R
for the statistical analysis of spatial point patterns.spatstat, see the workshop notes
by Baddeley (2008).
Both of these documents are available on the internet. Type demo(spatstat) for a demonstration
of the package's capabilities.
Type demo(data) to see all the datasets
available in the package.
latest.news() to read the news documentation about
changes to the current installed version of news(package="spatstat") to read news documentation about
all previous versions of the package.library(help=spatstat). For further information on any of these,
type help(name) where name is the name of the function
or dataset.
The main types of spatial data supported by
ppp point pattern
owin window (spatial region)
im pixel image
psp line segment pattern
tess tessellation
pp3 three-dimensional point pattern
ppx point pattern in any number of dimensions
}
To create a point pattern:
ppp create a point pattern from $(x,y)$ and window information
ppp(x, y, xlim, ylim) for rectangular window
ppp(x, y, poly) for polygonal window
ppp(x, y, mask) for binary image window
as.ppp convert other types of data to a ppp object
clickppp interactively add points to a plot
marks<-, %mark% attach/reassign marks to a point pattern
}
To simulate a random point pattern:
runifpoint generate $n$ independent uniform random points
rpoint generate $n$ independent random points
rmpoint generate $n$ independent multitype random points
rpoispp simulate the (in)homogeneous Poisson point process
rmpoispp simulate the (in)homogeneous multitype Poisson point process
runifdisc generate $n$ independent uniform random points in disc
rstrat stratified random sample of points
rsyst systematic random sample of points
rjitter apply random displacements to points in a pattern
rMaternI simulate the Mat'ern Model I inhibition process
rMaternII simulate the Mat'ern Model II inhibition process
rSSI simulate Simple Sequential Inhibition process
rStrauss simulate Strauss process (perfect simulation)
rNeymanScott simulate a general Neyman-Scott process
rMatClust simulate the Mat'ern Cluster process
rThomas simulate the Thomas process
rGaussPoisson simulate the Gauss-Poisson cluster process
rthin random thinning
rcell simulate the Baddeley-Silverman cell process
rmh simulate Gibbs point process using Metropolis-Hastings
simulate.ppm simulate Gibbs point process using Metropolis-Hastings
runifpointOnLines generate $n$ random points along specified line segments
rpoisppOnLines generate Poisson random points along specified line segments
}
To randomly change an existing point pattern:
rlabel random (re)labelling of a multitype
point pattern
rshift random shift (including toroidal shifts)
}
Standard point pattern datasets:
Remember to say data(amacrine) etc.
Type demo(data) to see all the datasets
installed with the package.
amacrine Austin Hughes' rabbit amacrine cells
anemones Upton-Fingleton sea anemones data
ants Harkness-Isham ant nests data
bei Tropical rainforest trees
betacells Waessle et al. cat retinal ganglia data
bramblecanes Bramble Canes data
bronzefilter Bronze Filter Section data
cells Crick-Ripley biological cells data
chorley Chorley-Ribble cancer data
copper Berman-Huntington copper deposits data
demopat Synthetic point pattern
finpines Finnish Pines data
hamster Aherne's hamster tumour data
humberside North Humberside childhood leukaemia data
japanesepines Japanese Pines data
lansing Lansing Woods data
longleaf Longleaf Pines data
murchison Murchison gold deposits
nbfires New Brunswick fires data
nztrees Mark-Esler-Ripley trees data
osteo Osteocyte lacunae (3D, replicated)
ponderosa Getis-Franklin ponderosa pine trees data
redwood Strauss-Ripley redwood saplings data
redwoodfull Strauss redwood saplings data (full set)
residualspaper Data from Baddeley et al (2005)
shapley Galaxies in an astronomical survey
simdat Simulated point pattern (inhomogeneous, with interaction)
spruces Spruce trees in Saxonia
swedishpines Strand-Ripley swedish pines data
urkiola Urkiola Woods data
}
To manipulate a point pattern:
plot.ppp plot a point pattern (e.g. plot(X))
iplot plot a point pattern interactively
[.ppp extract or replace a subset of a point pattern
pp[subset] or pp[subwindow]
superimpose combine several point patterns
by.ppp apply a function to sub-patterns of a point pattern
cut.ppp classify the points in a point pattern
unmark remove marks
marks attach marks or reset marks
npoints count the number of points
split.ppp divide pattern into sub-patterns
rotate rotate pattern
shift translate pattern
periodify make several translated copies
affine apply affine transformation
density.ppp kernel smoothing
identify.ppp interactively identify points
unique.ppp remove duplicate points
duplicated.ppp determine which points are duplicates
dirichlet compute Dirichlet-Voronoi tessellation
delaunay compute Delaunay triangulation
convexhull compute convex hull
discretise discretise coordinates
pixellate.ppp approximate point pattern by
pixel image
as.im.ppp approximate point pattern by
pixel image
}
See spatstat.options to control plotting behaviour.
