#' Multitype (cross-type) neighbourhood density function
#'
#' Computes the cross-type neighbourhood density function, a local version of
#' the \eqn{K}-function or \eqn{L}-function, defined by Getis and Franklin (1987).
#'
#' The command \code{local_l_cross} computes the \emph{neighbourhood density function},
#' a local version of the \eqn{L}-function (Besag's transformation of Ripley's
#' \eqn{K}-function) proposed by Getis and Franklin (1987), for 2 types in a multitype
#' spatial point pattern. The command \code{local_k_cross} computes the corresponding
#' local analogue of the cross-type K-function.
#'
#' Given a multitype spatial point pattern \code{X} with types \code{i} and \code{j},
#' the neighbourhood density function
#' \eqn{L_{ij}(r)}{L[ij](r)} associated with the \eqn{i}th point
#' in \code{X} is computed by
#' \deqn{
#' L_{ij}(r) = \sqrt{\frac a {(n_j) \pi} \sum_j e_{ij}}
#' }{
#' L[ij](r) = sqrt( (a/((n[j])* pi)) * sum[j] e[i,j])
#' }
#' where the sum is over all points \eqn{j}{j} that lie
#' within a distance \eqn{r} of the \eqn{i}th point,
#' \eqn{a} is the area of the observation window, \eqn{n_j}{n[j]} is the number
#' of \code{j} points in \code{X}, and \eqn{e_{ij}}{e[i,j]} is an edge correction
#' term (as described in \code{\link{Kest}}).
#' The value of \eqn{L_{ij}(r)}{L[ij](r)} can also be interpreted as one
#' of the summands that contributes to the global estimate of the cross-type
#' \eqn{L}-function.
#'
#' By default, the function \eqn{L_{ij}(r)}{L[ij](r)} or
#' \eqn{K_{ij}(r)}{K[ij](r)} is computed for a range of \eqn{r} values
#' for each point \eqn{i}. The results are stored as a function value
#' table (object of class \code{"fv"}) with a column of the table
#' containing the function estimates for each point \eqn{i} of the pattern
#' \code{X}.
#'
#' Alternatively, if the argument \code{rvalue} is given, and it is a
#' single number, then the function will only be computed for this value
#' of \eqn{r}, and the results will be returned as a numeric vector,
#' with one entry of the vector for each point \eqn{i} of the pattern \code{X}.
#'
#' Inhomogeneous counterparts of \code{local_k_cross} and \code{local_l_cross}
#' are computed by \code{local_k_cross_inhom} and \code{local_l_cross_inhom}.
#'
#' Computation can be done in parallel by registering a parallel backend for
#' the \code{\link{foreach}} package.
#'
#' @aliases local_l_cross
#' @seealso \code{\link{localL}}
#' \code{\link{localK}}
#' \code{\link{Lcross}}
#' \code{\link{Kcross}}
#' @param X Multitype point pattern (\code{\link{ppp}} object). It must be a multitype
#' point pattern (or marked point pattern).
#' @param i The type (mark value) of the points in X from which distances are
#' measured. A character string (or something that will be converted to a
#' character string). Defaults to the first level of \code{\link{marks}(X)}.
#' @param j The type (marks value) of the points in X to which distances are
#' measured. A character string (or something that will be converted to a
#' character string'). Defaults to the second level of \code{\link{marks}(X)}.
#' @param ... Ignored.
#' @param correction String sprecifying the edge correction to be applied.
#' Option are \code{"none"}, \code{"translate"}, \code{"translation"},
#' \code{"Ripley"}, \code{"isotropic"} or \code{"best"}. Only one correction
#' may be given.
#' @param verbose Logical flag indicating whether to print progress reports
#' during the calculation.
#' @param rvalue Optional. A \emph{single} value of the distance argument \eqn{r}
#' at which the function L or K should be computed.
#' @return If \code{rvalue} is given, the result is a numeric vector of equal
#' length to the number of points in X_i.
