# Jcross

0th

Percentile

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

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

Keywords
spatial
##### 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.
##### 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.

##### 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

• Jcross
##### Examples
# Lansing woods data: 6 types of trees
data(lansing)

<testonly>lansing <- lansing[seq(1,lansing$n, by=30), ]</testonly> Jhm <- Jcross(lansing, "hickory", "maple") # diagnostic plot for independence between hickories and maples plot(Jhm) # synthetic example with two marks "a" and "b" pp <- runifpoispp(50) pp <- pp %mark% sample(c("a","b"), pp$n, replace=TRUE)
J <- Jcross(pp, "a", "b")
Documentation reproduced from package spatstat, version 1.5-5, License: GPL version 2 or newer

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