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