#'
#' If the code{rvalue} is absent, the result is an object of class \code{"fv"},
#' see \code{\link{fv.object}}, which can be plotted directly using
#' \code{\link{plot.fv}}. Essentially a data frame containing columns:
#' \item{r}{the vector of values of the argument \eqn{r}
#' at which the function \eqn{K} has been estimated}
#' \item{theo}{the theoretical value \eqn{K(r) = \pi r^2}{K(r) = pi * r^2}
#' or \eqn{L(r)=r} for a stationary Poisson process
#' }
#' together with columns containing the values of the
#' neighbourhood density function for each point in the pattern.
#' Column \code{i} corresponds to the \code{i}th point.
#' The last two columns contain the \code{r} and \code{theo} values.
#' @references Getis, A. and Franklin, J. (1987)
#' Second-order neighbourhood analysis of mapped point patterns.
#' \emph{Ecology} \bold{68}, 473--477.
local_k_cross <- function(X, i, j, ..., correction="Ripley", verbose=TRUE, rvalue=NULL) {
spatstat::verifyclass(X, "ppp")
if(!spatstat::is.multitype(X, dfok=FALSE))
stop("Point pattern must be multitype")
marx <- spatstat::marks(X)
if(missing(i))
i <- levels(marx)[1]
if(missing(j))
j <- levels(marx)[2]
I <- (marx == i)
if(!any(I))
stop(paste("No points have mark i =", i))
J <- (marx == j)
if(!any(J))
stop(paste("No points have mark j =", j))
local_k_multi_engine(X=X, I=I, J=J, ..., correction=correction,
verbose=verbose, rvalue=rvalue)
}
#' @export
local_l_cross <- function(X, i, j, ..., correction="Ripley", verbose=TRUE, rvalue=NULL) {
local_k_cross(X=X, i=i, j=j, ..., wantL=TRUE, correction=correction,
verbose=verbose, rvalue=rvalue)
}
#' Inhomogeneous multitype (cross-type) neighbourhood density function
#'
#' Computes spatially-weighted versions of the local \eqn{K}-function or
#' \eqn{L}-function, defined by Getis and Franklin (1987).
#'
#' The command \code{local_l_cross_inhom} computes the \emph{neighbourhood density function},
#' a local version of the \eqn{L}-function (Besag's transformation of Ripley's
#' \eqn{K}-function) proposed by Getis and Franklin (1987), for 2 types in a multitype
#' spatial point pattern. The command \code{local_k_cross} computes the corresponding
#' local analogue of the cross-type K-function.
#'
#' Given a multitype spatial point pattern \code{X} with types \code{i} and \code{j},
#' the neighbourhood density function
#' \eqn{L_{ij}(r)}{L[ij](r)} associated with the \eqn{i}th point
#' in \code{X} is computed by
#' \deqn{
#' L_{ij}(r) = \sqrt{\frac a {(n_j) \pi} \sum_j e_{ij}}
#' }{
#' L[ij](r) = sqrt( (a/((n[j])* pi)) * sum[j] e[i,j])
#' }
#' where the sum is over all points \eqn{j}{j} that lie
#' within a distance \eqn{r} of the \eqn{i}th point,
#' \eqn{\lambda_j}{\lambda[j]} is the estimated intensity of type \eqn{j} of the
#' point pattern at the point \eqn{j}, and \eqn{e_{ij}}{e[i,j]} is an edge correction
#' term (as described in \code{\link{Kest}}).
#' The value of \eqn{L_{ij}(r)}{L[ij](r)} can also be interpreted as one
#' of the summands that contributes to the global estimate of the cross-type \eqn{L}-
#' function.
#'
#' By default, the function \eqn{L_{ij}(r)}{L[ij](r)} or
#' \eqn{K_{ij}(r)}{K[ij](r)} is computed for a range of \eqn{r} values
#' for each point \eqn{i}. The results are stored as a function value
#' table (object of class \code{"fv"}) with a column of the table
#' containing the function estimates for each point \eqn{i} of the pattern
#' \code{X}.
