# Smooth.ppp

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##### Spatial smoothing of observations at irregular points

Performs spatial smoothing of numeric values observed at a set of irregular locations. Uses Gaussian kernel smoothing and least-squares cross-validated bandwidth selection.

Keywords
methods, smooth, spatial
##### Usage
## S3 method for class 'ppp':
Smooth(X, sigma=NULL,
...,
weights = rep(1, npoints(X)),
at="pixels",
edge=TRUE, diggle=FALSE)markmean(X, ...)markvar(X, sigma=NULL, ..., weights=NULL, varcov=NULL)
##### Arguments
X
A marked point pattern (object of class "ppp").
sigma
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.
weights
Optional weights attached to the observations. A numeric vector, numeric matrix, or an expression. See density.ppp.
at
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").
edge,diggle
Arguments passed to density.ppp to determine the edge correction.
varcov
Variance-covariance matrix. An alternative to sigma. See density.ppp.
##### Details

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 Gaussian kernel weighting. 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 a Gaussian kernel. 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) $$g(u) = \frac{\sum_i k(u-x_i) v_i w_i}{\sum_i k(u-x_i) w_i}$$

An alternative to kernel smoothing is inverse-distance weighting, which is performed by idw.

##### Value

• If X has a single column of marks:
• Ifat="pixels"(the default), the result is a pixel image (object of class"im"). Pixel values are values of the interpolated function.
• Ifat="points", the result is a numeric vector of length equal to the number of points inX. Entries are values of the interpolated function at the points ofX.
If X has a data frame of marks:
• Ifat="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.
• Ifat="points", the result is a data frame with one row for each point ofX, and one column for each column of marks. Entries are values of the interpolated function at the points ofX.
The return value has attributes "sigma" and "varcov" which report the smoothing bandwidth that was used.

##### Very small bandwidth

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.

##### References

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.

• Smooth.ppp
• markmean
• markvar
##### Examples
# 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))
Documentation reproduced from package spatstat, version 1.43-0, License: GPL (>= 2)

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