# Jinhom

##### Inhomogeneous J-function

Estimates the inhomogeneous $J$ function of a non-stationary point pattern.

- Keywords
- spatial, nonparametric

##### Usage

```
Jinhom(X, lambda = NULL, lmin = NULL, ...,
sigma = NULL, varcov = NULL,
r = NULL, breaks = NULL)
```

##### Arguments

- X
- The observed data point pattern,
from which an estimate of the inhomogeneous $J$ function
will be computed.
An object of class
`"ppp"`

or in a format recognised by`as.ppp()`

- lambda
- Optional.
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern
`X`

, a pixel image (object of class`"im"`

) giving the intensity values at all locatio - lmin
- Optional. The minimum possible value of the intensity over the spatial domain. A positive numerical value.
- sigma,varcov
- Optional arguments passed to
`density.ppp`

to control the smoothing bandwidth, when`lambda`

is estimated by kernel smoothing. - ...
- Extra arguments passed to
`as.mask`

to control the pixel resolution, or passed to`density.ppp`

to control the smoothing bandwidth. - r
- vector of values for the argument $r$ at which the inhomogeneous $K$ function should be evaluated. Not normally given by the user; there is a sensible default.
- breaks
- An alternative to the argument
`r`

. Not normally invoked by the user. See Details.

##### Details

This command computes estimates of the
inhomogeneous $J$-function (Van Lieshout, 2010)
of a point pattern. It is the counterpart, for inhomogeneous
spatial point patterns, of the $J$ function
for homogeneous point patterns computed by `Jest`

.

The argument `X`

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

).

The inhomogeneous $J$ function is computed as
$Jinhom(r) = (1 - Ginhom(r))/(1-Finhom(r))$
where $Ginhom, Finhom$ are the inhomogeneous $G$ and $F$
functions computed using the border correction
(equations (7) and (6) respectively in Van Lieshout, 2010).
The argument `lambda`

should supply the
(estimated) values of the intensity function $\lambda$
of the point process. It may be either
[object Object],[object Object],[object Object],[object Object],[object Object]
If `lambda`

is a numeric vector, then its length should
be equal to the number of points in the pattern `X`

.
The value `lambda[i]`

is assumed to be the
the (estimated) value of the intensity
$\lambda(x_i)$ for
the point $x_i$ of the pattern $X$.
Each value must be a positive number; `NA`

's are not allowed.

If `lambda`

is a pixel image, the domain of the image should
cover the entire window of the point pattern. If it does not (which
may occur near the boundary because of discretisation error),
then the missing pixel values
will be obtained by applying a Gaussian blur to `lambda`

using
`blur`

, then looking up the values of this blurred image
for the missing locations.
(A warning will be issued in this case.)

If `lambda`

is a function, then it will be evaluated in the
form `lambda(x,y)`

where `x`

and `y`

are vectors
of coordinates of the points of `X`

. It should return a numeric
vector with length equal to the number of points in `X`

.

If `lambda`

is omitted, then it will be estimated using
a `leave-one-out' kernel smoother, as described in Baddeley, Moller
and Waagepetersen (2000). The estimate `lambda[i]`

for the
point `X[i]`

is computed by removing `X[i]`

from the
point pattern, applying kernel smoothing to the remaining points using
`density.ppp`

, and evaluating the smoothed intensity
at the point `X[i]`

. The smoothing kernel bandwidth is controlled
by the arguments `sigma`

and `varcov`

, which are passed to
`density.ppp`

along with any extra arguments.

##### Value

##### References

van Lieshout, M.N.M. and Baddeley, A.J. (1996)
A nonparametric measure of spatial interaction in point patterns.
*Statistica Neerlandica* **50**, 344--361.

van Lieshout, M.N.M. (2010)
A J-function for inhomogeneous point processes.
*Statistica Neerlandica* **65**, 183--201.

##### See Also

##### Examples

```
plot(Jinhom(swedishpines, sigma=bw.diggle, adjust=2))
plot(Jinhom(swedishpines, sigma=10))
```

*Documentation reproduced from package spatstat, version 1.31-3, License: GPL (>= 2)*