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 Rfunction 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.
...
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
  • rthe values of the argument $r$ at which the mark correlation integral $K_f(r)$ has been estimated
  • theothe 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).

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
# 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(X$n)
    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 1.42-2, License: GPL (>= 2)

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