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)
"ppp" or something acceptable to
as.ppp."isotropic", "Ripley" or "translate".
It specifies the edge correction(s) to be applied.f. If this argument is given,
then $f$ is assumed to take the form
$f(u,v)=f_1(u)f_1(v)$.normalise=FALSE,
compute only the numerator of the expression for the
mark correlation.returnL=FALSE
or the analogue of the L-function if returnL=TRUE.f or f1."fv" (see fv.object).
Essentially a data frame containing numeric columns"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).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.
Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008) Statistical analysis and modelling of spatial point patterns. Chichester: John Wiley.
markcorr to estimate the mark correlation function.# 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)Run the code above in your browser using DataLab