Multitype J Function (i-to-any)

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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

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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 Ji.(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 Ji.(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 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

Ji.(r) = (1 - Gi.(r))/(1-F.(r))

where Gi.(r) is the distribution function of the distance from a type i point to the nearest other point of the pattern, and F.(r) is the distribution function of the distance from a fixed point in space to the nearest point of the pattern.

An estimate of Ji.(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 Ji.(r) = 1. If the subprocess of type i points is independent of the subprocess of points of all types not equal to i, then Ji.(r) equals Jii(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 Ji.-Jii may suggest dependence between types.

This algorithm estimates Ji.(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 Ji.(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

J

the recommended estimator of Ji.(r), currently the Kaplan-Meier estimator.

r

the values of the argument r at which the function Ji.(r) has been estimated

km

the Kaplan-Meier estimator of Ji.(r)

rs

the “reduced sample” or “border correction” estimator of Ji.(r)

han

the Hanisch-style estimator of Ji.(r)

un

the “uncorrected” estimator of Ji.(r) formed by taking the ratio of uncorrected empirical estimators of 1 - Gi.(r) and 1 - F.(r), see Gdot and Fest.

theo

the theoretical value of Ji.(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.

Author(s)

\adrian

and \rolf

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

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

    
    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")
    
## End(Not run)

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