To create a window:
An object of class "owin" describes a spatial region
(a window of observation).
owin Create a window object
owin(xlim, ylim) for rectangular window
owin(poly) for polygonal window
owin(mask) for binary image window
as.owin Convert other data to a window object
square make a square window
disc make a circular window
ripras Ripley-Rasson estimator of window, given only the points
convexhull compute convex hull of something
letterR polygonal window in the shape of the Rlogo
}
To manipulate a window:
plot.owin plot a window.
plot(W)
bounding.box Find a tight bounding box for the window
erosion erode window by a distance r
dilation dilate window by a distance r
closing close window by a distance r
opening open window by a distance r
border difference between window and its erosion/dilation
complement.owin invert (swap inside and outside)
simplify.owin approximate a window by a simple polygon
rotate rotate window
shift translate window
periodify make several translated copies
affine apply affine transformation
}
Digital approximations:
as.mask Make a discrete pixel approximation of a given window
as.im.owin convert window to pixel image
pixellate.owin convert window to pixel image
nearest.raster.point map continuous coordinates to raster locations
raster.x raster x coordinates
raster.y raster y coordinates
as.polygonal convert pixel mask to polygonal window
}
See spatstat.options to control the approximation
Geometrical computations with windows:
intersect.owin intersection of two windows
union.owin union of two windows
setminus.owin set subtraction of two windows
inside.owin determine whether a point is inside a window
area.owin compute area
perimeter compute perimeter length
diameter.owin compute diameter
incircle find largest circle inside a window
connected find connected components of window
eroded.areas compute areas of eroded windows
dilated.areas compute areas of dilated windows
bdist.points compute distances from data points to window boundary
bdist.pixels compute distances from all pixels to window boundary
bdist.tiles boundary distance for each tile in tessellation
distmap.owin distance transform image
distfun.owin distance transform
centroid.owin compute centroid (centre of mass) of window
is.subset.owin determine whether one
window contains another
is.convex determine whether a window is convex
convexhull compute convex hull
as.mask pixel approximation of window
as.polygonal polygonal approximation of window
}
Pixel images:
An object of class "im" represents a pixel image.
Such objects are returned by some of the functions in
Kmeasure,
setcov and density.ppp.
im create a pixel image
as.im convert other data to a pixel image
pixellate convert other data to a pixel image
as.matrix.im convert pixel image to matrix
as.data.frame.im convert pixel image to data frame
plot.im plot a pixel image on screen as a digital image
contour.im draw contours of a pixel image
persp.im draw perspective plot of a pixel image
[.im extract a subset of a pixel image
[<-.im replace a subset of a pixel image
shift.im apply vector shift to pixel image
X print very basic information about image X
summary(X) summary of image X
hist.im histogram of image
mean.im mean pixel value of image
integral.im integral of pixel values
quantile.im quantiles of image
cut.im convert numeric image to factor image
is.im test whether an object is a pixel image
interp.im interpolate a pixel image
blur apply Gaussian blur to image
connected find connected components
compatible.im test whether two images have
compatible dimensions
eval.im evaluate any expression involving images
levelset level set of an image
solutionset region where an expression is true
}
Line segment patterns
An object of class "psp" represents a pattern of straight line
segments.
