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

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.- 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

`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`

.

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.

`Fest`

,
`Gest`

,
`Kest`

,
`kmrs`

,
`reduced.sample`

,
`kaplan.meier`

library(spatstat) data(cells) Jc <- Jest(cells, 0.01) plot(Jc$r, Jc$km, type="l") abline(h=1, lty=2) # values are far above J= 1, indicating regular pattern data(redwood) Jw <- Jest(redwood, 0.01) plot(Jw$r, Jw$km, type="l") abline(h=1, lty=2) # values are below J= 1, indicating clustered pattern Jc <- Jest(cells, 0.01) # alternative plot commands plot(km ~ r, type="l", data=Jc) # restrict the plot to values of r less than 0.1 plot(km ~ r, type="l", data=Fc[Fc$r <= 0.1, ]) plot(km ~ r, type="l", data=Fc, subset=(r <= 0.1)) # simulated data X <- runifpoint(50) J <- Jest(X, 0.01) plot(J$r, J$km, type="l", xlab="r", ylab="J(r)", ylim=c(0,1), main = "empty space function") abline(h=1, lty=2) legend(0.5, 2, c("Kaplan-Meier estimator", "Poisson process"), lty=c(1,2)) # functions F, G and J Jout <- Jest(X, 0.01) FF <- attr(Jout, "F") G <- attr(Jout, "G") plot(km ~ r, type="l", data=FF, main="Empty space function") plot(km ~ r, type="l", data=G, main="Nearest neighbour function") plot(km ~ r, type="l", data=Jout, main="J-function")