Estimate the marked connection function of a multitype point pattern.
markconnect(X, i, j, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", …, normalise=FALSE)The observed point pattern.
An object of class "ppp" or something acceptable to
as.ppp.
Number or character string identifying the type (mark value)
of the points in X from which distances are measured.
Number or character string identifying the type (mark value)
of the points in X to which distances are measured.
numeric vector. The values of the argument \(r\) at which the mark connection function \(p_{ij}(r)\) should be evaluated. There is a sensible default.
A character vector containing any selection of the
options "isotropic", "Ripley" or "translate".
It specifies the edge correction(s) to be applied.
A character vector indicating the user's choice of
density estimation technique to be used. Options are
"density",
"loess",
"sm" and "smrep".
If TRUE, normalise the pair connection function by
dividing it by \(p_i p_j\), the estimated probability
that randomly-selected points will have marks \(i\) and \(j\).
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
the values of the argument \(r\) at which the mark connection function \(p_{ij}(r)\) has been estimated
the theoretical value of \(p_{ij}(r)\) when the marks attached to different points are independent
The mark connection function \(p_{ij}(r)\) of a multitype point process \(X\) is a measure of the dependence between the types of two points of the process a distance \(r\) apart.
Informally \(p_{ij}(r)\) is defined as the conditional probability, given that there is a point of the process at a location \(u\) and another point of the process at a location \(v\) separated by a distance \(||u-v|| = r\), that the first point is of type \(i\) and the second point is of type \(j\). See Stoyan and Stoyan (1994).
If the marks attached to the points of X are independent
and identically distributed, then
\(p_{ij}(r) \equiv p_i p_j\) where
\(p_i\) denotes the probability that a point is of type
\(i\). Values larger than this,
\(p_{ij}(r) > p_i p_j\),
indicate positive association between the two types,
while smaller values indicate negative association.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
It must be a multitype point pattern (a marked point pattern
with factor-valued marks).
The argument r is the vector of values for the
distance \(r\) at which \(p_{ij}(r)\) is estimated.
There is a sensible default.
This algorithm assumes that X can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in X as Window(X))
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in Kest.
The edge corrections implemented here are
Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented only for rectangular and polygonal windows (not for binary masks) and is slow for complicated polygons.
Translation correction (Ohser, 1983). Implemented for all window geometries.
No edge correction.
The option correction="none" should only be used if the
number of data points is extremely large (otherwise an edge correction
is needed to correct bias).
Note that the estimator assumes the process is stationary (spatially homogeneous).
The mark connection function is estimated using density estimation techniques. The user can choose between
"density"which uses the standard kernel
density estimation routine density, and
works only for evenly-spaced r values;
"loess"which uses the function loess in the
package modreg;
"sm"which uses the function sm.density in the
package sm and is extremely slow;
"smrep"which uses the function sm.density in the
package sm and is relatively fast, but may require manual
control of the smoothing parameter hmult.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
Multitype pair correlation pcfcross
and multitype K-functions Kcross, Kdot.
Use alltypes to compute the mark connection functions
between all pairs of types.
Mark correlation markcorr and
mark variogram markvario
for numeric-valued marks.
# NOT RUN {
# Hughes' amacrine data
# Cells marked as 'on'/'off'
data(amacrine)
M <- markconnect(amacrine, "on", "off")
plot(M)
# Compute for all pairs of types at once
plot(alltypes(amacrine, markconnect))
# }
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