Gmulti: Marked Nearest Neighbour Distance Function

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

Marked Nearest Neighbour Distance Function

Description

For a marked point pattern, estimate the distribution of the distance from a typical point in subset I to the nearest point of subset J.

Usage

Gmulti(X, I, J, r=NULL, breaks=NULL, ...,
        disjoint=NULL, correction=c("rs", "km", "han"))

Arguments

X	The observed point pattern, from which an estimate of the multitype distance distribution function G_{IJ}(r) will be computed. It must be a marked point pattern. See under Details.
I	Subset of points of X from which distances are measured.
J	Subset of points in X to which distances are measured.
r	Optional. Numeric vector. The values of the argument r at which the distribution function G_{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.
disjoint	Optional flag indicating whether the subsets I and J are disjoint. If missing, this value will be computed by inspecting the vectors I and J.
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

The function Gmulti generalises Gest (for unmarked point patterns) and Gdot and Gcross (for multitype point patterns) to arbitrary marked point patterns.

Suppose X_I, X_J are subsets, possibly overlapping, of a marked point process. This function computes an estimate of the cumulative distribution function G_{IJ}(r) of the distance from a typical point of X_I to the nearest distinct point of X_J.

The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to as.ppp.

The arguments I and J specify two subsets of the point pattern. They may be any type of subset indices, for example, logical vectors of length equal to npoints(X), or integer vectors with entries in the range 1 to npoints(X), or negative integer vectors.

Alternatively, I and J may be functions that will be applied to the point pattern X to obtain index vectors. If I is a function, then evaluating I(X) should yield a valid subset index. This option is useful when generating simulation envelopes using envelope.

This algorithm estimates the distribution function G_{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 Gest.

The argument r is the vector of values for the distance r at which G_{IJ}(r) should be evaluated. It is also used to determine the breakpoints (in the sense of hist) for the computation of histograms of distances. The reduced-sample and Kaplan-Meier estimators are computed from histogram counts. In the case of the Kaplan-Meier estimator this introduces a discretisation error which is controlled by the fineness of the breakpoints.

First-time users would be strongly advised not to specify r. However, if it is specified, r must satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the window. Furthermore, the successive entries of r must be finely spaced.

The algorithm also returns an estimate of the hazard rate function, \lambda(r), of G_{IJ}(r). This estimate should be used with caution as G_{IJ}(r) is not necessarily differentiable.

The naive empirical distribution of distances from each point of the pattern X to the nearest other point of the pattern, is a biased estimate of G_{IJ}. However this is also returned by the algorithm, as it is sometimes useful in other contexts. Care should be taken not to use the uncorrected empirical G_{IJ} as if it were an unbiased estimator of G_{IJ}.

Value

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

Essentially a data frame containing six numeric columns

r	the values of the argument r at which the function G_{IJ}(r) has been estimated
rs	the “reduced sample” or “border correction” estimator of G_{IJ}(r)
han	the Hanisch-style estimator of G_{IJ}(r)
km	the spatial Kaplan-Meier estimator of G_{IJ}(r)
hazard	the hazard rate \lambda(r) of G_{IJ}(r) by the spatial Kaplan-Meier method
raw	the uncorrected estimate of G_{IJ}(r), i.e. the empirical distribution of the distances from each point of type i to the nearest point of type j
theo	the theoretical value of G_{IJ}(r) for a marked Poisson process with the same estimated intensity

Warnings

The function G_{IJ} does not necessarily have a density.

The reduced sample estimator of G_{IJ} is pointwise approximately unbiased, but need not be a valid distribution function; it may not be a nondecreasing function of r. Its range is always within [0,1].

The spatial Kaplan-Meier estimator of G_{IJ} is always nondecreasing but its maximum value may be less than 1.

Author(s)

\spatstatAuthors

.

References

Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.

Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial point pattern. J. Neurosci. Meth. 18, 115–125.

Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293–303

Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.

Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.

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

Gcross, Gdot, Gest

Examples

    trees <- longleaf
     # Longleaf Pine data: marks represent diameter
    
    Gm <- Gmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
    plot(Gm)

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