Jsphere: Estimate the J-function
In baddstats/spherstat: Spatial Point Pattern Analysis, Geometry and Trigonometry for the Sphere
Jsphere R Documentation
Estimate the J-function
Description
Estimates the summary function J(r) for a point pattern in a
window of arbitrary shape on a (subset of) a sphere.
Usage
Jsphere(X, refpoints, r=NULL, ..., correction=NULL)
Arguments
X
The observed point pattern, from which an estimate of J(r)
will be computed. An object of class sp2 or
sp3, or a 2 or 3 column matrix giving the locations of
points in spherical coordinates.
refpoints
The reference points in the window used to estimate the F function
(a prerequisite to calculating J(r), see Details
section). An object of class sp2 or
sp3, or a 2 or 3 column matrix giving the locations
of points in spherical coordinates. If NULL, a homogeneous Poisson process with expected number of points equal to 100 is generated inside the window.
r
Optional. Numeric vector. The values of the argument r at
which F(r) should be evaluated. There is a sensible default,
which works well the window is either the sphere or contains a
significant proportion of the sphere. First-time users are strongly
advised not to specify this argument.
...
Optional. Extra detail that can be passed to
eroded.areas.sphwin.
correction
The corrections to be applied. If set to NULL, all
estimators are given. Specific estimators can be called using the
arguments un, raw, rs, rs.modif,
cs, han and km.
Details
The J function (Van Lieshout and Baddeley, 1996) of a
stationary point process is defined as
J(r) = (1-G(r))/(1-F(r))
where G(r) is the nearest neighbour distance distribution
function of the point process (see Gsphere) and F(r) is
its empty space function (see Fsphere).
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$win) may have arbitrary shape.
The argument X is interpreted as a point pattern object (of
class sp2 or sp3).
The functions Fsphere and Gsphere
are called to compute
estimates of F(r) and G(r) respectively. These estimates
are then combined by simply taking the ratio
J(r) = (1-G(r))/(1-F(r)).
In fact several different estimates are computed using different edge
corrections (Baddeley, 1998).
The Kaplan-Meier estimate (returned as km) is the ratio
J = (1-G)/(1-F) of the Kaplan-Meier estimates of 1-F and
1-G computed by Fsphere and Gsphere
respectively. This is computed if correction=NULL or if
correction includes "km".
The Hanisch-style estimate (returned as han) is the ratio
J = (1-G)/(1-F) using the Chiu-Stoyan estimate of
F and the Hanisch estimate of G. This is
computed if correction=NULL or if correction includes
"cs" or "han".
The reduced-sample or border corrected estimate (returned as
rs) is the same ratio J = (1-G)/(1-F) of the border
corrected estimates. This is computed if correction=NULL or if
correction includes "rs" or "border".
These edge-corrected estimators are slightly biased for J,
since they are ratios of approximately unbiased estimators. The
logarithm of the Kaplan-Meier estimate is exactly unbiased for
log(J).
The uncorrected estimate (returned as un and computed only if
correction includes "none") is the ratio
J = (1-G)/(1-F)
of the uncorrected ("raw") estimates of the survival functions
of F and G, which are the empirical distribution
functions of the empty space distances Fsphere(X,...)$raw and
of the nearest neighbour distances Gsphere(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 Fsphere uses two discrete approximations
which are controlled by the parameter eps and by the spacing of
values of r respectively. See Fsphere for
details. First-time users are strongly advised not to specify these
arguments.
Note that the value returned by Jsphere includes the output of
Fsphere and Gsphere 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
Jsphere.
Value
An object of class "fv", see fv.object,
which can be plotted directly using plot.fv.
Essentially a data frame containing some or all of the following columns:
r
the values of the argument r at which the function J(r)
has been estimated
rs
the "reduced sample" (border correction) estimator of J(r)
km
the spatial Kaplan-Meier estimator of J(r)
hazard
the hazard rate lambda(r) of J(r) by the
spatial Kaplan-Meier method
raw
the uncorrected estimate of J(r), i.e. the empirical
distribution of the distance from a fixed point in the window to the
nearest point of X
han
the Hanisch correction estimator of J(r)
theo
the theoretical value of J(r) for a stationary
Poisson process of the same estimated intensity.
