Kdot.inhom(X, i, lambdaI=NULL, lambdadot=NULL, ..., r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL, lambdaIdot=NULL)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).i.
Either a pixel image (object of class "im"),
a numeric vector containing the intensity values
at each of the type i"im"),
a numeric vector containing the intensity values at each of the
points in X, or a functir.
Not normally invoked by the user. See the Details section."border", "bord.modif",
"isotropic", "Ripley", "translate",
"translation",
"none" or lambdaI, lambdadot if they are omitted.lambdaI, lambdadot if they are omitted.
Incompatible with sigma.lambdaI and lambdadot
for each pair of points, the first point of type i and
the second of any type."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 $K_{i\bullet}(r)$
obtained by the edge corrections named.i is interpreted as
a level of the factor X$marks. It is converted to a character
string if it is not already a character string.
The value i=1 does not
refer to the first level of the factor.Kdot
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function Kinhom.Briefly, given a multitype point process, consider the points without their types, and suppose this unmarked point process has intensity function $\lambda(u)$ at spatial locations $u$. Suppose we place a mass of $1/\lambda(\zeta)$ at each point $\zeta$ of the process. Then the expected total mass per unit area is 1. The inhomogeneous ``dot-type'' $K$ function $K_{i\bullet}^{\mbox{inhom}}(r)$ equals the expected total mass within a radius $r$ of a point of the process of type $i$, discounting this point itself. If the process of type $i$ points were independent of the points of other types, then $K_{i\bullet}^{\mbox{inhom}}(r)$ would equal $\pi r^2$. Deviations between the empirical $K_{i\bullet}$ curve and the theoretical curve $\pi r^2$ suggest dependence between the points of types $i$ and $j$ for $j\neq i$.
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$marks must be a factor.
The argument i will be interpreted as a
level of the factor X$marks. (Warning: this means that
an integer value i=3 will be interpreted as the number 3,
not the 3rd smallest level).
If i is missing, it defaults to the first
level of the marks factor, i = levels(X$marks)[1].
The argument lambdaI supplies the values
of the intensity of the sub-process of points of type i.
It may be either
[object Object],[object Object],[object Object],[object Object]
If lambdaI is omitted, then it will be estimated using
a `leave-one-out' kernel smoother, as described in Baddeley, Moller
and Waagepetersen (2000). The estimate of lambdaI for a given
point is computed by removing the point from the
point pattern, applying kernel smoothing to the remaining points using
density.ppp, and evaluating the smoothed intensity
at the point in question. The smoothing kernel bandwidth is controlled
by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Similarly the argument lambdadot should contain
estimated values of the intensity of the entire point process.
It may be either a pixel image, a numeric vector of length equal
to the number of points in X, a function, or omitted.
For advanced use only, the optional argument lambdaIdot
is a matrix containing estimated
values of the products of these two intensities for each pair of
points, the first point of type i and the second of any type.
The argument r is the vector of values for the
distance $r$ at which $K_{i\bullet}(r)$ 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 argument correction chooses the edge correction
as explained e.g. in Kest.
The pair correlation function can also be applied to the
result of Kcross.inhom; see pcf.
Kdot,
Kinhom,
Kcross.inhom,
Kmulti.inhom,
pcf# Lansing Woods data
data(lansing)
lansing <- lansing[seq(1,lansing$n, by=10)]
ma <- split(lansing)$maple
lg <- unmark(lansing)
# Estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdadot <- density.ppp(lg, sigma=0.15, at="points")
K <- Kdot.inhom(lansing, "maple", lambdaI=lambdaM,
lambdadot=lambdadot)
# Equivalent
K <- Kdot.inhom(lansing, "maple", sigma=0.15)
# synthetic example: type A points have intensity 50,
# type B points have intensity 50 + 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
lamdot <- as.im(function(x,y) { 100 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kdot.inhom(X, "B", lambdaI=lamB, lambdadot=lamdot)Run the code above in your browser using DataLab