# 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

the values of the argument \(r\) at which the mark variogram \(\gamma(r)\) has been estimated

the theoretical value of \(\gamma(r)\) when the marks attached to different points are independent; equal to the sample variance of the marks

##### 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

```
# NOT RUN {
# 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 1.55-1, License: GPL (>= 2)*