Jdot

From spatstat v1.9-2 by Adrian Baddeley

Multitype J Function (i-to-any)

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

Keywords: spatial, nonparametric

Usage

Jdot(X, i=1)
Jdot(X, i=1, eps, r)
Jdot(X, i=1, eps, breaks)

Arguments

X: The observed point pattern, from which an estimate of the multitype $J$ function $J_{i\bullet}(r)$ 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.
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 $J_{i\bullet}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important condit
breaks: An alternative to the argument r. Not normally invoked by the user. See the Details section.

Details

This function Jdot and its companions Jcross 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 any type'' multitype $J$ function of a stationary multitype point process $X$ was introduced by Van lieshout and Baddeley (1999). It is defined by $$J_{i\bullet}(r) = \frac{1 - G_{i\bullet}(r)}{1 - F_{\bullet}(r)}$$ where $G_{i\bullet}(r)$ is the distribution function of the distance from a type $i$ point to the nearest other point of the pattern, and $F_{\bullet}(r)$ is the distribution function of the distance from a fixed point in space to the nearest point of the pattern.

An estimate of $J_{i\bullet}(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the pattern is a marked Poisson point process, then $J_{i\bullet}(r) \equiv 1$. If the subprocess of type $i$ points is independent of the subprocess of points of all types not equal to $i$, then $J_{i\bullet}(r)$ equals $J_{ii}(r)$, the ordinary $J$ function (see Jest and Van Lieshout and Baddeley (1996)) of the points of type $i$. Hence deviations from zero of the empirical estimate of $J_{i\bullet} - J_{ii}$ may suggest dependence between types.

This algorithm estimates $J_{i\bullet}(r)$ 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 $J_{i\bullet}(r)$ 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.

Value

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

synopsis

Jdot(X, i=1, eps=NULL, r=NULL, breaks=NULL)

Warnings

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.

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

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

    <testonly>lansing <- lansing[seq(1,lansing$n, by=30), ]</testonly>
    Jh. <- Jdot(lansing, "hickory")
    plot(Jh.)
    # diagnostic plot for independence between hickories and other trees
    Jhh <- Jest(lansing[lansing$marks == "hickory", ])
    plot(Jhh, add=TRUE)

    # synthetic example with two marks "a" and "b"
    pp <- runifpoispp(50)
    pp <- pp %mark% sample(c("a","b"), pp$n, replace=TRUE)
    J <- Jdot(pp, "a")

Documentation reproduced from package spatstat, version 1.9-2, License: GPL version 2 or newer

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