# smooth.ppp

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

Performs spatial smoothing of numeric values observed at a set of irregular locations.

##### Usage

```
smooth.ppp(X, ..., weights = rep(1, X$n), at="pixels")
markmean(X, ...)
markvar(X, ...)
```

##### Arguments

- X
- A marked point pattern (object of class
`"ppp"`

). - ...
- Arguments passed to
`density.ppp`

to control the kernel smoothing and the pixel resolution of the result. - weights
- Optional weights attached to the observations.
- at
- 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"`

).

##### 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.

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.
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 numerator and denominator are computed by `density.ppp`

.
The arguments `...`

control the smoothing kernel parameters
and determine whether edge correction is applied.
See `density.ppp`

.

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)
$$g(u) = \frac{\sum_i k(u-x_i) v_i w_i}{\sum_i k(u-x_i) w_i}$$

##### Value

- By 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`

.

##### See Also

`density.ppp`

,
`ppp.object`

,
`im.object`

.
To perform interpolation, see the `akima`

package.

##### Examples

```
# Longleaf data - tree locations, marked by tree diameter
data(longleaf)
# Local smoothing of tree diameter
Z <- smooth.ppp(longleaf)
# Kernel bandwidth sigma=5
plot(smooth.ppp(longleaf, 5))
# mark variance
plot(markvar(longleaf))
```

*Documentation reproduced from package spatstat, version 1.18-4, License: GPL (>= 2)*