psp create a line segment pattern
as.psp convert other data into a line segment pattern
is.psp determine whether a dataset has class "psp"
plot.psp plot a line segment pattern
print.psp print basic information
summary.psp print summary information
[.psp extract a subset of a line segment pattern
as.data.frame.psp convert line segment pattern to data frame
marks.psp extract marks of line segments
marks<-.psp assign new marks to line segments
unmark.psp delete marks from line segments
midpoints.psp compute the midpoints of line segments
endpoints.psp extract the endpoints of line segments
lengths.psp compute the lengths of line segments
angles.psp compute the orientation angles of line segments
rotate.psp rotate a line segment pattern
shift.psp shift a line segment pattern
periodify make several shifted copies
affine.psp apply an affine transformation
pixellate.psp approximate line segment pattern
by pixel image
as.mask.psp approximate line segment pattern
by binary mask
distmap.psp compute the distance map of a line
segment pattern
distfun.psp compute the distance map of a line
segment pattern
density.psp kernel smoothing of line segments
selfcrossing.psp find crossing points between
line segments
crossing.psp find crossing points between
two line segment patterns
nncross find distance to nearest line segment
from a given point
nearestsegment find line segment closest to a
given point
project2segment find location along a line segment
closest to a given point
pointsOnLines generate points evenly spaced
along line segment
rpoisline generate a realisation of the
Poisson line process inside a window
rlinegrid generate a random array of parallel
lines through a window
}
Tessellations
An object of class "tess" represents a tessellation.
tess create a tessellation
quadrats create a tessellation of rectangles
as.tess convert other data to a tessellation
plot.tess plot a tessellation
tiles extract all the tiles of a tessellation
[.tess extract some tiles of a tessellation
[<-.tess change some tiles of a tessellation
intersect.tess intersect two tessellations
or restrict a tessellation to a window
chop.tess subdivide a tessellation by a line
dirichlet compute Dirichlet-Voronoi tessellation of points
delaunay compute Delaunay triangulation of points
rpoislinetess generate tessellation using Poisson line
process
}
Three-dimensional point patterns
An object of class "pp3" represents a three-dimensional
point pattern in a rectangular box. The box is represented by
an object of class "box3".
pp3 create a 3-D point pattern
plot.pp3 plot a 3-D point pattern
coords extract coordinates
as.hyperframe extract coordinates
unitname.pp3 name of unit of length
npoints count the number of points
runifpoint3 generate uniform random points in 3-D
rpoispp3 generate Poisson random points in 3-D
envelope.pp3 generate simulation envelopes for
3-D pattern
box3 create a 3-D rectangular box
as.box3 convert data to 3-D rectangular box
unitname.box3 name of unit of length
diameter.box3 diameter of box
volume.box3 volume of box
shortside.box3 shortest side of box
eroded.volumes volumes of erosions of box
}
Multi-dimensional space-time point patterns
An object of class "ppx" represents a
point pattern in multi-dimensional space and/or time.
ppx create a multidimensional space-time point pattern
coords extract coordinates
as.hyperframe extract coordinates
unitname.ppx name of unit of length
npoints count the number of points
runifpointx generate uniform random points
rpoisppx generate Poisson random points
boxx define multidimensional box
diameter.boxx diameter of box
volume.boxx volume of box
shortside.boxx shortest side of box
eroded.volumes.boxx volumes of erosions of box
}
Hyperframes
A hyperframe is like a data frame, except that the entries may be objects of any kind.
