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

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

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

- 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 \(G\)

- han
the Hanisch-style estimator of \(J(r)\) computed from the Hanisch estimate of \(G\) and the Chiu-Stoyan estimate of \(F\)

- 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 function- G
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

- 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()`

.- ...
Ignored.

- 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
This argument is for internal use only.

- correction
Optional. Character string specifying the choice of edge correction(s) in

`Fest`

and`Gest`

. See Details.

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

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 `Window(X)`

)
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 several 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.
This is computed if `correction=NULL`

or if `correction`

includes `"km"`

.

The Hanisch-style estimate (returned as `han`

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

where `F`

is the Chiu-Stoyan estimate of
\(F\) and `G`

is the Hanisch estimate of \(G\).
This is computed if `correction=NULL`

or if `correction`

includes `"cs"`

or `"han"`

.

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

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

of the border corrected estimates.
This is computed if `correction=NULL`

or if `correction`

includes `"rs"`

or `"border"`

.

These edge-corrected estimators are slightly biased for \(J\), since they are ratios of approximately unbiased estimators. The logarithm of the Kaplan-Meier estimate is exactly unbiased for \(\log J\).

The uncorrected estimate (returned as `un`

and computed only if `correction`

includes `"none"`

)
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`

.

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.

`Jinhom`

,
`Fest`

,
`Gest`

,
`Kest`

,
`km.rs`

,
`reduced.sample`

,
`kaplan.meier`

```
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, legendpos="center")
plot(J, main="redwood data")
# values are below J = 1, indicating clustered pattern
```

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