| Kinhom {spatstat} | R Documentation |
Estimates the inhomogeneous K function of a non-stationary point pattern.
Kinhom(X, lambda, r = NULL, breaks = NULL, slow = FALSE,
correction=c("border", "bord.modif", "isotropic", "translate"),
..., lambda2
)
X |
The observed data point pattern,
from which an estimate of the inhomogeneous K function
will be computed.
An object of class "ppp"
or in a format recognised by as.ppp()
|
lambda |
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern X,
or a pixel image (object of class "im") giving the
intensity values at all locations.
|
r |
vector of values for the argument r at which the inhomogeneous K function should be evaluated. Not normally given by the user; there is a sensible default. |
breaks |
An alternative to the argument r.
Not normally invoked by the user.
See Details.
|
slow |
Not normally given by the user.
Logical flag which selects the algorithm used to compute the
inhomogeneous K function. The default (slow=FALSE)
is faster than the alternative (slow=TRUE). The slow
algorithm is retained for internal purposes as a check on
validity.
|
correction |
A character vector containing any selection of the
options "border", "bord.modif",
"isotropic", "Ripley" or "translate".
It specifies the edge correction(s) to be applied.
|
... |
Currently ignored. |
lambda2 |
Advanced use only. Matrix containing estimates of the products lambda(x[i]) * lambda(x[j]) of the intensities at each pair of data points x[i] and x[j]. |
This computes a generalisation of the K function for inhomogeneous point patterns, proposed by Baddeley, Moller and Waagepetersen (2000).
The ``ordinary'' K function
(variously known as the reduced second order moment function
and Ripley's K function), is
described under Kest. It is defined only
for stationary point processes.
The inhomogeneous K function Kinhom(r) is a direct generalisation to nonstationary point processes. Suppose x is a point process with non-constant intensity lambda(u) at each location u. Define Kinhom(r) to be the expected value, given that u is a point of x, of the sum of all terms 1/lambda(u)lambda(x[j]) over all points x[j] in the process separated from u by a distance less than r. This reduces to the ordinary K function if lambda() is constant. If x is an inhomogeneous Poisson process with intensity function lambda(u), then Kinhom(r) = pi * r^2.
This allows us to inspect a point pattern for evidence of interpoint interactions after allowing for spatial inhomogeneity of the pattern. Values Kinhom(r) > pi * r^2 are suggestive of clustering.
The argument lambda must supply the
(estimated) values of the intensity function lambda.
It may be either
X.
The length of the vector lambda should
be equal to the number of points in the pattern X.
The value lambda[i] is assumed to be the
the (estimated) value of the intensity
lambda(x[i]) for
the point x[i] of the pattern X.
"im")
assumed to contain the values of the intensity function
at all locations in the window.
Edge corrections are used to correct bias in the estimation of Kinhom. Each edge-corrected estimate of Kinhom(r) is of the form
K^inhom(r)= sum[i] sum[j] 1(d[i,j] <= r) * e(x[i],x[j],r)/(lambda(x[i]) * lambda(x[j]))
where d[i,j] is the distance between points x[i] and x[j], and e(x[i],x[j],r) is an edge correction factor. For the `border' correction,
1(b[i] > r)/(sum[j] 1(b[j] > r)/lambda(x[j]))
where b[i] is the distance from x[i] to the boundary of the window. For the `modified border' correction,
1(b[i] > r)/area(W [-] r)
where W [-] r is the eroded window obtained by trimming a margin of width r from the border of the original window. For the `translation' correction,
1/area(W intersect (W + x[j]-x[i]))
and for the `isotropic' correction,
1/(area(W) g(x[i],x[j]))
where g(x[i],x[j]) is the fraction of the circumference of the circle with centre x[i] and radius ||x[i]-x[j]|| which lies inside the window.
The pair correlation function can also be applied to the
result of Kinhom; see pcf.
An object of class "fv" (see fv.object).
Essentially a data frame containing at least the following columns,
r |
the vector of values of the argument r at which the pair correlation function g(r) has been estimated |
theo |
vector of values of pi * r^2, the theoretical value of Kinhom(r) for an inhomogeneous Poisson process |
and containing additional columns
according to the choice specified in the correction
argument. The additional columns are named
border, trans and iso
and give the estimated values of
Kinhom(r)
using the border correction, translation correction,
and Ripley isotropic correction, respectively.
Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf
Baddeley, A., Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
data(lansing)
# inhomogeneous pattern of maples
X <- unmark(split(lansing)$maple)
# (1) intensity function estimated by model-fitting
# Fit spatial trend: polynomial in x and y coordinates
fit <- ppm(X, ~ polynom(x,y,2), Poisson())
# predict intensity values at points themselves
lambda <- predict(fit, locations=X, type="trend")
# inhomogeneous K function
Ki <- Kinhom(X, lambda)
plot(Ki)
# (2) intensity function estimated by heavy smoothing
smo <- density.ppp(X, sigma=0.1)
# evaluate smoothed intensity at data points
lambda <- smo[X]
# inhomogeneous K function
Ki <- Kinhom(X, lambda)
plot(Ki)
# (3) simulated data: known intensity function
lamfun <- function(x,y) { 100 * x }
# inhomogeneous Poisson process
Y <- rpoispp(lamfun, 100, owin())
# evaluate intensity at points of pattern
lambda <- lamfun(Y$x, Y$y)
# inhomogeneous K function
Ki <- Kinhom(Y, lambda)
plot(Ki)
# How to make simulation envelopes:
# Example shows method (2)
## Not run:
smo <- density.ppp(X, sigma=0.1)
myKfun <- function(X, ..., lam) { Kinhom(X, lambda=lam[X], ...) }
Ken <- envelope(X, myKfun, nsim=99,
simulate=expression(rpoispp(smo)),
lam=smo, correction="trans")
plot(Ken)
## End(Not run)