#'
#' Alternatively, if the argument \code{rvalue} is given, and it is a
#' single number, then the function will only be computed for this value
#' of \eqn{r}, and the results will be returned as a numeric vector,
#' with one entry of the vector for each point \eqn{i} of the pattern \code{X}.
#'
#' Computation can be done in parallel by registering a parallel backend for
#' the \code{\link{foreach}} package.
#'
#' @aliases local_l_cross_inhom
#' @seealso \code{\link{localLinhom}}
#' \code{\link{localKinhom}}
#' \code{\link{Lcross.inhom}}
#' \code{\link{Kcross.inhom}}
#' @param X Multitype point pattern (\code{\link{ppp}} object). It must be a multitype
#' point pattern (or marked point pattern).
#' @param i The type (mark value) of the points in X from which distances are
#' measured. A character string (or something that will be converted to a
#' character string). Defaults to the first level of \code{\link{marks}(X)}.
#' @param j The type (marks value) of the points in X to which distances are
#' measured. A character string (or something that will be converted to a
#' character string'). Defaults to the second level of \code{\link{marks}(X)}.
#' @param lambdaI Optional. Values of the estimated intensity of the sub-process
#' of points of type \code{i}. Either a pixel image (object of class \code{"im"}),
#' a numeric vector containing the intensity values at each of the type
#' \code{i} points in \code{X}, or a \code{function(x,y)} which can be
#' evaluated to give the intensity value at any location.
#' @param lambdaJ Optional. Values of the estimated intensity of the sub-process
#' of points of type \code{j}. Either a pixel image (object of class \code{"im"}),
#' a numeric vector containing the intensity values at each of the type
#' \code{j} points in \code{X}, or a \code{function(x,y)} which can be
#' evaluated to give the intensity value at any location.
#' @param ... Ignored.
#' @param correction String sprecifying the edge correction to be applied.
#' Option are \code{"none"}, \code{"translate"}, \code{"translation"},
#' \code{"Ripley"}, \code{"isotropic"} or \code{"best"}. Only one correction
#' may be given.
#' @param verbose Logical flag indicating whether to print progress reports
#' during the calculation.
#' @param rvalue Optional. A \emph{single} value of the distance argument \eqn{r}
#' at which the function L or K should be computed.
#' @param lambdaIJ Optional. A matrix containing estimates of the product of the
#' intensities \code{lambdaI} and \code{lambdaJ} for each pair of points
#' of types \code{i} and \code{j} respectively.
#' @param sigma Optional argument passed to \code{\link{density.ppp}} to control
#' the kernel smoothing procedure for estimating \code{lambda}, if
#' \code{lambda} is missing.
#' @param varcov Optional argument passed to \code{\link{density.ppp}} to control
#' the kernel smoothing procedure for estimating \code{lambda}, if
#' \code{lambda} is missing.
#' @return If \code{rvalue} is given, the result is a numeric vector of equal
#' length to the number of points in X_i.
#'
#' If the code{rvalue} is absent, the result is an object of class \code{"fv"},
#' see \code{\link{fv.object}}, which can be plotted directly using
#' \code{\link{plot.fv}}. Essentially a data frame containing columns:
#' \item{r}{the vector of values of the argument \eqn{r}
#' at which the function \eqn{K} has been estimated }
#' \item{theo}{the theoretical value \eqn{K(r) = \pi r^2}{K(r) = pi * r^2}
#' or \eqn{L(r)=r} for a stationary Poisson process
#' }
#' together with columns containing the values of the
#' neighbourhood density function for each point in the pattern.
#' Column \code{i} corresponds to the \code{i}th point.
#' The last two columns contain the \code{r} and \code{theo} values.
