Estimate the summary functions \(E(r)\) and \(V(r)\) for a marked point pattern, proposed by Schlather et al (2004) as diagnostics for dependence between the points and the marks.

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

If `marks(X)`

is a numeric vector, the result is
an object of class `"fv"`

(see `fv.object`

).
If `marks(X)`

is a data frame, the result is
a list of objects of class `"fv"`

, one for each column of marks.

An object of class `"fv"`

is essentially
a data frame containing numeric columns

- r
the values of the argument \(r\) at which the function \(E(r)\) or \(V(r)\) has been estimated

- theo
the theoretical, constant value of \(E(r)\) or \(V(r)\) when the marks attached to different points are independent

together with a column or columns named

`"iso"`

and/or `"trans"`

,
according to the selected edge corrections. These columns contain
estimates of the function \(E(r)\) or \(V(r)\)

obtained by the edge corrections named.

- X
The observed point pattern. An object of class

`"ppp"`

or something acceptable to`as.ppp`

. The pattern should have numeric marks.- r
Optional. Numeric vector. The values of the argument \(r\) at which the function \(E(r)\) or \(V(r)\) should be evaluated. There is a sensible default.

- correction
A character vector containing any selection of the options

`"isotropic"`

,`"Ripley"`

or`"translate"`

. It specifies the edge correction(s) to be applied.- 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 estimate of \(E(r)\) or \(V(r)\) so that it would have value equal to 1 if the marks are independent of the points.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

For a marked point process, Schlather et al (2004) defined the functions \(E(r)\) and \(V(r)\) to be the conditional mean and conditional variance of the mark attached to a typical random point, given that there exists another random point at a distance \(r\) away from it.

More formally, $$ E(r) = E_{0u}[M(0)] $$ and $$ V(r) = E_{0u}[(M(0)-E(u))^2] $$ where \(E_{0u}\) denotes the conditional expectation given that there are points of the process at the locations \(0\) and \(u\) separated by a distance \(r\), and where \(M(0)\) denotes the mark attached to the point \(0\).

These functions may serve as diagnostics for dependence between the points and the marks. If the points and marks are independent, then \(E(r)\) and \(V(r)\) should be constant (not depending on \(r\)). See Schlather et al (2004).

The argument `X`

must be a point pattern (object of class
`"ppp"`

) or any data that are acceptable to `as.ppp`

.
It must be a marked point pattern with numeric marks.

The argument `r`

is the vector of values for the
distance \(r\) at which \(k_f(r)\) is estimated.

This algorithm assumes that `X`

can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in `X`

as `Window(X)`

)
may have arbitrary shape.

Biases due to edge effects are
treated in the same manner as in `Kest`

.
The edge corrections implemented here are

- isotropic/Ripley
Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented only for rectangular and polygonal windows (not for binary masks).

- translate
Translation correction (Ohser, 1983). Implemented for all window geometries, but slow for complex windows.

Note that the estimator assumes the process is stationary (spatially homogeneous).

The numerator and denominator of the mark correlation function (in the expression above) are estimated using density estimation techniques. The user can choose between

`"density"`

which uses the standard kernel density estimation routine

`density`

, and works only for evenly-spaced`r`

values;`"loess"`

which uses the function

`loess`

in the package modreg;`"sm"`

which uses the function

`sm.density`

in the package sm and is extremely slow;`"smrep"`

which uses the function

`sm.density`

in the package sm and is relatively fast, but may require manual control of the smoothing parameter`hmult`

.

Schlather, M. and Ribeiro, P. and Diggle, P. (2004)
Detecting dependence between marks and locations of
marked point processes.
*Journal of the Royal Statistical Society, series B*
**66** (2004) 79-83.

Mark correlation `markcorr`

,
mark variogram `markvario`

for numeric marks.

Mark connection function `markconnect`

and
multitype K-functions `Kcross`

, `Kdot`

for factor-valued marks.

```
plot(Emark(spruces))
E <- Emark(spruces, method="density", kernel="epanechnikov")
plot(Vmark(spruces))
plot(Emark(finpines))
V <- Vmark(finpines)
```

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