hyperframe create a hyperframe
as.hyperframe convert data to hyperframe
plot.hyperframe plot hyperframe
with.hyperframe evaluate expression using each row
of hyperframe
cbind.hyperframe combine hyperframes by columns
rbind.hyperframe combine hyperframes by rows
as.data.frame.hyperframe convert hyperframe to
data frame
}
summary(X) print useful summary of point pattern X
X print basic description of point pattern X
any(duplicated(X)) check for duplicated points in pattern X
istat(X) Interactive exploratory analysis
} Classical exploratory tools:
clarkevans Clark and Evans aggregation index
fryplot Fry plot
miplot Morishita Index plot
}
Modern exploratory tools:
nnclean Byers-Raftery feature detection
}
Summary statistics for a point pattern:
quadratcount Quadrat counts
Fest empty space function $F$
Gest nearest neighbour distribution function $G$
Kest Ripley's $K$-function
Lest Ripley's $L$-function
Jest $J$-function $J = (1-G)/(1-F)$
allstats all four functions $F$, $G$, $J$, $K$
localL Getis-Franklin neighbourhood density function
localK neighbourhood K-function
pcf pair correlation function
Kinhom $K$ for inhomogeneous point patterns
Linhom $L$ for inhomogeneous point patterns
pcfinhom pair correlation for inhomogeneous patterns
Kest.fft fast $K$-function using FFT for large datasets
Kmeasure reduced second moment measure
envelope simulation envelopes for a summary
function
}
Related facilities:
plot.fv plot a summary function
eval.fv evaluate any expression involving
summary functions
eval.fasp evaluate any expression involving
an array of functions
with.fv evaluate an expression for a
summary function
smooth.fv apply smoothing to a summary function
nndist nearest neighbour distances
nnwhich find nearest neighbours
pairdist distances between all pairs of points
crossdist distances between points in two patterns
nncross nearest neighbours between two point patterns
exactdt distance from any location to nearest data point
distmap distance map image
distfun distance map function
density.ppp kernel smoothed density
smooth.ppp spatial interpolation of marks
rknn theoretical distribution of nearest
neighbour distance
}
Summary statistics for a multitype point pattern:
A multitype point pattern is represented by an object X
of class "ppp" with a component X$marks which is a factor.
Gcross,Gdot,Gmulti multitype nearest neighbour distributions
$G_{ij}, G_{i\bullet}$
Kcross,Kdot, Kmulti multitype $K$-functions
$K_{ij}, K_{i\bullet}$
Lcross,Ldot multitype $L$-functions
$L_{ij}, L_{i\bullet}$
Jcross,Jdot,Jmulti multitype $J$-functions
$J_{ij}, J_{i\bullet}$
pcfcross multitype pair correlation function $g_{ij}$
pcfdot multitype pair correlation function $g_{i\bullet}$
markconnect marked connection function $p_{ij}$
alltypes estimates of the above
for all $i,j$ pairs
Iest multitype $I$-function
Kcross.inhom,Kdot.inhom inhomogeneous counterparts of Kcross, Kdot
Lcross.inhom,Ldot.inhom inhomogeneous counterparts of Lcross, Ldot
pcfcross.inhom,pcfdot.inhom inhomogeneous counterparts of pcfcross, pcfdot
}
Summary statistics for a marked point pattern:
A marked point pattern is represented by an object X
of class "ppp" with a component X$marks.
The entries in the vector X$marks may be numeric, complex,
string or any other atomic type. For numeric marks, there are the
following functions:
markmean smoothed local average of marks
markvar smoothed local variance of marks
markcorr mark correlation function
markvario mark variogram
markcorrint mark correlation integral
Emark mark independence diagnostic $E(r)$
Vmark mark independence diagnostic $V(r)$
nnmean nearest neighbour mean index
nnvario nearest neighbour mark variance index
}
For marks of any type, there are the following:
Gmulti multitype nearest neighbour distribution
Kmulti multitype $K$-function
Jmulti multitype $J$-function
}
Alternatively use cut.ppp to convert a marked point pattern
to a multitype point pattern.
Programming tools:
applynbd apply function to every neighbourhood
in a point pattern
markstat apply function to the marks of neighbours
in a point pattern
marktable tabulate the marks of neighbours
in a point pattern
pppdist find the optimal match between two point
patterns
}
Summary statistics for a three-dimensional point pattern:
These are for 3-dimensional point pattern objects (class pp3).
F3est empty space function $F$
G3est nearest neighbour function $G$
K3est $K$-function
}
Related facilities:
envelope.pp3 simulation envelopes
pairdist.pp3 distances between all pairs of
points
crossdist.pp3 distances between points in
two patterns
nndist.pp3 nearest neighbour distances
nnwhich.pp3 find nearest neighbours
}
Computations for multi-dimensional point pattern:
These are for multi-dimensional space-time
point pattern objects (class ppx).
pairdist.ppx distances between all pairs of
points
crossdist.ppx distances between points in
two patterns
nndist.ppx nearest neighbour distances
nnwhich.ppx find nearest neighbours
}
Summary statistics for random sets:
These work for point patterns (class ppp),
line segment patterns (class psp)
or windows (class owin).
