Performs spatial smoothing of numeric values observed at a set of locations on a network. Uses kernel smoothing.
# S3 method for lpp
Smooth(X, sigma,
...,
at=c("pixels", "points"),
weights=rep(1, npoints(X)),
leaveoneout=TRUE)If X has a single column of marks:
If at="pixels" (the default), the result is
a pixel image on the network (object of class "linim").
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 on the network (objects of class "linim"). 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 attribute
"sigma" which reports the smoothing
bandwidth that was used.
A marked point pattern on a linear network
(object of class "lpp").
Smoothing bandwidth.
A single positive number.
See density.lpp.
Further arguments passed to
density.lpp
to control the kernel smoothing and
the pixel resolution of the result.
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").
Optional numeric vector of weights attached to the observations.
Logical value indicating whether to compute a leave-one-out
estimator. Applicable only when at="points".
If the chosen bandwidth sigma is very small,
kernel smoothing is mathematically equivalent
to nearest-neighbour interpolation.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk.
The function Smooth.lpp
performs spatial smoothing of numeric values
observed at a set of irregular locations on a linear network.
Smooth.lpp is a method for the generic function
Smooth for the class "lpp" 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
$$
g(u) = \frac{\sum_i k(u, x_i) v_i}{\sum_i k(u, x_i)}
$$
where \(k\) is the kernel.
This is known as the Nadaraya-Watson smoother
(Nadaraya, 1964, 1989; Watson, 1964).
The type of kernel is determined by further arguments ...
which are passed to density.lpp
The argument X must be a marked point pattern on a linear
network (object of class "lpp").
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. Then the smoothing procedure is applied to each column of marks.
The numerator and denominator are computed by density.lpp.
The arguments ... control the smoothing kernel parameters.
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)
$$
g(u) = \frac{\sum_i k(u, x_i) v_i w_i}{\sum_i k(u, x_i) w_i}
$$
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.lpp.
X <- spiders
if(!interactive()) X <- X[owin(c(0,1100), c(0, 500))]
marks(X) <- coords(X)$x
plot(Smooth(X, 50))
Smooth(X, 50, at="points")
Run the code above in your browser using DataLab