Gcross: Multitype Nearest Neighbour Distance Function (i-to-j)

Description

For a multitype point pattern, estimate the distribution of the distance from a point of type $i$ to the nearest point of type $j$ .

Usage

Gcross(X, i, j, r=NULL, breaks=NULL, …, correction=c("rs", "km", "han"))

Arguments

The observed point pattern, from which an estimate of the cross type distance distribution function $G_{i j} (r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.

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).

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).

Optional. Numeric vector. The values of the argument $r$ at which the distribution function $G_{i j} (r)$ 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

the values of the argument $r$ at which the function $G_{i j} (r)$ has been estimated

the ``reduced sample'' or ``border correction'' estimator of $G_{i j} (r)$

han

the Hanisch-style estimator of $G_{i j} (r)$

the spatial Kaplan-Meier estimator of $G_{i j} (r)$

hazard

the hazard rate $λ (r)$ of $G_{i j} (r)$ by the spatial Kaplan-Meier method

raw

the uncorrected estimate of $G_{i j} (r)$ , 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 $G_{i j} (r)$ for a marked Poisson process with the same estimated intensity (see below).

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.

The function $G_{i j}$ does not necessarily have a density.

The reduced sample estimator of $G_{i j}$ 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 $G_{i j}$ is always nondecreasing but its maximum value may be less than $1$ .

Details

This function Gcross and its companions Gdot and Gmulti are generalisations of the function Gest 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$marks must be a factor. The arguments i and j will be interpreted as levels of the factor X$marks. (Warning: this means that an integer value i=3 will be interpreted as the number 3, not the 3rd smallest level).

The ``cross-type'' (type $i$ to type $j$ ) nearest neighbour distance distribution function of a multitype point process is the cumulative distribution function $G_{i j} (r)$ of the distance from a typical random point of the process with type $i$ the nearest point of type $j$ .

An estimate of $G_{i j} (r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the process of type $i$ points were independent of the process of type $j$ points, then $G_{i j} (r)$ would equal $F_{j} (r)$ , the empty space function of the type $j$ points. For a multitype Poisson point process where the type $i$ points have intensity $λ_{i}$ , we have $G_{i j} (r) = 1 - e^{- λ_{j} π r^{2}}$ Deviations between the empirical and theoretical $G_{i j}$ curves may suggest dependence between the points of types $i$ and $j$ .

This algorithm estimates the distribution function $G_{i j} (r)$ 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 $G_{i j} (r)$ 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, $λ (r)$ , of $G_{i j} (r)$ . This estimate should be used with caution as $G_{i j} (r)$ 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 $G_{i j}$ . 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 $G_{i j}$ as if it were an unbiased estimator of $G_{i j}$ .

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.

Examples

Run this code

# NOT RUN {
    # amacrine cells data
    G01 <- Gcross(amacrine)

    # equivalent to:
    # G01 <- Gcross(amacrine, "off", "on")

    plot(G01)

    # empty space function of `on' points
    if(interactive()) {
       F1 <- Fest(split(amacrine)$on, r = G01$r)
       lines(F1$r, F1$km, lty=3)
    }

    # synthetic example    
    pp <- runifpoispp(30)
    pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
    G <- Gcross(pp, "0", "1")   # note: "0" not 0
# }

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