Kcross
Multitype K Function (Cross-type)
For a multitype point pattern, estimate the multitype $K$ function which counts the expected number of points of type $j$ within a given distance of a point of type $i$.
- Keywords
- spatial, nonparametric
Usage
Kcross(X, i, j, r=NULL, breaks=NULL, correction, ...)Arguments
- X
- The observed point pattern, from which an estimate of the cross type $K$ function $K_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.
- i
- Number or character string identifying the type (mark value)
of the points in
Xfrom which distances are measured. Defaults to the first level ofmarks(X). - j
- Number or character string identifying the type (mark value)
of the points in
Xto which distances are measured. Defaults to the second level ofmarks(X). - r
- numeric vector. The values of the argument $r$ at which the distribution function $K_{ij}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important
- breaks
- An alternative to the argument
r. Not normally invoked by the user. See the Details section. - correction
- A character vector containing any selection of the
options
"border","bord.modif","isotropic","Ripley","translate","none"or"best". It specifie - ...
- Ignored.
Details
This function 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.
Value
- An object of class
"fv"(seefv.object).Essentially a data frame containing numeric columns
r the values of the argument $r$ at which the function $K_{ij}(r)$ has been estimated theo the theoretical value of $K_{ij}(r)$ for a marked Poisson process, namely $\pi r^2$ - together 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 $K_{ij}(r)$ obtained by the edge corrections named.
Warnings
The arguments 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 first example.
The reduced sample estimator of $K_{ij}$ is pointwise approximately unbiased, but need not be a valid distribution function; it may not be a nondecreasing function of $r$. Its range is always within $[0,1]$.
References
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.
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.
See Also
Examples
data(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 0