Inhomogeneous Linear K Function
Computes an estimate of the inhomogeneous linear K function for a point pattern on a linear network.
linearKinhom(X, lambda=NULL, r=NULL, ..., correction="Ang",
normalise=TRUE, normpower=1,
update=TRUE, leaveoneout=TRUE, ratio=FALSE)X |
Point pattern on linear network (object of class |
lambda |
Intensity values for the point pattern. Either a numeric vector,
a |
r |
Optional. Numeric vector of values of the function argument r. There is a sensible default. |
... |
Ignored. |
correction |
Geometry correction.
Either |
normalise |
Logical. If |
normpower |
Integer (usually either 1 or 2). Normalisation power. See Details. |
update |
Logical value indicating what to do when |
leaveoneout |
Logical value (passed to |
ratio |
Logical.
If |
This command computes the inhomogeneous version of the linear K function from point pattern data on a linear network.
If lambda = NULL the result is equivalent to the
homogeneous K function linearK.
If lambda is given, then it is expected to provide estimated values
of the intensity of the point process at each point of X.
The argument lambda may be a numeric vector (of length equal to
the number of points in X), or a function(x,y) that will be
evaluated at the points of X to yield numeric values,
or a pixel image (object of class "im") or a fitted point
process model (object of class "ppm" or "lppm").
If lambda is a fitted point process model,
the default behaviour is to update the model by re-fitting it to
the data, before computing the fitted intensity.
This can be disabled by setting update=FALSE.
If correction="none", the calculations do not include
any correction for the geometry of the linear network.
If correction="Ang", the pair counts are weighted using
Ang's correction (Ang, 2010).
Each estimate is initially computed as
K^inhom(r)= (1/length(L)) sum[i] sum[j] 1(d[i,j] <= r) * e(x[i],x[j])/(lambda(x[i]) * lambda(x[j]))
where L is the linear network,
d[i,j] is the distance between points
x[i] and x[j], and
e(x[i],x[j]) is a weight.
If correction="none" then this weight is equal to 1,
while if correction="Ang" the weight is
e(x[i],x[j],r) = 1/m(x[i],d[i,j])
where m(u,t) is the number of locations on the network that lie
exactly t units distant from location u by the shortest
path.
If normalise=TRUE (the default), then the estimates
described above
are multiplied by c^normpower where
c = length(L)/sum[i] (1/lambda(x[i])).
This rescaling reduces the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity.
The default value of normpower is 1 (for consistency with
previous versions of spatstat)
but the most sensible value is 2, which would correspond to rescaling
the lambda values so that
sum[i] (1/lambda(x[i])) = area(W).
Function value table (object of class "fv").
Ang Qi Wei aqw07398@hotmail.com and Adrian Baddeley Adrian.Baddeley@curtin.edu.au
Ang, Q.W. (2010) Statistical methodology for spatial point patterns on a linear network. MSc thesis, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of events on a linear network, with applications to ecology and criminology. Scandinavian Journal of Statistics 39, 591–617.
data(simplenet) X <- rpoislpp(5, simplenet) fit <- lppm(X ~x) K <- linearKinhom(X, lambda=fit) plot(K)
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