Kcross(X, i, j, r=NULL, breaks=NULL, correction, ..., ratio=FALSE, from, to )X from which distances are measured.
A character string (or something that will be converted to a
character string).
Defaults to the first level of marks(X).
X to which distances are measured.
A character string (or something that will be
converted to a character string).
Defaults to the second level of marks(X).
"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.
TRUE, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
i and j respectively.
"fv" (see fv.object).Essentially a data frame containing numeric columnstogether with a column or columns named
"border", "bord.modif",
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function $Kij(r)$
obtained by the edge corrections named.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 $K(r)$.
i and j are always interpreted as
levels of the factor X$marks. They are converted to character
strings if they are not already character strings.
The value i=1 does not
refer to the first level of the factor.Kcross and its companions
Kdot and Kmulti
are generalisations of the function Kest
to multitype point patterns. A multitype point pattern is a spatial pattern of
points classified into a finite number of possible
``colours'' or ``types''. In the spatstat package,
a multitype pattern is represented as a single
point pattern object in which the points carry marks,
and the mark value attached to each point
determines the type of that point.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
It must be a marked point pattern, and the mark vector
X$marks must be a factor.
The arguments i and j will be interpreted as
levels of the factor X$marks.
If i and j are missing, they default to the first
and second level of the marks factor, respectively.
The ``cross-type'' (type $i$ to type $j$)
$K$ function
of a stationary multitype point process $X$ is defined so that
$lambda[j] Kij(r)$ equals the expected number of
additional random points of type $j$
within a distance $r$ of a
typical point of type $i$ in the process $X$.
Here $lambda[j]$
is the intensity of the type $j$ points,
i.e. the expected number of points of type $j$ per unit area.
The function $Kij$ is determined by the
second order moment properties of $X$.
An estimate of $Kij(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the process of type $i$ points were independent of the process of type $j$ points, then $Kij(r)$ would equal $pi * r^2$. Deviations between the empirical $Kij$ curve and the theoretical curve $pi * r^2$ may suggest dependence between the points of types $i$ and $j$.
This algorithm estimates the distribution function $Kij(r)$
from the point pattern X. It 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,
using the border correction.
The argument r is the vector of values for the
distance $r$ at which $Kij(r)$ should be evaluated.
The values of $r$ must be increasing nonnegative numbers
and the maximum $r$ value must not exceed the radius of the
largest disc contained in the window.
The pair correlation function can also be applied to the
result of Kcross; see pcf.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
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.
Kdot,
Kest,
Kmulti,
pcf
# amacrine cells data
K01 <- Kcross(amacrine, "off", "on")
plot(K01)
## Not run:
# K10 <- Kcross(amacrine, "on", "off")
#
# # synthetic example: point pattern with marks 0 and 1
# pp <- runifpoispp(50)
# pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
# K <- Kcross(pp, "0", "1")
# K <- Kcross(pp, 0, 1) # equivalent
# ## End(Not run)
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