The data frame also has attributes:
F
the output of Fsphere 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 Gsphere for this point pattern, containing
three estimates of the nearest neighbour distance distribution
function G(r) and an estimate of its hazard function
Note
Sizeable amounts of memory may be needed during the calculation
Note
This function is the analogue for point processes on the sphere of the
function Jest in spatstat, which is the
corresponding function for point processes in R^2. Hence
elements of the code for Jsphere and help page have been taken
from Jest with the permission of
A. J. Baddeley. This enables the code to be highly efficient and give
corresponding output to, and for the information on this help page to
be consistent with that for the function Jest.
It is hoped that this will minimise or remove any confusion for users
of both spatstat and spherstat.
Author(s)
Tom Lawrence <email:tjlawrence@bigpond.com>
References
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.
Lawrence, T.J. (2017) Master's Thesis, University of Western Australia.
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.
See Also
Jest, Ksphere
Fsphere, Gsphere.
Examples
sph <- sphwin(type="sphere")
sph.pp <- rpoispp.sphwin(win=sph, lambda=10)
sph.ref <- rpoispp.sphwin(win=sph, lambda=150)
sph.Jest <- Jsphere(X=sph.pp, refpoints=sph.ref)
baddstats/spherstat documentation built on Feb. 6, 2023, 1:45 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
| Jsphere | R Documentation |
Estimate the J-function
Description
Estimates the summary function J(r) for a point pattern in a window of arbitrary shape on a (subset of) a sphere.
Usage
Jsphere(X, refpoints, r=NULL, ..., correction=NULL)
Arguments
X |
The observed point pattern, from which an estimate of J(r)
will be computed. An object of class |
refpoints |
The reference points in the window used to estimate the F function
(a prerequisite to calculating J(r), see Details
section). An object of class |
r |
Optional. Numeric vector. The values of the argument r at which F(r) should be evaluated. There is a sensible default, which works well the window is either the sphere or contains a significant proportion of the sphere. First-time users are strongly advised not to specify this argument. |
... |
Optional. Extra detail that can be passed to
|
correction |
The corrections to be applied. If set to |
Details
The J function (Van Lieshout and Baddeley, 1996) of a stationary point process is defined as
J(r) = (1-G(r))/(1-F(r))
where G(r) is the nearest neighbour distance distribution
function of the point process (see Gsphere) and F(r) is
its empty space function (see Fsphere).
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$win) may have arbitrary shape.
The argument X is interpreted as a point pattern object (of
class sp2 or sp3).
The functions Fsphere and Gsphere
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 Fsphere and Gsphere
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) using the Chiu-Stoyan estimate of
F and 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 Fsphere(X,...)$raw and
of the nearest neighbour distances Gsphere(X,...)$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 Fsphere uses two discrete approximations
which are controlled by the parameter eps and by the spacing of
values of r respectively. See Fsphere for
details. First-time users are strongly advised not to specify these
arguments.
Note that the value returned by Jsphere includes the output of
Fsphere and Gsphere 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
Jsphere.
Value
An object of class "fv", see fv.object,
which can be plotted directly using plot.fv.
Essentially a data frame containing some or all of the following columns:
r |
the values of the argument r at which the function J(r) has been estimated |
rs |
the "reduced sample" (border correction) estimator of J(r) |
km |
the spatial Kaplan-Meier estimator of J(r) |
hazard |
the hazard rate lambda(r) of J(r) by the spatial Kaplan-Meier method |
raw |
the uncorrected estimate of J(r), i.e. the empirical
distribution of the distance from a fixed point in the window to the
nearest point of |
han |
the Hanisch correction estimator of J(r) |
theo |
the theoretical value of J(r) for a stationary Poisson process of the same estimated intensity. |
The data frame also has attributes:
F |
the output of |
G |
the output of |
Note
Sizeable amounts of memory may be needed during the calculation
Note
This function is the analogue for point processes on the sphere of the
function Jest in spatstat, which is the
corresponding function for point processes in R^2. Hence
elements of the code for Jsphere and help page have been taken
from Jest with the permission of
A. J. Baddeley. This enables the code to be highly efficient and give
corresponding output to, and for the information on this help page to
be consistent with that for the function Jest.
It is hoped that this will minimise or remove any confusion for users
of both spatstat and spherstat.
Author(s)
Tom Lawrence <email:tjlawrence@bigpond.com>
References
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.
Lawrence, T.J. (2017) Master's Thesis, University of Western Australia.
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.
See Also
Jest, Ksphere
Fsphere, Gsphere.
Examples
sph <- sphwin(type="sphere") sph.pp <- rpoispp.sphwin(win=sph, lambda=10) sph.ref <- rpoispp.sphwin(win=sph, lambda=150) sph.Jest <- Jsphere(X=sph.pp, refpoints=sph.ref)
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.