent random numbers in already selected seed.
xibuffer <- placegrainsfromlib(pp, grainlib, prob = weights, w = win, xy = xy)
#plot(w)
#plot(xibuffer, add = TRUE, col = "black")
#plot(xibuffer[square(r = 50)], col = "black")
xi <- intersect.owin(xibuffer, win)
return(xi)
}
#' @describeIn rbpto The coverage probability of the Boolean model with grain size distributed according to a truncated Pareto distribution.
bpto.coverageprob <- function(lambda, grain, xm, alpha,
lengthscales = 1:500){
#first get mean area. Need weight for each discrete grain.
#check that smallest scale is larger than xm
stopifnot(lengthscales[1] >= xm)
#get weights of grain sizes from pmf
weights <- alpha * xm ^ alpha / (lengthscales ^ (alpha + 1) )
weights <- weights / sum(weights) #standardise
meangrainarea <- sum(lengthscales * lengthscales * area.owin(grain) * weights)
return(1 - exp(- lambda * meangrainarea))
}
#' @describeIn rbpto The germ intensity of the Boolean model with grain size distributed according to a truncated Pareto distribution.
bpto.germintensity <- function(coverp, grain, xm, alpha,
lengthscales = 1:500){
#first get mean area. Need weight for each discrete grain.
#check that smallest scale is larger than xm
stopifnot(lengthscales[1] >= xm)
#get weights of grain sizes from pmf
weights <- alpha * xm ^ alpha / (lengthscales ^ (alpha + 1) )
weights <- weights / sum(weights) #standardise
meangrainarea <- sum(lengthscales * lengthscales * area.owin(grain) * weights)
#coverp formula: p = 1 - exp(- lambda * meangrainarea)
lambda <- - 1 * log(1 - coverp)/ meangrainarea
return(lambda)
}
#' @describeIn rbpto The covariance of the Boolean model with grain size distributed according to a truncated Pareto distribution.
#' \code{xy} is required to specify resolution and offset of pixel grid.
bpto.covar <- function(lambda, grain, xm, alpha, lengthscales = 1:500, xy){
#check that smallest scale is larger than xm
stopifnot(lengthscales[1] >= xm)
#first get grainlib with weights
#get scaled versions of grain
grainlib <- mapply(scalardilate, X = list(grain), f = lengthscales, SIMPLIFY = FALSE)
grainlib <- as.solist(grainlib)
#get weights of these grains from pmf
weights <- alpha * xm ^ alpha / (lengthscales ^ (alpha + 1) )
weights <- weights / sum(weights) #standardise
covar <- covar.grainlib(lambda, grainlib, weights, xy)
return(covar)
}
```

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