Kcross(X, i, j, r=NULL, breaks=NULL, correction, ...)X from which distances are measured.
Defaults to the first level of marks(X).X to which distances are measured.
Defaults to the second level of marks(X).r.
Not normally invoked by the user. See the Details section."border", "bord.modif",
"isotropic", "Ripley", "translate",
"none" or "best".
It specifie"fv" (see fv.object).Essentially a data frame containing numeric columns
"border", "bord.modif",
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function $K_{ij}(r)$
obtained by the edge corrections named.i and j are interpreted as
levels of the factor X$marks. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values. See the last example.Kcross and its companions
Kdot and Kmulti
are generalisations of the function Kest
to multitype point patterns. A multitype point pattern is a spatial pattern of
points classified into a finite number of possible
``colours'' or ``types''. In the X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
It must be a marked point pattern, and the mark vector
X$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
$\lambda_j K_{ij}(r)$ 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 $K_{ij}$ is determined by the
second order moment properties of $X$.
An estimate of $K_{ij}(r)$ 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 $K_{ij}(r)$ would equal $\pi r^2$. Deviations between the empirical $K_{ij}$ 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 $K_{ij}(r)$
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 X$window)
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 $K_{ij}(r)$ should be evaluated.
The values of $r$ must be increasing nonnegative numbers
and the maximum $r$ value must 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,
pcfdata(betacells)
# cat retina data
K01 <- Kcross(betacells, "off", "on")
plot(K01)
K10 <- Kcross(betacells, "on", "off")
# synthetic example
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(0:1, pp$n, replace=TRUE))
K <- Kcross(pp, "0", "1") # note: "0" not 0Run the code above in your browser using DataLab