spatstatpackage
The Spatstat Package
This is a summary of the features of spatstat, a package in R for the statistical analysis of spatial point patterns.
Details
spatstat is a package for the statistical analysis of spatial data. Its main focus is the analysis of spatial patterns of points in twodimensional space. The points may carry auxiliary data (`marks'), and the spatial region in which the points were recorded may have arbitrary shape.
The package is designed to support a complete statistical analysis of spatial data. It supports
creation, manipulation and plotting of point patterns;
exploratory data analysis;
spatial random sampling;
simulation of point process models;
parametric modelfitting;
nonparametric smoothing and regression;
formal inference (hypothesis tests, confidence intervals);
model diagnostics.
Apart from twodimensional point patterns and point processes, spatstat also supports point patterns in three dimensions, point patterns in multidimensional spacetime, point patterns on a linear network, patterns of line segments in two dimensions, and spatial tessellations and random sets in two dimensions.
The package can fit several types of point process models to a point pattern dataset:
Poisson point process models (by BermanTurner approximate maximum likelihood or by spatial logistic regression)
Gibbs/Markov point process models (by BaddeleyTurner approximate maximum pseudolikelihood, CoeurjollyRubak logistic likelihood, or HuangOgata approximate maximum likelihood)
Cox/cluster point process models (by Waagepetersen's twostep fitting procedure and minimum contrast, composite likelihood, or Palm likelihood)
determinantal point process models (by Waagepetersen's twostep fitting procedure and minimum contrast, composite likelihood, or Palm likelihood)
The models may include spatial trend,
dependence on covariates, and complicated interpoint interactions.
Models are specified by
a formula
in the R language, and are fitted using
a function analogous to lm
and glm
.
Fitted models can be printed, plotted, predicted, simulated and so on.
Getting Started
For a quick introduction to spatstat, read the package vignette Getting started with spatstat installed with spatstat. To read that document, you can either
visit cran.rproject.org/web/packages/spatstat and click on
Getting Started with Spatstat
start R, type
library(spatstat)
andvignette('getstart')
start R, type
help.start()
to open the help browser, and navigate toPackages > spatstat > Vignettes
.
Once you have installed spatstat, start R and type
library(spatstat)
. Then type beginner
for a beginner's introduction, or
demo(spatstat)
for a demonstration of the package's capabilities.
For a complete course on spatstat, and on statistical analysis of spatial point patterns, read the book by Baddeley, Rubak and Turner (2015). Other recommended books on spatial point process methods are Diggle (2014), Gelfand et al (2010) and Illian et al (2008).
The spatstat package includes over 50 datasets,
which can be useful when learning the package.
Type demo(data)
to see plots of all datasets
available in the package.
Type vignette('datasets')
for detailed background information
on these datasets, and plots of each dataset.
For information on converting your data into spatstat format, read Chapter 3 of Baddeley, Rubak and Turner (2015). This chapter is available free online, as one of the sample chapters at the book companion website, spatstat.github.io/book.
For information about handling data in shapefiles,
see Chapter 3, or the Vignette
Handling shapefiles in the spatstat package,
installed with spatstat, accessible as
vignette('shapefiles')
.
Updates
New versions of spatstat are released every 8 weeks. Users are advised to update their installation of spatstat regularly.
Type latest.news
to read the news documentation about
changes to the current installed version of spatstat.
See the Vignette Summary of recent updates,
installed with spatstat, which describes the main changes
to spatstat since the book (Baddeley, Rubak and Turner, 2015)
was published. It is accessible as vignette('updates')
.
Type news(package="spatstat")
to read news documentation about
all previous versions of the package.
FUNCTIONS AND DATASETS
Following is a summary of the main functions and datasets
in the spatstat package.
Alternatively an alphabetical list of all functions and
datasets is available by typing library(help=spatstat)
.
For further information on any of these,
type help(name)
or ?name
where name
is the name of the function
or dataset.
