Estimates the inhomogeneous $J$ function of a non-stationary point pattern.
Jinhom(X, lambda = NULL, lmin = NULL, ..., sigma = NULL, varcov = NULL, r = NULL, breaks = NULL)
- The observed data point pattern,
from which an estimate of the inhomogeneous $J$ function
will be computed.
An object of class
"ppp"or in a format recognised by
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern
X, a pixel image (object of class
"im") giving the intensity values at all locatio
- Optional. The minimum possible value of the intensity over the spatial domain. A positive numerical value.
- Optional arguments passed to
density.pppto control the smoothing bandwidth, when
lambdais estimated by kernel smoothing.
- Extra arguments passed to
as.maskto control the pixel resolution, or passed to
density.pppto control the smoothing bandwidth.
- 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.
- An alternative to the argument
r. Not normally invoked by the user. See Details.
This command computes estimates of the
inhomogeneous $J$-function (Van Lieshout, 2010)
of a point pattern. It is the counterpart, for inhomogeneous
spatial point patterns, of the $J$ function
for homogeneous point patterns computed by
X should be a point pattern
(object of class
The inhomogeneous $J$ function is computed as
$Jinhom(r) = (1 - Ginhom(r))/(1-Finhom(r))$
where $Ginhom, Finhom$ are the inhomogeneous $G$ and $F$
functions computed using the border correction
(equations (7) and (6) respectively in Van Lieshout, 2010).
lambda should supply the
(estimated) values of the intensity function $\lambda$
of the point process. It may be either
[object Object],[object Object],[object Object],[object Object],[object Object]
lambda is a numeric vector, then its length should
be equal to the number of points in the pattern
lambda[i] is assumed to be the
the (estimated) value of the intensity
the point $x_i$ of the pattern $X$.
Each value must be a positive number;
NA's are not allowed.
lambda is a pixel image, the domain of the image should
cover the entire window of the point pattern. If it does not (which
may occur near the boundary because of discretisation error),
then the missing pixel values
will be obtained by applying a Gaussian blur to
blur, then looking up the values of this blurred image
for the missing locations.
(A warning will be issued in this case.)
lambda is a function, then it will be evaluated in the
y are vectors
of coordinates of the points of
X. It should return a numeric
vector with length equal to the number of points in
lambda is omitted, then it will be estimated using
a `leave-one-out' kernel smoother, as described in Baddeley, Moller
and Waagepetersen (2000). The estimate
lambda[i] for the
X[i] is computed by removing
X[i] from the
point pattern, applying kernel smoothing to the remaining points using
density.ppp, and evaluating the smoothed intensity
at the point
X[i]. The smoothing kernel bandwidth is controlled
by the arguments
varcov, which are passed to
density.ppp along with any extra arguments.
van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344--361.
van Lieshout, M.N.M. (2010) A J-function for inhomogeneous point processes. Statistica Neerlandica 65, 183--201.
plot(Jinhom(swedishpines, sigma=bw.diggle, adjust=2)) plot(Jinhom(swedishpines, sigma=10))