Kmark: Mark-Weighted K Function

Description

Estimates the mark-weighted $K$ function of a marked point pattern.

Usage

Kmark(X, f = NULL, r = NULL,
        correction = c("isotropic", "Ripley", "translate"), ...,
        f1 = NULL, normalise = TRUE, returnL = FALSE, fargs = NULL)
  markcorrint(X, f = NULL, r = NULL,
              correction = c("isotropic", "Ripley", "translate"), ...,
              f1 = NULL, normalise = TRUE, returnL = FALSE, fargs = NULL)

Value

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

Essentially a data frame containing numeric columns

r: the values of the argument $r$ at which the mark correlation integral $K_{f} (r)$ has been estimated
theo: the theoretical value of $K_{f} (r)$ when the marks attached to different points are independent, namely $π r^{2}$

together with a column or columns named

"iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the mark-weighted $K$ function $K_{f} (r)$

obtained by the edge corrections named (if returnL=FALSE).

Arguments

X: The observed point pattern. An object of class "ppp" or something acceptable to as.ppp.
f: Optional. Test function $f$ used in the definition of the mark correlation function. An R function with at least two arguments. There is a sensible default.
r: Optional. Numeric vector. The values of the argument $r$ at which the mark correlation function $k_{f} (r)$ should be evaluated. There is a sensible default.
correction: A character vector containing any selection of the options "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.
...: Ignored.
f1: An alternative to f. If this argument is given, then $f$ is assumed to take the form $f (u, v) = f_{1} (u) f_{1} (v)$ .
normalise: If normalise=FALSE, compute only the numerator of the expression for the mark correlation.
returnL: Compute the analogue of the K-function if returnL=FALSE or the analogue of the L-function if returnL=TRUE.
fargs: Optional. A list of extra arguments to be passed to the function f or f1.

Author

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

Details

The functions Kmark and markcorrint are identical. (Eventually markcorrint will be deprecated.)

The mark-weighted $K$ function $K_{f} (r)$ of a marked point process (Penttinen et al, 1992) is a generalisation of Ripley's $K$ function, in which the contribution from each pair of points is weighted by a function of their marks. If the marks of the two points are $m_{1}, m_{2}$ then the weight is proportional to $f (m_{1}, m_{2})$ where $f$ is a specified test function.

The mark-weighted $K$ function is defined so that $λ K_{f} (r) = \frac{C_{f} (r)}{E [f (M_{1}, M_{2})]}$ where $C_{f} (r) = E [\sum_{x \in X} f (m (u), m (x)) 1 0 < | | u - x | | \leq r | u \in X]$ for any spatial location $u$ taken to be a typical point of the point process $X$ . Here $| | u - x | |$ is the euclidean distance between $u$ and $x$ , so that the sum is taken over all random points $x$ that lie within a distance $r$ of the point $u$ . The function $C_{f} (r)$ is the unnormalised mark-weighted $K$ function. To obtain $K_{f} (r)$ we standardise $C_{f} (r)$ by dividing by $E [f (M_{1}, M_{2})]$ , the expected value of $f (M_{1}, M_{2})$ when $M_{1}$ and $M_{2}$ are independent random marks with the same distribution as the marks in the point process.

Under the hypothesis of random labelling, the mark-weighted $K$ function is equal to Ripley's $K$ function, $K_{f} (r) = K (r)$ .

The mark-weighted $K$ function is sometimes called the mark correlation integral because it is related to the mark correlation function $k_{f} (r)$ and the pair correlation function $g (r)$ by $K_{f} (r) = 2 π \int_{0}^{r} s k_{f} (s) g (s) d s$ See markcorr for a definition of the mark correlation function.

Given a marked point pattern X, this command computes edge-corrected estimates of the mark-weighted $K$ function. If returnL=FALSE then the estimated function $K_{f} (r)$ is returned; otherwise the function $L_{f} (r) = \sqrt{K_{f} (r) / π}$ is returned.

References

Penttinen, A., Stoyan, D. and Henttonen, H. M. (1992) Marked point processes in forest statistics. Forest Science 38 (1992) 806-824.

Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008) Statistical analysis and modelling of spatial point patterns. Chichester: John Wiley.

Examples

Run this code

    # CONTINUOUS-VALUED MARKS:
    # (1) Spruces
    # marks represent tree diameter
    # mark correlation function
    ms <- Kmark(spruces)
    plot(ms)

    # (2) simulated data with independent marks
    X <- rpoispp(100)
    X <- X %mark% runif(npoints(X))
    Xc <- Kmark(X)
    plot(Xc)
    
    # MULTITYPE DATA:
    # Hughes' amacrine data
    # Cells marked as 'on'/'off'
    M <- Kmark(amacrine, function(m1,m2) {m1==m2},
                         correction="translate")
    plot(M)

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