CONTENTS:
I.  Creating and manipulating data 
II.  Exploratory Data Analysis 
III.  Model fitting (Cox and cluster models) 
IV.  Model fitting (Poisson and Gibbs models) 
V.  Model fitting (determinantal point processes) 
VI.  Model fitting (spatial logistic regression) 
VII.  Simulation 
VIII.  Tests and diagnostics 
I. CREATING AND MANIPULATING DATA
Types of spatial data:
The main types of spatial data supported by spatstat are:
ppp 
point pattern 
owin 
window (spatial region) 
im 
pixel image 
psp 
line segment pattern 
tess 
tessellation 
pp3 
threedimensional 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 
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 Matern Model I inhibition process 
rMaternII 
simulate the Matern Model II inhibition process 
rSSI 
simulate Simple Sequential Inhibition process 
rStrauss 
simulate Strauss process (perfect simulation) 
rHardcore 
simulate Hard Core process (perfect simulation) 
rStraussHard 
simulate Strausshard core process (perfect simulation) 
rDiggleGratton 
simulate DiggleGratton process (perfect simulation) 
rDGS 
simulate DiggleGatesStibbard process (perfect simulation) 
rPenttinen 
simulate Penttinen process (perfect simulation) 
rNeymanScott 
simulate a general NeymanScott process 
rPoissonCluster 
simulate a general Poisson cluster process 
rMatClust 
simulate the Matern Cluster process 
rThomas 
simulate the Thomas process 
rGaussPoisson 
simulate the GaussPoisson cluster process 
rCauchy 
simulate NeymanScott Cauchy cluster process 
rVarGamma 
simulate NeymanScott Variance Gamma cluster process 
rthin 
random thinning 
rcell 
simulate the BaddeleySilverman cell process 
rmh 
simulate Gibbs point process using MetropolisHastings 
simulate.ppm 
simulate Gibbs point process using MetropolisHastings 
runifpointOnLines 
generate \(n\) random points along specified line segments 
To randomly change an existing point pattern:
rshift 
random shifting of points 
rjitter 
apply random displacements to points in a pattern 
rthin 
random thinning 
rlabel 
random (re)labelling of a multitype point pattern 
Standard point pattern datasets:
Datasets in spatstat are lazyloaded, so you can simply
type the name of the dataset to use it; there is no need
to type data(amacrine)
etc.
Type demo(data)
to see a display of all the datasets
installed with the package.
Type vignette('datasets')
for a document giving an overview
of all datasets, including background information, and plots.
amacrine 
Austin Hughes' rabbit amacrine cells 
anemones 
UptonFingleton sea anemones data 
ants 
HarknessIsham ant nests data 
bdspots 
Breakdown spots in microelectrodes 
bei 
Tropical rainforest trees 
betacells 
Waessle et al. cat retinal ganglia data 
bramblecanes 
Bramble Canes data 
bronzefilter 
Bronze Filter Section data 
cells 
CrickRipley biological cells data 
chicago 
Chicago street crimes 
chorley 
ChorleyRibble cancer data 
clmfires 
CastillaLa Mancha forest fires 
copper 
BermanHuntington copper deposits data 
dendrite 
Dendritic spines 
demohyper 
Synthetic point patterns 
demopat 
Synthetic point pattern 
finpines 
Finnish Pines data 
flu 
Influenza virus proteins 
gordon 
People in Gordon Square, London 
gorillas 
Gorilla nest sites 
hamster 
Aherne's hamster tumour data 
humberside 
North Humberside childhood leukaemia data 
hyytiala 
Mixed forest in Hyytiala, Finland 
japanesepines 
Japanese Pines data 
lansing 
Lansing Woods data 
longleaf 
Longleaf Pines data 
mucosa 
Cells in gastric mucosa 
murchison 
Murchison gold deposits 
nbfires 
New Brunswick fires data 
nztrees 
MarkEslerRipley trees data 
osteo 
Osteocyte lacunae (3D, replicated) 
paracou 
Kimboto trees in Paracou, French Guiana 
ponderosa 
GetisFranklin ponderosa pine trees data 
pyramidal 
Pyramidal neurons from 31 brains 
redwood 
StraussRipley 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) 
spiders 
Spider webs on mortar lines of brick wall 
sporophores 
Mycorrhizal fungi around a tree 
spruces 
Spruce trees in Saxonia 
swedishpines 
StrandRipley Swedish pines data 
urkiola 
Urkiola Woods data 
waka 
Trees in Waka national park 
To manipulate a point pattern:
plot.ppp 
plot a point pattern (e.g. plot(X) ) 
iplot 
plot a point pattern interactively 
edit.ppp 
interactive text editor 
[.ppp 
extract or replace a subset of a point pattern 
pp[subset] or pp[subwindow] 

subset.ppp 
extract subset of point pattern satisfying a condition 
superimpose 
combine several point patterns 
by.ppp 
apply a function to subpatterns of a point pattern 