#' @references Getis, A. and Franklin, J. (1987)
#' Second-order neighbourhood analysis of mapped point patterns.
#' \emph{Ecology} \bold{68}, 473--477.
local_k_cross_inhom <-
function(X, i, j, lambdaI=NULL, lambdaJ=NULL, ..., correction="Ripley",
verbose=TRUE, rvalue=NULL, lambdaIJ= NULL, sigma=NULL, varcov=NULL) {
spatstat::verifyclass(X, "ppp")
if(!spatstat::is.multitype(X, dfok=FALSE))
stop("Point pattern must be multitype")
marx <- spatstat::marks(X)
if(missing(i))
i <- levels(marx)[1]
if(missing(j))
j <- levels(marx)[2]
I <- (marx == i)
if(!any(I))
stop(paste("No points have mark i =", i))
nI <- sum(I)
J <- (marx == j)
if(!any(J))
stop(paste("No points have mark j =", j))
nJ <- sum(J)
dangerous <- c("lambdaI", "lambdaJ", "lambdaIJ")
dangerI <- dangerJ <- TRUE
## intensity data
if(is.null(lambdaI)) {
# estimate intensity
dangerI <- FALSE
lambdaI <- density(X[I], ..., sigma=sigma, varcov=varcov,
at="points", leaveoneout=TRUE)
} else if(spatstat::is.im(lambdaI)) {
# look up intensity values
lambdaI <- spatstat::safelookup(lambdaI, X[I])
} else if(is.function(lambdaI)) {
# evaluate function at locations
XI <- X[I]
lambdaI <- lambdaI(XI$x, XI$y)
} else if(is.numeric(lambdaI) && is.vector(as.numeric(lambdaI))) {
# validate intensity vector
if(length(lambdaI) != nI)
stop(paste("The length of", sQuote("lambdaI"),
"should equal the number of", i))
} else
stop(paste(sQuote("lambdaI"), "should be a vector or an image"))
if(is.null(lambdaJ)) {
# estimate intensity
dangerJ <- FALSE
lambdaJ <- density(X[J], ..., sigma=sigma, varcov=varcov,
at="points", leaveoneout=TRUE)
} else if(spatstat::is.im(lambdaJ)) {
# look up intensity values
lambdaJ <- spatstat::safelookup(lambdaJ, X[J])
} else if(is.function(lambdaJ)) {
# evaluate function at locations
XJ <- X[J]
lambdaJ <- lambdaJ(XJ$x, XJ$y)
} else if(is.numeric(lambdaJ) && is.vector(as.numeric(lambdaJ))) {
# validate intensity vector
if(length(lambdaJ) != nJ)
stop(paste("The length of", sQuote("lambdaJ"),
"should equal the number of", j))
} else
stop(paste(sQuote("lambdaJ"), "should be a vector or an image"))
# Weight for each pair
if(!is.null(lambdaIJ)) {
dangerIJ <- TRUE
if(!is.matrix(lambdaIJ))
stop("lambdaIJ should be a matrix")
if(nrow(lambdaIJ) != nI)
stop(paste("nrow(lambdaIJ) should equal the number of", i))
if(ncol(lambdaIJ) != nJ)
stop(paste("ncol(lambdaIJ) should equal the number of", j))
} else dangerIJ <- FALSE
danger <- dangerI || dangerJ || dangerIJ
if(danger) {
d <- dangerous[c(dangerI, dangerJ, dangerIJ)]
warning(paste("User specified: ", d))
}
local_k_multi_engine(X=X, I=I, J=J, lambdaI=lambdaI, lambdaJ=lambdaJ,
..., correction=correction, lambdaIJ=lambdaIJ,
verbose=verbose, rvalue=rvalue)
}
#' @export
local_l_cross_inhom <-
function(X, i, j, lambdaI=NULL, lambdaJ=NULL, ..., correction="Ripley",