Hest spherical contact distribution $H$
Gfox Foxall $G$-function
Jfox Foxall $J$-function
}
kppm.
Its result is an object of class "kppm".
The fitted model can be printed, plotted, predicted, simulated
and updated. plot.kppm Plot the fitted model
predict.kppm Compute fitted intensity
update.kppm Update the model
simulate.kppm Generate simulated realisations
}
Lower-level fitting functions include:
thomas.estK fit the Thomas process model
thomas.estpcf fit the Thomas process model
matclust.estK fit the Matern Cluster process model
matclust.estpcf fit the Matern Cluster process model
lgcp.estK fit a log-Gaussian Cox process model
lgcp.estpcf fit a log-Gaussian Cox process model
mincontrast low-level algorithm for fitting models
by the method of minimum contrast
}
The Thomas and Matern models can also be simulated,
using rThomas and rMatClust respectively.
Model fitting in ppm. Its result is an object of class "ppm".
Manipulating the fitted model:
plot.ppm Plot the fitted model
predict.ppm
Compute the spatial trend and conditional intensity
of the fitted point process model
coef.ppm Extract the fitted model coefficients
formula.ppm Extract the trend formula
fitted.ppm Compute fitted conditional intensity at quadrature points
residuals.ppm Compute point process residuals at quadrature points
update.ppm Update the fit
vcov.ppm Variance-covariance matrix of estimates
rmh.ppm Simulate from fitted model
simulate.ppm Simulate from fitted model
print.ppm Print basic information about a fitted model
summary.ppm Summarise a fitted model
effectfun Compute the fitted effect of one covariate
logLik.ppm log-likelihood or log-pseudolikelihood
anova.ppm Analysis of deviance
}
For model selection, you can also use
the generic functions step, drop1
and AIC on fitted point process models.
See spatstat.options to control plotting of fitted model.
To specify a point process model:
The first order ``trend'' of the model is determined by an R
language formula. The formula specifies the form of the
logarithm of the trend.
~1 No trend (stationary)
~x Loglinear trend
$\lambda(x,y) = \exp(\alpha + \beta x)$
where $x,y$ are Cartesian coordinates
~polynom(x,y,3) Log-cubic polynomial trend
~harmonic(x,y,2) Log-harmonic polynomial trend
}
The higher order (``interaction'') components are described by
an object of class "interact". Such objects are created by:
Poisson() the Poisson point process
AreaInter() Area-interaction process
BadGey() multiscale Geyer process
DiggleGratton() Diggle-Gratton potential
DiggleGatesStibbard() Diggle-Gates-Stibbard potential
Fiksel() Fiksel pairwise interaction process
Geyer() Geyer's saturation process
Hardcore() Hard core process
LennardJones() Lennard-Jones potential
MultiStrauss() multitype Strauss process
MultiStraussHard() multitype Strauss/hard core process
OrdThresh() Ord process, threshold potential
Ord() Ord model, user-supplied potential
PairPiece() pairwise interaction, piecewise constant
Pairwise() pairwise interaction, user-supplied potential
SatPiece() Saturated pair model, piecewise constant potential
Saturated() Saturated pair model, user-supplied potential
Softcore() pairwise interaction, soft core potential
Strauss() Strauss process
StraussHard() Strauss/hard core point process
}
Finer control over model fitting:
A quadrature scheme is represented by an object of
class "quad". To create a quadrature scheme, typically
use quadscheme.
