Performs spatial smoothing of numeric values observed at a set of irregular locations. Uses kernel smoothing and least-squares cross-validated bandwidth selection.
# S3 method for ppp
Smooth(X, sigma=NULL,
...,
weights = rep(1, npoints(X)),
at="pixels",
adjust=1, varcov=NULL,
edge=TRUE, diggle=FALSE,
kernel="gaussian", scalekernel=is.character(kernel),
geometric=FALSE)markmean(X, ...)
markvar(X, sigma=NULL, ..., weights=NULL, varcov=NULL)
If X has a single column of marks:
If at="pixels" (the default), the result is
a pixel image (object of class "im").
Pixel values are values of the interpolated function.
If at="points", the result is a numeric vector
of length equal to the number of points in X.
Entries are values of the interpolated function at the points of X.
If X has a data frame of marks:
If at="pixels" (the default), the result is a named list of
pixel images (object of class "im"). There is one
image for each column of marks. This list also belongs to
the class "solist", for which there is a plot method.
If at="points", the result is a data frame
with one row for each point of X,
and one column for each column of marks.
Entries are values of the interpolated function at the points of X.
The return value has attributes
"sigma" and "varcov" which report the smoothing
bandwidth that was used.
A marked point pattern (object of class "ppp").
Smoothing bandwidth.
A single positive number, a numeric vector of length 2,
or a function that selects the bandwidth automatically.
See density.ppp.
Further arguments passed to
bw.smoothppp and density.ppp
to control the kernel smoothing and
the pixel resolution of the result.
Optional weights attached to the observations.
A numeric vector, a function(x,y), a pixel image,
or an expression. See density.ppp.
String specifying whether to compute the smoothed values
at a grid of pixel locations (at="pixels") or
only at the points of X (at="points").
Arguments passed to density.ppp to
determine the edge correction.
Optional. Adjustment factor for the bandwidth sigma.
Variance-covariance matrix. An alternative
to sigma. See density.ppp.
The smoothing kernel.
A character string specifying the smoothing kernel
(current options are "gaussian", "epanechnikov",
"quartic" or "disc"),
or a pixel image (object of class "im")
containing values of the kernel, or a function(x,y) which
yields values of the kernel.
Logical value.
If scalekernel=TRUE, then the kernel will be rescaled
to the bandwidth determined by sigma and varcov:
this is the default behaviour when kernel is a character string.
If scalekernel=FALSE, then sigma and varcov
will be ignored: this is the default behaviour when kernel is a
function or a pixel image.
Logical value indicating whether to perform geometric mean smoothing instead of arithmetic mean smoothing. See Details.
If the chosen bandwidth sigma is very small,
kernel smoothing is mathematically equivalent
to nearest-neighbour interpolation; the result will
be computed by nnmark. This is
unless at="points" and leaveoneout=FALSE,
when the original mark values are returned.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
The function Smooth.ppp
performs spatial smoothing of numeric values
observed at a set of irregular locations. The functions
markmean and markvar are wrappers for Smooth.ppp
which compute the spatially-varying mean and variance of the marks of
a point pattern.
Smooth.ppp is a method for the generic function
Smooth for the class "ppp" of point patterns.
Thus you can type simply Smooth(X).
Smoothing is performed by kernel weighting, using the Gaussian kernel by default. If the observed values are \(v_1,\ldots,v_n\) at locations \(x_1,\ldots,x_n\) respectively, then the smoothed value at a location \(u\) is (ignoring edge corrections) $$ g(u) = \frac{\sum_i k(u-x_i) v_i}{\sum_i k(u-x_i)} $$ where \(k\) is the kernel (a Gaussian kernel by default). This is known as the Nadaraya-Watson smoother (Nadaraya, 1964, 1989; Watson, 1964). By default, the smoothing kernel bandwidth is chosen by least squares cross-validation (see below).
The argument X must be a marked point pattern (object
of class "ppp", see ppp.object).
The points of the pattern are taken to be the
observation locations \(x_i\), and the marks of the pattern
are taken to be the numeric values \(v_i\) observed at these
locations.
The marks are allowed to be a data frame (in
Smooth.ppp
and markmean). Then the smoothing procedure is applied to each
column of marks.
The numerator and denominator are computed by density.ppp.
The arguments ... control the smoothing kernel parameters
and determine whether edge correction is applied.
The smoothing kernel bandwidth can be specified by either of the arguments
sigma or varcov which are passed to density.ppp.
If neither of these arguments is present, then by default the
bandwidth is selected by least squares cross-validation,
using bw.smoothppp.
The optional argument weights allows numerical weights to
be applied to the data. If a weight \(w_i\)
is associated with location \(x_i\), then the smoothed
function is
(ignoring edge corrections)
$$
g(u) = \frac{\sum_i k(u-x_i) v_i w_i}{\sum_i k(u-x_i) w_i}
$$
If geometric=TRUE then geometric mean smoothing
is performed instead of arithmetic mean smoothing.
The mark values must be non-negative numbers.
The logarithm of the mark values is computed; these logarithmic values are
kernel-smoothed as described above; then the exponential function
is applied to the smoothed values.
An alternative to kernel smoothing is inverse-distance weighting,
which is performed by idw.
Nadaraya, E.A. (1964) On estimating regression. Theory of Probability and its Applications 9, 141--142.
Nadaraya, E.A. (1989) Nonparametric estimation of probability densities and regression curves. Kluwer, Dordrecht.
Watson, G.S. (1964) Smooth regression analysis. Sankhya A 26, 359--372.
Smooth,
density.ppp,
bw.smoothppp,
nnmark,
ppp.object,
im.object.
See idw for inverse-distance weighted smoothing.
To perform interpolation, see also the akima package.
# Longleaf data - tree locations, marked by tree diameter
# Local smoothing of tree diameter (automatic bandwidth selection)
Z <- Smooth(longleaf)
# Kernel bandwidth sigma=5
plot(Smooth(longleaf, 5))
# mark variance
plot(markvar(longleaf, sigma=5))
# data frame of marks: trees marked by diameter and height
plot(Smooth(finpines, sigma=2))
head(Smooth(finpines, sigma=2, at="points"))
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