verbose=TRUE, rvalue=NULL, lambdaIJ=NULL, sigma=NULL, varcov=NULL) {
local_k_cross_inhom(X=X, i=i, j=j, lambdaI=lambdaI, lambdaJ=lambdaJ, ...,
wantL=TRUE, correction=correction, verbose=verbose, rvalue=rvalue,
lambdaIJ=lambdaIJ, sigma=sigma, varcov=varcov)
}
local_k_multi_engine <-
function(X, I, J, lambdaI=NULL, lambdaJ=NULL, ..., wantL=FALSE,
correction="Ripley", lambdaIJ=NULL, verbose=TRUE, rvalue=NULL) {
spatstat::verifyclass(X, "ppp")
npts <- spatstat::npoints(X)
x <- X$x
y <- X$y
W <- X$window
areaW <- spatstat::area(W)
I <- spatstat::ppsubset(X, I)
J <- spatstat::ppsubset(X, J)
if(is.null(I) || is.null(J))
stop("I and J must be valid subset indices")
if(!any(I)) stop("no points belong to subset I")
if(!any(J)) stop("no points belong to subset J")
nI <- sum(I)
nJ <- sum(J)
lambdaI.ave <- nI/areaW
lambdaJ.ave <- nJ/areaW
if(is.null(rvalue))
rmaxdefault <- spatstat::rmax.rule("K", W, lambdaJ.ave)
else {
stopifnot(is.numeric(rvalue))
stopifnot(length(rvalue) == 1)
stopifnot(rvalue >= 0)
rmaxdefault <- rvalue
}
breaks <- spatstat::handle.r.b.args(NULL, NULL, W, rmaxdefault=rmaxdefault)
r <- breaks$r
rmax <- breaks$max
correction.given <- !missing(correction)
correction <- spatstat::pickoption("correction", correction,
c(none="none",
isotropic="isotropic",
Ripley="isotropic",
trans="translate",
translate="translate",
translation="translate",
best="best"),
multi=FALSE)
correction <- spatstat::implemented.for.K(correction, W$type, correction.given)
# recommended range of r values
alim <- c(0, min(rmax, rmaxdefault))
#find close pairs
XI <- X[I]
XJ <- X[J]
inpts <- spatstat::npoints(XI)
close <- spatstat::crosspairs(XI, XJ, rmax)
# close$i and close$j are serial numbers in XI and XJ respectively;
# map them to original serial numbers in X
orig <- seq_len(npts)
imap <- orig[I]
jmap <- orig[J]
iX <- imap[close$i]
jX <- jmap[close$j]
# eliminate any identical pairs
if(any(I & J)) {
ok <- (iX != jX)
if(!all(ok)) {
close$i <- close$i[ok]
close$j <- close$j[ok]
close$xi <- close$xi[ok]
close$yi <- close$yi[ok]
close$xj <- close$xj[ok]
close$yj <- close$yj[ok]
close$dx <- close$dx[ok]
close$dy <- close$dy[ok]
close$d <- close$d[ok]
}
}
# extract information for these pairs (relative to orderings of XI, XJ)
dcloseIJ <- close$d
icloseI <- close$i
jcloseJ <- close$j
weighted <- FALSE
# Form weight for each pair
if(!is.null(lambdaI) && !is.null(lambdaJ)) {
weight <- 1/(lambdaI[icloseI] * lambdaJ[jcloseJ])
weighted <- TRUE
} else if(!is.null(lambdaIJ)) {
weight <- 1/lambdaIJ[cbind(icloseI, jcloseJ)]
weighted <- TRUE
}
# initialise
iseq <- 1:inpts
icode <- sprintf("%03d", iseq)
bkt <- function(x) { paste("[", x, "]", sep="") }
parallel <- check_parallel(verbose)
fe <- foreach::foreach(i = iseq, .export = c("local_k_calc"))
