spatstat (version 1.3-2)

# Jcross: Multitype J Function (i-to-j)

## Description

For a multitype point pattern, estimate the multitype $J$ function summarising the interpoint dependence between points of type $i$ and of type $j$.

## Usage

Jcross(X, i=1, j=2)
Jcross(X, i=1, j=2, eps, r)
Jcross(X, i=1, j=2, eps, breaks)

## Arguments

X
The observed point pattern, from which an estimate of the multitype $J$ function $J_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.
i
Number or character string identifying the type (mark value) of the points in X from which distances are measured.
j
Number or character string identifying the type (mark value) of the points in X to which distances are measured.
eps
A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
r
numeric vector. The values of the argument $r$ at which the function $J_{ij}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions o
breaks
An alternative to the argument r. Not normally invoked by the user. See the Details section.

## Value

• A data frame containing six numeric columns
• Jthe recommended estimator of $J_{ij}(r)$, currently the Kaplan-Meier estimator.
• rthe values of the argument $r$ at which the function $J_{ij}(r)$ has been estimated
• kmthe Kaplan-Meier estimator of $J_{ij}(r)$
• rsthe reduced sample'' or border correction'' estimator of $J_{ij}(r)$
• unthe uncorrected'' estimator of $J_{ij}(r)$ formed by taking the ratio of uncorrected empirical estimators of $1 - G_{ij}(r)$ and $1 - F_{j}(r)$, see Gdot and Fest.
• theothe theoretical value of $J_{ij}(r)$ for a marked Poisson process, namely 1.
• The result also has two attributes "G" and "F" which are respectively the outputs of Gcross and Fest for the point pattern.

## synopsis

Jcross(X, i=1, j=2, eps=NULL, r=NULL, breaks=NULL)

The argument i is interpreted as a level of the factor X$marks. Beware of the usual trap with factors: numerical values are not interpreted in the same way as character values. See the first example. ## Details This function Jcross and its companions Jdot and Jmulti are generalisations of the function Jest to multitype point patterns. A multitype point pattern is a spatial pattern of points classified into a finite number of possible colours'' or types''. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point. The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor. The argument i will be interpreted as a level of the factor X$marks. (Warning: this means that an integer value i=3 will be interpreted as the 3rd smallest level, not the number 3). The type$i$to type$j$'' multitype$J$function of a stationary multitype point process$X$was introduced by Van lieshout and Baddeley (1999). It is defined by $$J_{ij}(r) = \frac{1 - G_{ij}(r)}{1 - F_{j}(r)}$$ where$G_{ij}(r)$is the distribution function of the distance from a type$i$point to the nearest point of type$j$, and$F_{j}(r)$is the distribution function of the distance from a fixed point in space to the nearest point of type$j$in the pattern. An estimate of$J_{ij}(r)$is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the subprocess of type$i$points is independent of the subprocess of points of type$j$, then$J_{ij}(r) \equiv 1$. Hence deviations of the empirical estimate of$J_{ij}$from the value 1 may suggest dependence between types. This algorithm estimates$J_{ij}(r)$from the point pattern X. It assumes that X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as X$window) may have arbitrary shape. Biases due to edge effects are treated in the same manner as in Jest, using the Kaplan-Meier and border corrections. The main work is done by Gmulti and Fest.

The argument r is the vector of values for the distance $r$ at which $J_{ij}(r)$ should be evaluated. The values of $r$ must be increasing nonnegative numbers and the maximum $r$ value must exceed the radius of the largest disc contained in the window.

## References

Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344--361.

Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511--532.

Jdot, Jest, Jmulti
library(spatstat)
<testonly>sub <- lansing[seq(1,lansing$n, by=30), ] Jhm <- Jcross(sub, "hickory", "maple")</testonly> # diagnostic plot for independence between hickories and maples plot(Jhm$r, Jhm$J, xlab="r", ylab="Jhm(r)", type="l") abline(h=1, lty=2) # synthetic example with two marks "a" and "b" pp <- runifpoispp(50) pp$marks <- sample(c("a","b"), pp\$n, replace=TRUE)
J <- Jcross(pp, "a", "b")