Jest(X) Jest(X, eps) Jest(X, eps, r) Jest(X, eps, breaks)
"ppp", or data in any format acceptable to
r. Not normally invoked by the user. See Details section.
Festfor this point pattern, containing three estimates of the empty space function $F(r)$ and an estimate of its hazard function
Gestfor 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
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
may have arbitrary shape.
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
The reduced-sample or border corrected estimate
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
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
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
uses two discrete approximations which are controlled
by the parameter
eps and by the spacing of values of
Fest for details.
First-time users are strongly advised not to specify these arguments.
Note that the value returned by
the output of
as attributes (see the last example below).
If the user is intending to compute the
functions for the point pattern, it is only necessary to
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.
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")