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, j, eps=NULL, r=NULL, breaks=NULL, ..., correction=NULL)

Value

An object of class "fv" (see fv.object).

Essentially a data frame containing six numeric columns

J: the recommended estimator of $J_{i j} (r)$ , currently the Kaplan-Meier estimator.
r: the values of the argument $r$ at which the function $J_{i j} (r)$ has been estimated
km: the Kaplan-Meier estimator of $J_{i j} (r)$
rs: the ``reduced sample'' or ``border correction'' estimator of $J_{i j} (r)$
han: the Hanisch-style estimator of $J_{i j} (r)$
un: the ``uncorrected'' estimator of $J_{i j} (r)$ formed by taking the ratio of uncorrected empirical estimators of $1 - G_{i j} (r)$ and $1 - F_{j} (r)$ , see Gdot and Fest.
theo: the theoretical value of $J_{i j} (r)$ 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.

Arguments

X: The observed point pattern, from which an estimate of the multitype $J$ function $J_{i j} (r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.
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 marks(X).
j: The type (mark 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 marks(X).
eps: A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
r: Optional. Numeric vector. The values of the argument $r$ at which the function $J_{i j} (r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on $r$ .
breaks: This argument is for internal use only.
...: Ignored.
correction: Optional. Character string specifying the edge correction(s) to be used. Options are "none", "rs", "km", "Hanisch" and "best". Alternatively correction="all" selects all options.

Warnings

The arguments i and j are always interpreted as levels of the factor X$marks. They are converted to character strings if they are not already character strings. The value i=1 does not refer to the first level of the factor.

Author

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

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 number 3, not the 3rd smallest level).

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 $J_{i j} (r) = \frac{1 - G_{i j} (r)}{1 - F_{j} (r)}$ where $G_{i j} (r)$ 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 $J_{i j} (r)$ 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 $J_{i j} (r) \equiv 1$ . Hence deviations of the empirical estimate of $J_{i j}$ from the value 1 may suggest dependence between types.

This algorithm estimates $J_{i j} (r)$ 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 Window(X)) 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 $J_{i j} (r)$ should be evaluated. The values of $r$ must be increasing nonnegative numbers and the maximum $r$ value must not 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.

Examples

Run this code

     # Lansing woods data: 6 types of trees
    woods <- lansing
    # \testonly{
       woods <- woods[seq(1,npoints(woods), by=30)]
    # }
    Jhm <- Jcross(woods, "hickory", "maple")
    # diagnostic plot for independence between hickories and maples
    plot(Jhm)

    # synthetic example with two types "a" and "b"
    pp <- runifpoint(30) %mark% factor(sample(c("a","b"), 30, replace=TRUE))
    J <- Jcross(pp)

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