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
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 data frame also has attributes
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 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.
the resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
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.
This argument is for internal use only.
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 rolfturner@posteo.net
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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