| markcorr {spatstat} | R Documentation |
Estimate the marked correlation function of a marked point pattern.
markcorr(X, f = function(m1, m2) { m1 * m2}, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ...)
X |
The observed point pattern.
An object of class "ppp" or something acceptable to
as.ppp.
|
f |
Function f used in the definition of the
mark correlation function. There is a sensible default
that depends on the kind of marks in X.
|
r |
numeric vector. The values of the argument r at which the mark correlation function rho_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.
|
method |
A character vector indicating the user's choice of
density estimation technique to be used. Options are
"density",
"loess",
"sm" and "smrep".
|
... |
Arguments passed to the density estimation routine
(density, loess or sm.density)
selected by method.
|
The mark correlation function rho_f(r) of a marked point process X is a measure of the dependence between the marks of two points of the process a distance r apart. It is informally defined as
rho_f(r) = E[f(M1,M2)]/E[f(M,M')]
where E[ ] denotes expectation and M1,M2 are the marks attached to two points of the process separated by a distance r, while M,M' are independent realisations of the marginal distribution of marks.
Here f is any function f(m1,m2) with two arguments which are possible marks of the pattern, and which returns a nonnegative real value. Common choices of f are: for continuous real-valued marks,
f(m1,m2)= m1 * m2
for discrete marks (multitype point patterns),
f(m1,m2)= (m1 == m2)
and for marks taking values in [0,2 * pi),
f(m1,m2) = sin(m1-m2)
.
Note that rho_f(r) is not a ``correlation''
in the usual statistical sense. It can take any
nonnegative real value. The value 1 suggests ``lack of correlation'':
if the marks attached to the points of X are independent
and identically distributed, then
rho_f(r) = 1.
The interpretation of values larger or smaller than 1 depends
on the choice of function f.
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 marked point pattern.
The argument f determines the function to be applied to
pairs of marks. It has a sensible default, which depends on the
kind of marks in X. If the marks
are numeric values, then f <- function(m1, m2) { m1 * m2}
computes the product of two marks.
If the marks are a factor (i.e. if X is a multitype point
pattern) then f <- function(m1, m2) { m1 == m2} yields
the value 1 when the two marks are equal, and 0 when they are unequal.
These are the conventional definitions for numerical
marks and multitype points respectively.
Alternatively the argument f may be specified by the user.
It must be a function, accepting two arguments m1
and m2 which are vectors of equal length containing mark
values (of the same type as the marks of X).
It must return a vector of numeric
values of the same length as m1 and m2.
The values must be non-negative, and NA values are not permitted.
The argument r is the vector of values for the
distance r at which rho_f(r) is estimated.
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 X$window)
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in Kest.
The edge corrections implemented here are
Note that the estimator assumes the process is stationary (spatially homogeneous).
The numerator and denominator of the mark correlation function (in the expression above) are estimated using density estimation techniques. The user can choose between
"density"density, and
works only for evenly-spaced r values;
"loess"loess in the
package modreg;
"sm"sm.density in the
package sm and is extremely slow;
"smrep"sm.density in the
package sm and is relatively fast, but may require manual
control of the smoothing parameter hmult.
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r |
the values of the argument r at which the mark correlation function rho_f(r) has been estimated |
theo |
the theoretical value of rho_f(r) when the marks attached to different points are independent, namely 1 |
together with a column or columns named
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function rho_f(r)
obtained by the edge corrections named.
Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
# CONTINUOUS-VALUED MARKS:
# (1) Longleaf Pine data
# marks represent tree diameter
data(longleaf)
# Subset of this large pattern
swcorner <- owin(c(0,100),c(0,100))
sub <- longleaf[ , swcorner]
# mark correlation function
mc <- markcorr(sub)
plot(mc)
# (2) simulated data with independent marks
X <- rpoispp(100)
X <- X %mark% runif(X$n)
Xc <- markcorr(X)
plot(Xc)
# MULTITYPE DATA:
# Hughes' amacrine data
# Cells marked as 'on'/'off'
data(amacrine)
# (3) Kernel density estimate with Epanecnikov kernel
# (as proposed by Stoyan & Stoyan)
M <- markcorr(amacrine, function(m1,m2) {m1==m2},
correction="translate", method="density",
kernel="epanechnikov")
plot(M)
# Note: kernel="epanechnikov" comes from help(density)
# (4) Same again with explicit control over bandwidth
M <- markcorr(amacrine,
correction="translate", method="density",
kernel="epanechnikov", bw=0.02)
# see help(density) for correct interpretation of 'bw'