# idw

##### Inverse-distance weighted smoothing of observations at irregular points

Performs spatial smoothing of numeric values observed at a set of irregular locations using inverse-distance weighting.

##### Usage

`idw(X, power=2, at=c("pixels", "points"), ..., se=FALSE)`

##### Arguments

- X
A marked point pattern (object of class

`"ppp"`

).- power
Numeric. Power of distance used in the weighting.

- at
Character 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"`

). String is partially matched.- …
Arguments passed to

`as.mask`

to control the pixel resolution of the result.- se
Logical value specifying whether to calculate a standard error.

##### Details

This function performs spatial smoothing of numeric values observed at a set of irregular locations.

Smoothing is performed by inverse distance 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 $$ g(u) = \frac{\sum_i w_i v_i}{\sum_i w_i} $$ where the weights are the inverse \(p\)-th powers of distance, $$ w_i = \frac 1 {d(u,x_i)^p} $$ where \(d(u,x_i) = ||u - x_i||\) is the Euclidean distance from \(u\) to \(x_i\).

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. Then the smoothing procedure is applied to each column of marks.

If `at="pixels"`

(the default), the smoothed mark value
is calculated at a grid of pixels, and the result is a pixel image.
The arguments `…`

control the pixel resolution.
See `as.mask`

.

If `at="points"`

, the smoothed mark values are calculated
at the data points only, using a leave-one-out rule (the mark value
at a data point is excluded when calculating the smoothed value
for that point).

An estimate of standard error is also calculated, if `se=TRUE`

.
The calculation assumes that the data point locations are fixed,
that is, the standard error only takes into account the variability
in the mark values, and not the variability due to randomness of the
data point locations.

An alternative to inverse-distance weighting is kernel smoothing,
which is performed by `Smooth.ppp`

.

##### Value

*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`

.

If `se=TRUE`

, then the result is a list
with two entries named `estimate`

and `SE`

, which
each have the format described above.

##### References

Shepard, D. (1968) A two-dimensional interpolation function for
irregularly-spaced data.
*Proceedings of the 1968 ACM National Conference*,
1968, pages 517--524. DOI: 10.1145/800186.810616

##### See Also

`density.ppp`

,
`ppp.object`

,
`im.object`

.

See `Smooth.ppp`

for kernel smoothing
and `nnmark`

for nearest-neighbour interpolation.

To perform other kinds of interpolation, see also the `akima`

package.

##### Examples

```
# NOT RUN {
# data frame of marks: trees marked by diameter and height
plot(idw(finpines))
idw(finpines, at="points")[1:5,]
plot(idw(finpines, se=TRUE)$SE)
idw(finpines, at="points", se=TRUE)$SE[1:5, ]
# }
```

*Documentation reproduced from package spatstat, version 1.63-0, License: GPL (>= 2)*