Kmark: Mark-Weighted K Function

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

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)

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.

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

\lambda K_f(r) = \frac{C_f(r)}{E[ f(M_1, M_2) ]}

where

C_f(r) = E \left[ \sum_{x \in X} f(m(u), m(x)) 1{0 < ||u - x|| \le r} \; \big| \; u \in X \right]

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 \pi \int_0^r s k_f(s) \, g(s) \, {\rm 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)/\pi}

is returned.

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 \pi 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).

Author(s)

\adrian

and \rolf

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.

See Also

markcorr to estimate the mark correlation function.

Examples

    # 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)

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