cut.ppp 
classify the points in a point pattern 
split.ppp 
divide pattern into subpatterns 
unmark 
remove marks 
npoints 
count the number of points 
coords 
extract coordinates, change coordinates 
marks 
extract marks, change marks or attach marks 
rotate 
rotate pattern 
shift 
translate pattern 
flipxy 
swap \(x\) and \(y\) coordinates 
reflect 
reflect in the origin 
periodify 
make several translated copies 
affine 
apply affine transformation 
scalardilate 
apply scalar dilation 
density.ppp 
kernel estimation of point pattern intensity 
Smooth.ppp 
kernel smoothing of marks of point pattern 
nnmark 
mark value of nearest data point 
sharpen.ppp 
data sharpening 
identify.ppp 
interactively identify points 
unique.ppp 
remove duplicate points 
duplicated.ppp 
determine which points are duplicates 
connected.ppp 
find clumps of points 
dirichlet 
compute DirichletVoronoi tessellation 
delaunay 
compute Delaunay triangulation 
delaunayDistance 
graph distance in Delaunay triangulation 
convexhull 
compute convex hull 
discretise 
discretise coordinates 
pixellate.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 

Window 
Extract window of another object 
Frame 
Extract the containing rectangle ('frame') of another object 
as.owin 
Convert other data to a window object 
square 
make a square window 
disc 
make a circular window 
ellipse 
make an elliptical window 
ripras 
RipleyRasson estimator of window, given only the points 
convexhull 
compute convex hull of something 
letterR 
polygonal window in the shape of the R logo 
clickpoly 
interactively draw a polygonal window 
To manipulate a window:
plot.owin 
plot a window. 
plot(W) 

boundingbox 
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 
flipxy 
swap \(x\) and \(y\) coordinates 
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 
commonGrid 
find common pixel grid for windows 
nearest.raster.point 
map continuous coordinates to raster locations 
raster.x 
raster x coordinates 
raster.y 
raster y coordinates 
raster.xy 
raster x and y coordinates 
See spatstat.options
to control the approximation
Geometrical computations with windows:
edges 
extract boundary edges 
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 
inradius 
radius of incircle 
connected.owin 
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 
triangulate.owin 
decompose into triangles 
as.mask 
pixel approximation of window 
as.polygonal 
polygonal approximation of window 
is.rectangle 
test whether window is a rectangle 
is.polygonal 
test whether window is polygonal 
is.mask 
test whether window is a mask 
setcov 
spatial covariance function of window 
pixelcentres 
extract centres of pixels in mask 
Pixel images:
An object of class "im"
represents a pixel image.
Such objects are returned by some of the functions in
spatstat including 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 
as.function.im 
convert pixel image to function 
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 
rgbim 
create colourvalued pixel image 
hsvim 
create colourvalued pixel image 
[.im 
extract a subset of a pixel image 
[<.im 
replace a subset of a pixel image 
rotate.im 
rotate pixel image 
shift.im 
apply vector shift to pixel image 
affine.im 
apply affine transformation to 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 
Smooth.im 
apply Gaussian blur to image 
connected.im 
find connected components 
compatible.im 
test whether two images have compatible dimensions 
harmonise.im 
make images compatible 
commonGrid 
find a common pixel grid for images 
eval.im 
evaluate any expression involving images 
scaletointerval 
rescale pixel values 
zapsmall.im 
set very small pixel values to zero 
levelset 
level set of an image 
solutionset 
region where an expression is true 
imcov 
spatial covariance function of image 
convolve.im 
spatial convolution of images 
transect.im 
line transect of image 
pixelcentres 
extract centres of pixels 
transmat 
convert matrix of pixel values 
to a different indexing convention 
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 
edges 
extract edges of a window 
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 
superimpose 
combine several line segment patterns 
flipxy 
swap \(x\) and \(y\) coordinates 
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 
selfcut.psp 
cut segments where they cross 
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 
Tessellations
An object of class "tess"
represents a tessellation.
