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

In spatstat.explore: Exploratory Data Analysis for the 'spatstat' Family

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)

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	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_{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 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.

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_{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 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_{ij}(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.

Value

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

Essentially a data frame containing six numeric columns

J	the recommended estimator of J_{ij}(r), currently the Kaplan-Meier estimator.
r	the values of the argument r at which the function J_{ij}(r) has been estimated
km	the Kaplan-Meier estimator of J_{ij}(r)
rs	the “reduced sample” or “border correction” estimator of J_{ij}(r)
han	the Hanisch-style estimator of J_{ij}(r)
un	the “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.
theo	the 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.

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(s)

\spatstatAuthors

.

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.

See Also

Jdot, Jest, Jmulti

Examples

     # Lansing woods data: 6 types of trees
    woods <- lansing
    
    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)

spatstat.explore documentation built on Oct. 22, 2024, 9:07 a.m.