Computes an adaptive estimate of the intensity function of a point pattern using a variable-bandwidth smoothing kernel.
densityAdaptiveKernel(X, …)# S3 method for ppp
densityAdaptiveKernel(X, bw, …,
weights=NULL,
at=c("pixels", "points"),
edge=TRUE, ngroups)
Point pattern (object of class "ppp").
Numeric vector of smoothing bandwidths for each point in X,
or a pixel image giving the smoothing bandwidth at each spatial
location, or a spatial function of class "funxy" giving the
smoothing bandwidth at each location.
The default is to compute bandwidths using bw.abram.
Optional vector of numeric weights for the points of X.
String specifying whether to compute the intensity values
at a grid of pixel locations (at="pixels") or
only at the points of x (at="points").
Logical value indicating whether to perform edge correction.
Number of groups into which the bandwidth values should be partitioned and discretised.
If at="pixels" (the default), the result is a pixel image.
If at="points", the result is a numeric vector with one entry
for each data point in X.
The function densityAdaptiveKernel
computes one adaptive estimate of the intensity,
determined by the smoothing bandwidth values bw.
Typically the bandwidth values are computed by first computing
a pilot estimate of the intensity, then using bw.abram
to compute the vector of bandwidths according to Abramson's rule.
This involves specifying a global bandwidth h0.
The default bandwidths may work well in many contexts, but for optimal
bandwidth selection, this calculation should be performed repeatedly with
different values of h0 to optimise the value of h0.
This can be computationally demanding; we recommend
the function multiscale.density in the sparr package
which supports much faster bandwidth selection, using the FFT
method of Davies and Baddeley (2018).
This function computes a spatially-adaptive kernel estimate of the
spatially-varying intensity from the point pattern X
using the partitioning technique of Davies and Baddeley (2018).
The argument bw specifies the smoothing bandwidths to be
applied to each of the points in X. It may be a numeric vector
of bandwidth values, or a pixel image or function yielding the
bandwidth values.
If the points of X are \(x_1,\ldots,x_n\)
and the corresponding bandwidths are
\(\sigma_1,\ldots,\sigma_n\)
then the adaptive kernel estimate of intensity at a location \(u\) is
$$
\hat\lambda(u) = \sum_{i=1}^n k(u, x_i, \sigma_i)
$$
where \(k(u, v, \sigma)\) is the value at \(u\)
of the (possibly edge-corrected) smoothing kernel with bandwidth \(\sigma\)
induced by a data point at \(v\).
Exact computation of the estimate above can be time-consuming: it takes \(n\) times longer than fixed-bandwidth smoothing.
The partitioning method of Davies and Baddeley (2018)
accelerates this computation by partitioning the range of
bandwidths into ngroups intervals,
correspondingly subdividing the points of the pattern X into
ngroups sub-patterns according to bandwidth,
and applying fixed-bandwidth smoothing to each sub-pattern.
The default value of ngroups is the integer part of the square root of
the number of points in X, so that the computation time is
only about \(\sqrt{n}\) times slower than fixed-bandwidth
smoothing. Any positive value of ngroups
can be specified by the user. Specifying ngroups=Inf enforces exact
computation of the estimate without partitioning. Specifying
ngroups=1 is the same as fixed-bandwidth smoothing with
bandwidth sigma=median(bw).
Davies, T.M. and Baddeley, A. (2018) Fast computation of spatially adaptive kernel estimates. Statistics and Computing, 28(4), 937-956.
Hall, P. and Marron, J.S. (1988) Variable window width kernel density estimates of probability densities. Probability Theory and Related Fields, 80, 37-49.
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York.
density.ppp,
adaptive.density,
densityVoronoi,
im.object.
See the function bivariate.density in the sparr package
for a more flexible implementation, and
multiscale.density for an implementation that is more
efficient for bandwidth selection.
# NOT RUN {
Z <- densityAdaptiveKernel(redwood, h0=0.1)
plot(Z, main="Adaptive kernel estimate")
points(redwood, col="white")
# }
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