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.
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
}
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
setmarks, %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
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(bramblecanes) 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
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
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))
subset.ppp
[.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
setmarks attach marks or reset marks
split.ppp divide pattern into sub-patterns
rotate rotate pattern
shift translate pattern
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
}
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
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
erode.owin erode window by a distance r
dilate.owin dilate window by a distance r
closing.owin close window by a distance r
opening.owin open window by a distance r
complement.owin invert (swap inside and outside)
rotate rotate window
shift translate window
affine apply affine transformation
}
Digital approximations:
as.mask Make a discrete pixel approximation of a given window
nearest.raster.point map continuous coordinates to raster locations
raster.x raster x coordinates
raster.y raster y coordinates
}
See spatstat.options to control the approximation
Geometrical computations with windows:
intersect.owin intersection of two windows
union.owin union of two windows
inside.owin determine whether a point is inside a window
area.owin compute window's area
diameter compute window frame's diameter
incircle find largest circle inside a window
connected find connected components of window
eroded.areas compute areas of eroded windows
bdist.points compute distances from data points to window boundary
bdist.pixels compute distances from all pixels to window boundary
distmap.owin distance transform image
centroid.owin compute centroid (centre of mass) of window
is.subset.owin determine whether one window contains another
}
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
as.matrix.im convert pixel image to matrix
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
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
subset.psp
[.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
affine.psp apply an affine transformation
distmap.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
}
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
} Classical exploratory tools:
clarkevans Clark and Evans aggregation index
fryplot Fry plot
miplot Morishita Index plot
}
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 $J = (1-G)/(1-F)$
localL Getis-Franklin neighbourhood density function
localK neighbourhood K-function
pcf pair correlation function
Kinhom $K$ for inhomogeneous point patterns
Kest.fft fast $K$-function using FFT for large datasets
Kmeasure reduced second moment measure
allstats all four functions $F$, $G$, $J$, $K$
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
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
density.ppp kernel smoothed density
smooth.ppp spatial interpolation of marks
}
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
$K_{ij}, K_{i\bullet}$
Jcross,Jdot,Jmulti multitype $J$-functions
$J_{ij}, J_{i\bullet}$
pcfcross multitype pair correlation function $g_{ij}$
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
}
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
}
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
matclust.estK fit the Matern Cluster process model
lgcp.estK 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
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
Strauss() the Strauss process
StraussHard() the Strauss/hard core point process
Softcore() pairwise interaction, soft core potential
PairPiece() pairwise interaction, piecewise constant
DiggleGratton() Diggle-Gratton potential
LennardJones() Lennard-Jones potential
Pairwise() pairwise interaction, user-supplied potential
AreaInter() Area-interaction process
Geyer() Geyer's saturation process
BadGey() multiscale Geyer process
SatPiece() Saturated pair model, piecewise constant potential
Saturated() Saturated pair model, user-supplied potential
OrdThresh() Ord process, threshold potential
Ord() Ord model, user-supplied potential
MultiStrauss() multitype Strauss process
MultiStraussHard() multitype Strauss/hard core 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
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 one or more simulated
realisations of this fitted model, use rmh.
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
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.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.