# markvario

##### Mark Variogram

Estimate the mark variogram of a marked point pattern.

- Keywords
- spatial, nonparametric

##### Usage

```
markvario(X, correction = c("isotropic", "Ripley", "translate"),
r = NULL, method = "density", ..., normalise=FALSE)
```

##### Arguments

- X
- The observed point pattern.
An object of class
`"ppp"`

or something acceptable to`as.ppp`

. It must have marks which are numeric. - correction
- A character vector containing any selection of the
options
`"isotropic"`

,`"Ripley"`

or`"translate"`

. It specifies the edge correction(s) to be applied. - r
- numeric vector. The values of the argument $r$ at which the mark variogram $\gamma(r)$ should be evaluated. There is a sensible default.
- method
- A character vector indicating the user's choice of
density estimation technique to be used. Options are
`"density"`

,`"loess"`

,`"sm"`

and`"smrep"`

. - ...
- Arguments passed to the density estimation routine
(
`density`

,`loess`

or`sm.density`

) selected by`method`

. - normalise
- If
`TRUE`

, normalise the variogram by dividing it by the estimated mark variance.

##### Details

The mark variogram $\gamma(r)$ of a marked point process $X$ is a measure of the dependence between the marks of two points of the process a distance $r$ apart. It is informally defined as $$\gamma(r) = E[\frac 1 2 (M_1 - M_2)^2]$$ where $E[ ]$ denotes expectation and $M_1,M_2$ are the marks attached to two points of the process a distance $r$ apart.

The mark variogram of a marked point process is analogous,
but **not equivalent**, to the variogram of a random field
in geostatistics. See Waelder and Stoyan (1996).

##### Value

- An object of class
`"fv"`

(see`fv.object`

). Essentially a data frame containing numeric columns r the values of the argument $r$ at which the mark variogram $\gamma(r)$ has been estimated theo the theoretical value of $\gamma(r)$ when the marks attached to different points are independent; equal to the sample variance of the marks - together with a column or columns named
`"iso"`

and/or`"trans"`

, according to the selected edge corrections. These columns contain estimates of the function $\gamma(r)$ obtained by the edge corrections named.

##### References

Cressie, N.A.C. (1991)
*Statistics for spatial data*.
John Wiley and Sons, 1991.
Mase, S. (1996)
The threshold method for estimating annual rainfall.
*Annals of the Institute of Statistical Mathematics*
**48** (1996) 201-213.

Waelder, O. and Stoyan, D. (1996)
On variograms in point process statistics.
*Biometrical Journal* **38** (1996) 895-905.

##### See Also

Mark correlation function `markcorr`

for numeric marks.

Mark connection function `markconnect`

and
multitype K-functions `Kcross`

, `Kdot`

for factor-valued marks.

##### Examples

```
# Longleaf Pine data
# marks represent tree diameter
data(longleaf)
# Subset of this large pattern
swcorner <- owin(c(0,100),c(0,100))
sub <- longleaf[ , swcorner]
# mark correlation function
mv <- markvario(sub)
plot(mv)
```

*Documentation reproduced from package spatstat, version 1.34-1, License: GPL (>= 2)*