quadscheme default quadrature scheme
using rectangular cells or Dirichlet cells
pixelquad quadrature scheme based on image pixels
quad create an object of class "quad"
}
To inspect a quadrature scheme:
plot(Q) plot quadrature scheme Q
print(Q) print basic information about quadrature scheme Q
summary(Q) summary of quadrature scheme Q
}
A quadrature scheme consists of data points, dummy points, and
weights. To generate dummy points:
default.dummy default pattern of dummy points
gridcentres dummy points in a rectangular grid
rstrat stratified random dummy pattern
spokes radial pattern of dummy points
corners dummy points at corners of the window
}
To compute weights:
gridweights quadrature weights by the grid-counting rule
dirichlet.weights quadrature weights are
Dirichlet tile areas
}
Simulation and goodness-of-fit for fitted models:
rmh.ppm simulate realisations of a fitted model
simulate.ppm simulate realisations of a fitted model
envelope compute simulation envelopes for a
fitted model
}
Random point patterns:
runifpoint generate $n$ independent uniform random points
rpoint generate $n$ independent random points
rmpoint generate $n$ independent multitype random points
rpoispp simulate the (in)homogeneous Poisson point process
rmpoispp simulate the (in)homogeneous multitype Poisson point process
runifdisc generate $n$ independent uniform random points in disc
rstrat stratified random sample of points
rsyst systematic random sample (grid) of points
rMaternI simulate the Mat'ern Model I inhibition process
rMaternII simulate the Mat'ern Model II inhibition process
rSSI simulate Simple Sequential Inhibition process
rStrauss simulate Strauss process (perfect simulation)
rNeymanScott simulate a general Neyman-Scott process
rMatClust simulate the Mat'ern Cluster process
rThomas simulate the Thomas process
rGaussPoisson simulate the Gauss-Poisson cluster process
rcell simulate the Baddeley-Silverman cell process
runifpointOnLines generate $n$ random points along specified line segments
rpoisppOnLines generate Poisson random points along specified line segments
}
Resampling a point pattern:
quadratresample block resampling
rjitter apply random displacements to points in a pattern
rshift random shifting of (subsets of) points
rthin random thinning
}
Fitted point process models:
If you have fitted a point process model to a point pattern dataset, the fitted model can be simulated.
Cluster process models
are fitted by the function kppm yielding an
object of class "kppm". To generate one or more simulated
realisations of this fitted model, use
simulate.kppm.
Gibbs point process models
are fitted by the function ppm yielding an
object of class "ppm". To generate a simulated
realisation of this fitted model, use rmh.
To generate one or more simulated realisations of the fitted model,
use simulate.ppm.
Other random patterns:
rlinegrid generate a random array of parallel lines through a window
rpoisline simulate the Poisson line process within a window
rpoislinetess generate random tessellation using Poisson line process
rMosaicSet generate random set by selecting some tiles of a tessellation
rMosaicField generate random pixel image by assigning random values
in each tile of a tessellation
}
Simulation-based inference
envelope critical envelope for Monte Carlo
test of goodness-of-fit
qqplot.ppm diagnostic plot for interpoint
interaction
}
quadrat.test $\chi^2$ goodness-of-fit
test on quadrat counts
kstest Kolmogorov-Smirnov goodness-of-fit test
bermantest Berman's goodness-of-fit tests
envelope critical envelope for Monte Carlo
test of goodness-of-fit
anova.ppm Analysis of Deviance for
point process models
} Diagnostic plots:
Residuals for a fitted point process model, and diagnostic plots
based on the residuals, were introduced in Baddeley et al (2005).
Type demo(diagnose)
for a demonstration of the diagnostics features.
diagnose.ppm diagnostic plots for spatial trend
qqplot.ppm diagnostic plot for interpoint interaction
residualspaper examples from Baddeley et al (2005)
}
Resampling and randomisation procedures
You can build your own tests based on randomisation
and resampling using the following capabilities:
quadratresample block resampling
rjitter apply random displacements to points in a pattern
rshift random shifting of (subsets of) points
rthin random thinning
}
spatstat. Type citation("spatstat") to get these references.
The package supports
formula in the R language, and are fitted using
a single function ppm analogous to
lm and glm.
It is also possible to fit cluster process models by the method of
minimum contrast. Apart from two-dimensional point patterns and point processes,
www.csiro.au/resources/pf16h.html
Baddeley, A. and Turner, R. (2005a)
Spatstat: an R package for analyzing spatial point patterns.
Journal of Statistical Software 12:6, 1--42.
URL: www.jstatsoft.org, ISSN: 1548-7660.Baddeley, A. and Turner, R. (2005b) Modelling spatial point patterns in R. In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, editors, Case Studies in Spatial Point Pattern Modelling, Lecture Notes in Statistics number 185. Pages 23--74. Springer-Verlag, New York, 2006. ISBN: 0-387-28311-0.
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617--666.