Given a point pattern `X`

and a spatial object `Y`

,
compute estimates of Foxall's \(G\) and \(J\) functions.

```
Gfox(X, Y, r=NULL, breaks=NULL, correction=c("km", "rs", "han"), W, ...)
Jfox(X, Y, r=NULL, breaks=NULL, correction=c("km", "rs", "han"), W, ...,
warn.trim=TRUE)
```

A function value table (object of class `"fv"`

)
which can be printed, plotted, or converted to a data frame of values.

- X
A point pattern (object of class

`"ppp"`

) from which distances will be measured.- Y
An object of class

`"ppp"`

,`"psp"`

or`"owin"`

to which distances will be measured. Alternatively a pixel image (class`"im"`

) with logical values.- r
Optional. Numeric vector. The values of the argument \(r\) at which \(Gfox(r)\) or \(Jfox(r)\) should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on \(r\).

- breaks
This argument is for internal use only.

- correction
Optional. The edge correction(s) to be used to estimate \(Gfox(r)\) or \(Jfox(r)\). A vector of character strings selected from

`"none"`

,`"rs"`

,`"km"`

,`"cs"`

and`"best"`

. Alternatively`correction="all"`

selects all options.- W
Optional. A window (object of class

`"owin"`

) to be taken as the window of observation. The distribution function will be estimated from data inside`W`

. The default is`W=Frame(Y)`

when`Y`

is a window, and`W=Window(Y)`

otherwise.- ...
Extra arguments affecting the discretisation of distances. These arguments are ignored by

`Gfox`

, but`Jfox`

passes them to`Hest`

to determine the discretisation of the spatial domain.- warn.trim
Logical value indicating whether a warning should be issued by

`Jfox`

when the window of`X`

had to be trimmed in order to be a subset of the frame of`Y`

.

Rob Foxall and Adrian Baddeley Adrian.Baddeley@curtin.edu.au

Given a point pattern `X`

and another spatial object `Y`

,
these functions compute two nonparametric measures of association
between `X`

and `Y`

, introduced by Foxall
(Foxall and Baddeley, 2002).

Let the random variable \(R\) be the distance from a typical point
of `X`

to the object `Y`

.
Foxall's \(G\)-function is the cumulative distribution function
of \(R\):
$$G(r) = P(R \le r)$$

Let the random variable \(S\) be the distance from a *fixed* point
in space to the object `Y`

. The cumulative distribution function
of \(S\) is the (unconditional) spherical contact distribution
function
$$H(r) = P(S \le r)$$
which is computed by `Hest`

.

Foxall's \(J\)-function is the ratio $$ J(r) = \frac{1-G(r)}{1-H(r)} $$ For further interpretation, see Foxall and Baddeley (2002).

Accuracy of `Jfox`

depends on the pixel resolution,
which is controlled by the
arguments `eps`

, `dimyx`

and `xy`

passed to
`as.mask`

. For example, use `eps=0.1`

to specify
square pixels of side 0.1 units, and `dimyx=256`

to specify a
256 by 256 grid of pixels.

Foxall, R. and Baddeley, A. (2002)
Nonparametric measures of association between a
spatial point process and a random set, with
geological applications. *Applied Statistics* **51**, 165--182.

`Gest`

,
`Hest`

,
`Jest`

,
`Fest`

```
X <- copper$SouthPoints
Y <- copper$SouthLines
G <- Gfox(X,Y)
J <- Jfox(X,Y, correction="km")
# \testonly{
J <- Jfox(X,Y, correction="km", eps=1)
# }
```

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