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)`

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.

An object of class `"fv"`

, see `fv.object`

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

.

Essentially a data frame containing

the vector of values of the argument \(r\) at which the function \(J\) has been estimated

the ``reduced sample'' or ``border correction'' estimator of \(J(r)\) computed from the border-corrected estimates of \(F\) and \(G\)

the spatial Kaplan-Meier estimator of \(J(r)\) computed from the Kaplan-Meier estimates of \(F\) and \(G\)

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

the uncorrected estimate of \(J(r)\) computed from the uncorrected estimates of \(F\) and \(G\)

the theoretical value of \(J(r)\) for a stationary Poisson process: identically equal to \(1\)

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

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

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`

# NOT RUN { 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, legendpos="center") plot(J, main="redwood data") # values are below J = 1, indicating clustered pattern # }