switch(correction,
none={
# uncorrected! For demonstration purposes only!
df <- foreach::`%dopar%`(fe, local_k_calc(i, closeI=icloseI, distIJ=dcloseIJ, breaks=breaks,
weight=if (weighted) weight else NULL,
pb = if(verbose && !parallel) function(i) spatstat::progressreport(i, n=npts) else NULL))
# Hack to quickly convert list to data.frame
attributes(df) <- list(row.names=c(NA_integer_, length(r)),
class="data.frame",
names=make.names(paste("un", icode, sep=""),
unique=TRUE))
labl <- paste("%s", bkt(icode), "(r)", sep="")
desc <- paste("uncorrected estimate of %s",
"for point", icode)
if (!weighted) df <- df / lambdaJ.ave
},
translate={
edgewt <- spatstat::edge.Trans(XI[icloseI], XJ[jcloseJ], paired=TRUE)
if(weighted) {
edgewt <- edgewt * weight
}
df <- foreach::`%dopar%`(fe, local_k_calc(i, closeI=icloseI, distIJ=dcloseIJ,
breaks=breaks, weight=edgewt,
pb = if(verbose && !parallel) function(i) spatstat::progressreport(i, n=npts) else NULL))
# Hack to quickly convert list to data.frame
attributes(df) <- list(row.names=c(NA_integer_, length(r)),
class="data.frame",
names=make.names(paste("trans", icode, sep=""),
unique=TRUE))
desc <- paste("translation-corrected estimate of %s",
"for point", icode)
labl <- paste("%s", bkt(icode), "(r)", sep="")
if (!weighted) df <- df / lambdaJ.ave
h <- spatstat::diameter(W) / 2
df[r >= h, ] <- NA
},
isotropic={
edgewt <- spatstat::edge.Ripley(XI[icloseI], matrix(dcloseIJ, ncol=1))
if (weighted) {
edgewt <- edgewt * weight
}
df <- foreach::`%dopar%`(fe, local_k_calc(i, closeI=icloseI, distIJ=dcloseIJ,
breaks=breaks, weight=edgewt,
pb = if(verbose && !parallel) function(i) spatstat::progressreport(i, n=npts) else NULL))
# Hack to quickly convert list to data.frame
attributes(df) <- list(row.names=c(NA_integer_, length(r)),
class="data.frame",
names=make.names(paste("iso", icode, sep=""),
unique=TRUE))
desc <- paste("Ripley isotropic correction estimate of %s",
"for point", icode)
labl <- paste("%s", bkt(icode), "(r)", sep="")
if (!weighted) df <- df / lambdaJ.ave
h <- spatstat::diameter(W) / 2
df[r >= h, ] <- NA
})
if (wantL) {
df <- sqrt(df / pi)
}
if(!is.null(rvalue)) {
nr <- length(r)
if(r[nr] != rvalue)
stop("Internal error - rvalue not attained")
return(as.numeric(df[nr,]))
}
# function value table required
# add r and theo
if (!wantL) {
df <- cbind(df, data.frame(r=r, theo=pi * r^2))
if (!weighted) {
ylab <- quote(K[ij](r))
fnam <- "K[ij][',']"
} else {
ylab <- quote(Kinhom[ij](r))
fnam <- "Kinhom[ij][',']"
}
} else {
df <- cbind(df, data.frame(r=r, theo=r))
if (!weighted) {
ylab <- quote(L[ij](r))
fnam <- "L[ij][',']"
} else {
ylab <- quote(Linhom[ij](r))
fnam <- "Linhom[ij][',']"
}
}
desc <- c(desc, c("distance argument r", "theoretical Poisson %s"))
labl <- c(labl, c("r", "%s[pois](r)"))
# create fv object
K <- spatstat::fv(df, "r", ylab, "theo", "", alim, labl, desc, fname=fnam)
# default is to display them all
spatstat::formula(K) <- (. ~ r)
spatstat::unitname(K) <- spatstat::unitname(X)
attr(K, "correction") <- correction
return(K)
}
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