# Jest

##### Estimate the J-function

Estimates the summary function $J(r)$ for a point pattern in a window of arbitrary shape.

- Keywords
- spatial, nonparametric

##### Usage

`Jest(X, eps=NULL, r=NULL, breaks=NULL)`

##### Arguments

- X
- The observed point pattern,
from which an estimate of $J(r)$ will be computed.
An object of class
`"ppp"`

, or data in any format acceptable to`as.ppp()`

. - eps
- the resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
- r
- vector of values for the argument $r$ at which $J(r)$
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
- An alternative to the argument
`r`

. Not normally invoked by the user. See Details section.

##### Details

The $J$ function (Van Lieshout and Baddeley ,1996)
of a stationary point process is defined as
$$J(r) = \frac{1-G(r)}{1-F(r)}$$
where $G(r)$ is the nearest neighbour distance distribution
function of the point process (see `Gest`

)
and $F(r)$ is its empty space function (see `Fest`

).

For a completely random (uniform Poisson) point process, the $J$-function is identically equal to $1$. Deviations $J(r) < 1$ or $J(r) > 1$ typically indicate spatial clustering or spatial regularity, respectively. The $J$-function is one of the few characteristics that can be computed explicitly for a wide range of point processes. See Van Lieshout and Baddeley (1996), Baddeley et al (2000), Thonnes and Van Lieshout (1999) for further information.

An estimate of $J$ derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern. The estimate of $J(r)$ is compared against the constant function $1$. Deviations $J(r) < 1$ or $J(r) > 1$ may suggest spatial clustering or spatial regularity, respectively.

This algorithm estimates the $J$-function
from the point pattern `X`

. It 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 `X$window`

)
may have arbitrary shape.

The argument `X`

is interpreted as a point pattern object
(of class `"ppp"`

, see `ppp.object`

) and can
be supplied in any of the formats recognised by
`as.ppp()`

.

The functions `Fest`

and `Gest`

are called to
compute estimates of $F(r)$ and $G(r)$ respectively.
These estimates are then combined by simply taking the ratio
$J(r) = (1-G(r))/(1-F(r))$.

In fact three different estimates are computed
using different edge corrections (Baddeley, 1998).
The Kaplan-Meier estimate (returned as `km`

) is the ratio
`J = (1-G)/(1-F)`

of the Kaplan-Meier estimates of
$1-F$ and $1-G$ computed by
`Fest`

and `Gest`

respectively.
The reduced-sample or border corrected estimate
(returned as `rs`

) is
the same ratio `J = (1-G)/(1-F)`

of the border corrected estimates.
These estimators are slightly biased for $J$,
since they are ratios
of approximately unbiased estimators. The logarithm of the
Kaplan-Meier estimate is unbiased for $\log J$.

The uncorrected estimate (returned as `un`

)
is the ratio `J = (1-G)/(1-F)`

of the uncorrected (``raw'') estimates of the survival functions
of $F$ and $G$,
which are the empirical distribution functions of the
empty space distances `Fest(X,...)$raw`

and of the nearest neighbour distances
`Gest(X,...)$raw`

. The uncorrected estimates
of $F$ and $G$ are severely biased.
However the uncorrected estimate of $J$
is approximately unbiased (if the process is close to Poisson);
it is insensitive to edge effects, and should be used when
edge effects are severe (see Baddeley et al, 2000).

The algorithm for `Fest`

uses two discrete approximations which are controlled
by the parameter `eps`

and by the spacing of values of `r`

respectively. See `Fest`

for details.
First-time users are strongly advised not to specify these arguments.

Note that the value returned by `Jest`

includes
the output of `Fest`

and `Gest`

as attributes (see the last example below).
If the user is intending to compute the `F,G`

and `J`

functions for the point pattern, it is only necessary to
call `Jest`

.

##### Value

- An object of class
`"fv"`

, see`fv.object`

, which can be plotted directly using`plot.fv`

.Essentially a data frame containing

r the vector of values of the argument $r$ at which the function $J$ has been estimated J the recommended estimate of $J(r)$, which is the Kaplan-Meier estimate `km`

rs the ``reduced sample'' or ``border correction'' estimator of $J(r)$ computed from the border-corrected estimates of $F$ and $G$ km the spatial Kaplan-Meier estimator of $J(r)$ computed from the Kaplan-Meier estimates of $F$ and $G$ un the uncorrected estimate of $J(r)$ computed from the uncorrected estimates of $F$ and $G$ theo the theoretical value of $J(r)$ for a stationary Poisson process: identically equal to $1$ - The data frame also has
**attributes** F the output of `Fest`

for this point pattern, containing three estimates of the empty space function $F(r)$ and an estimate of its hazard functionG the output of `Gest`

for this point pattern, containing three estimates of the nearest neighbour distance distribution function $G(r)$ and an estimate of its hazard function

##### Note

Sizeable amounts of memory may be needed during the calculation.

##### References

Baddeley, A.J. Spatial sampling and censoring.
In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds)
*Stochastic Geometry: Likelihood and Computation*.
Chapman and Hall, 1998.
Chapter 2, pages 37--78.
Baddeley, A.J. and Gill, R.D.
The empty space hazard of a spatial pattern.
Research Report 1994/3, Department of Mathematics,
University of Western Australia, May 1994.

Baddeley, A.J. and Gill, R.D.
Kaplan-Meier estimators of interpoint distance
distributions for spatial point processes.
*Annals of Statistics* **25** (1997) 263--292.

Baddeley, A., Kerscher, M., Schladitz, K. and Scott, B.T.
Estimating the *J* function without edge correction.
*Statistica Neerlandica* **54** (2000) 315--328.

Borgefors, G.
Distance transformations in digital images.
*Computer Vision, Graphics and Image Processing*
**34** (1986) 344--371.

Cressie, N.A.C. *Statistics for spatial data*.
John Wiley and Sons, 1991.

Diggle, P.J. *Statistical analysis of spatial point patterns*.
Academic Press, 1983.

Ripley, B.D. *Statistical inference for spatial processes*.
Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J.
*Stochastic geometry and its applications*.
2nd edition. Springer Verlag, 1995.

Thonnes, E. and Van Lieshout, M.N.M,
A comparative study on the power of Van Lieshout and Baddeley's J-function.
*Biometrical Journal* **41** (1999) 721--734.

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

##### See Also

##### Examples

```
data(cells)
J <- Jest(cells, 0.01)
plot(J, main="cells data")
# values are far above J= 1, indicating regular pattern
data(redwood)
J <- Jest(redwood, 0.01)
plot(J, main="redwood data")
# values are below J= 1, indicating clustered pattern
```

*Documentation reproduced from package spatstat, version 1.10-2, License: GPL version 2 or newer*