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For a marked point pattern,
estimate the multitype I
.
Kmulti(X, I, J, r=NULL, breaks=NULL, correction, …, ratio=FALSE)
The observed point pattern,
from which an estimate of the multitype
Subset index specifying the points of X
from which distances are measured. See Details.
Subset index specifying the points in X
to which
distances are measured. See Details.
numeric vector. The values of the argument
This argument is for internal use only.
A character vector containing any selection of the
options "border"
, "bord.modif"
,
"isotropic"
, "Ripley"
, "translate"
,
"translation"
,
"none"
or "best"
.
It specifies the edge correction(s) to be applied.
Alternatively correction="all"
selects all options.
Ignored.
Logical.
If TRUE
, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
An object of class "fv"
(see fv.object
).
Essentially a data frame containing numeric columns
the values of the argument
the theoretical value of
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator" which are "fv" objects containing the numerators and denominators of each estimate of K(r).
The function
The border correction (reduced sample) estimator of
The function Kmulti
generalises Kest
(for unmarked point
patterns) and Kdot
and Kcross
(for
multitype point patterns) to arbitrary marked point patterns.
Suppose
The argument X
must be a point pattern (object of class
"ppp"
) or any data that are acceptable to as.ppp
.
The arguments I
and J
specify two subsets of the
point pattern. They may be any type of subset indices, for example,
logical vectors of length equal to npoints(X)
,
or integer vectors with entries in the range 1 to
npoints(X)
, or negative integer vectors.
Alternatively, I
and J
may be functions
that will be applied to the point pattern X
to obtain
index vectors. If I
is a function, then evaluating
I(X)
should yield a valid subset index. This option
is useful when generating simulation envelopes using
envelope
.
The argument r
is the vector of values for the
distance hist
)
for the computation of histograms of distances.
First-time users would be strongly advised not to specify r
.
However, if it is specified, r
must satisfy r[1] = 0
,
and max(r)
must be larger than the radius of the largest disc
contained in the window.
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
the border method or ``reduced sample'' estimator (see Ripley, 1988). This is the least efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary shape.
Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is currently implemented only for rectangular and polygonal windows.
Translation correction (Ohser, 1983). Implemented for all window geometries.
The pair correlation function pcf
can also be applied to the
result of Kmulti
.
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial point pattern. J. Neurosci. Meth. 18, 115--125.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293--303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406--413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511--532.
# NOT RUN {
# Longleaf Pine data: marks represent diameter
trees <- longleaf
# }
# NOT RUN {
K <- Kmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(K)
# functions determining subsets
f1 <- function(X) { marks(X) <= 15 }
f2 <- function(X) { marks(X) >= 15 }
K <- Kmulti(trees, f1, f2)
# }
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