Kmark

0th

Percentile

Mark-Weighted K Function

Estimates the mark-weighted \(K\) function of a marked point pattern.

Keywords
spatial, nonparametric
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)K[f](r) obtained by the edge corrections named (if returnL=FALSE).

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.

Aliases
  • Kmark
  • markcorrint
Examples
# NOT RUN {
    # 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)
# }
Documentation reproduced from package spatstat, version 1.60-1, License: GPL (>= 2)

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