spatstat (version 1.54-0)

Jdot: Multitype J Function (i-to-any)

Description

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.

Usage

Jdot(X, i, eps=NULL, r=NULL, breaks=NULL, …, correction=NULL)

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

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).

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 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.

Value

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

Essentially a data frame containing six numeric columns

J

the recommended estimator of $J_{i\bullet }(r)$, currently the Kaplan-Meier estimator.

r

the values of the argument $r$ at which the function $J_{i\bullet }(r)$ has been estimated

km

the Kaplan-Meier estimator of $J_{i\bullet }(r)$

rs

the ``reduced sample'' or ``border correction'' estimator of $J_{i\bullet }(r)$

han

the Hanisch-style estimator of $J_{i\bullet }(r)$

un

the ``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.

theo

the 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.

Warnings

The argument i is interpreted as a level of the factor X$marks. It is converted to a character string if it is not already a character string. The value i=1 does not refer to the first level of the factor.

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 number 3, not the 3rd smallest level.)

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 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 $J_{i\bullet }(r)$ 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.

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

Jcross, Jest, Jmulti

Examples

# NOT RUN {
     # Lansing woods data: 6 types of trees
   woods <- lansing

    
# }
# NOT RUN {
    Jh. <- Jdot(woods, "hickory")
    plot(Jh.)
    # diagnostic plot for independence between hickories and other trees
    Jhh <- Jest(split(woods)$hickory)
    plot(Jhh, add=TRUE, legendpos="bottom")

    
# }
# NOT RUN {
    # synthetic example with two marks "a" and "b"
    pp <- runifpoint(30) %mark% factor(sample(c("a","b"), 30, replace=TRUE))
    J <- Jdot(pp, "a")
    
# }