Inhomogeneous J-function
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,
update = TRUE, warn.bias=TRUE, savelambda=FALSE)X |
The observed data point pattern,
from which an estimate of the inhomogeneous J function
will be computed.
An object of class |
lambda |
Optional.
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern |
lmin |
Optional. The minimum possible value of the intensity over the spatial domain. A positive numerical value. |
sigma,varcov |
Optional arguments passed to |
... |
Extra arguments passed to |
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 |
This argument is for internal use only. |
update |
Logical. If |
warn.bias |
Logical value specifying whether to issue a warning when the inhomogeneity correction factor takes extreme values, which can often lead to biased results. This usually occurs when insufficient smoothing is used to estimate the intensity. |
savelambda |
Logical value specifying whether to save the values of
|
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 Jest.
The argument X should be a point pattern
(object of class "ppp").
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).
The argument lambda should supply the
(estimated) values of the intensity function lambda
of the point process. It may be either
containing the values
of the intensity function at the points of the pattern X.
(object of class "im")
assumed to contain the values of the intensity function
at all locations in the window.
(object of class "ppm" or "kppm")
whose fitted trend can be used as the fitted intensity.
(If update=TRUE the model will first be refitted to the
data X before the trend is computed.)
which can be evaluated to give values of the intensity at any locations.
if lambda is omitted, then it will be estimated using
a ‘leave-one-out’ kernel smoother.
If lambda is a numeric vector, then its length 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.
Each value must be a positive number; NA's are not allowed.
If 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 lambda using
blur, then looking up the values of this blurred image
for the missing locations.
(A warning will be issued in this case.)
If lambda is a function, then it will be evaluated in the
form lambda(x,y) where x and 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 X.
If lambda is omitted, then it will be estimated using
a ‘leave-one-out’ kernel smoother.
The estimate lambda[i] for the
point 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 sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Original code by Marie-Colette van Lieshout. C implementation and R adaptation by Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Ege Rubak rubak@math.aau.dk.
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))
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