tess 
create a tessellation 
quadrats 
create a tessellation of rectangles 
hextess 
create a tessellation of hexagons 
quantess 
quantile tessellation 
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 DirichletVoronoi tessellation of points 
delaunay 
compute Delaunay triangulation of points 
rpoislinetess 
generate tessellation using Poisson line process 
tile.areas 
area of each tile in tessellation 
Threedimensional point patterns
An object of class "pp3"
represents a threedimensional
point pattern in a rectangular box. The box is represented by
an object of class "box3"
.
pp3 
create a 3D point pattern 
plot.pp3 
plot a 3D point pattern 
coords 
extract coordinates 
as.hyperframe 
extract coordinates 
subset.pp3 
extract subset of 3D point pattern 
unitname.pp3 
name of unit of length 
npoints 
count the number of points 
runifpoint3 
generate uniform random points in 3D 
rpoispp3 
generate Poisson random points in 3D 
envelope.pp3 
generate simulation envelopes for 3D pattern 
box3 
create a 3D rectangular box 
as.box3 
convert data to 3D 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 
Multidimensional spacetime point patterns
An object of class "ppx"
represents a
point pattern in multidimensional space and/or time.
ppx 
create a multidimensional spacetime point pattern 
coords 
extract coordinates 
as.hyperframe 
extract coordinates 
subset.ppx 
extract subset 
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 
Point patterns on a linear network
An object of class "linnet"
represents a linear network
(for example, a road network).
linnet 
create a linear network 
clickjoin 
interactively join vertices in network 
iplot.linnet 
interactively plot network 
simplenet 
simple example of network 
lineardisc 
disc in a linear network 
delaunayNetwork 
network of Delaunay triangulation 
dirichletNetwork 
network of Dirichlet edges 
methods.linnet 
methods for linnet objects 
vertices.linnet 
nodes of network 
An object of class "lpp"
represents a
point pattern on a linear network (for example,
road accidents on a road network).
lpp 
create a point pattern on a linear network 
methods.lpp 
methods for lpp objects 
subset.lpp 
method for subset 
rpoislpp 
simulate Poisson points on linear network 
runiflpp 
simulate random points on a linear network 
chicago 
Chicago street crime data 
dendrite 
Dendritic spines data 
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 
subset.hyperframe 
method for subset 
head.hyperframe 
first few rows of hyperframe 
Layered objects
A layered object represents data that should be plotted in successive layers, for example, a background and a foreground.
layered 
create layered object 
plot.layered 
plot layered object 
Colour maps
A colour map is a mechanism for associating colours with data.
It can be regarded as a function, mapping data to colours.
Using a colourmap
object in a plot command
ensures that the mapping from numbers to colours is
the same in different plots.
colourmap 
create a colour map 
plot.colourmap 
plot the colour map only 
tweak.colourmap 
alter individual colour values 
interp.colourmap 
make a smooth transition between colours 
II. EXPLORATORY DATA ANALYSIS
Inspection of data:
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 
Smoothing:
density.ppp 
kernel smoothed density/intensity 
relrisk 
kernel estimate of relative risk 
Smooth.ppp 
spatial interpolation of marks 
bw.diggle 
crossvalidated bandwidth selection
for density.ppp 
bw.ppl 
likelihood crossvalidated bandwidth selection
for density.ppp 
bw.scott 
Scott's rule of thumb for density estimation 
bw.relrisk 
crossvalidated bandwidth selection
for relrisk 
bw.smoothppp 
crossvalidated bandwidth selection
for Smooth.ppp 
bw.frac 
bandwidth selection using window geometry 
Modern exploratory tools:
clusterset 
AllardFraley feature detection 
nnclean 
ByersRaftery feature detection 
sharpen.ppp 
ChoiHall data sharpening 
rhohat 
Kernel estimate of covariate effect 
rho2hat 
Kernel estimate of effect of two covariates 
spatialcdf 
Spatial cumulative distribution function 
Summary statistics for a point pattern:
Type demo(sumfun)
for a demonstration of many
of the summary statistics.
intensity 
Mean intensity 
quadratcount 
Quadrat counts 
intensity.quadratcount 
Mean intensity in quadrats 
Fest 
empty space function \(F\) 
Gest 
nearest neighbour distribution function \(G\) 
Jest 
\(J\)function \(J = (1G)/(1F)\) 
Kest 
Ripley's \(K\)function 
Lest 
Besag \(L\)function 
Tstat 
Third order \(T\)function 
allstats 
all four functions \(F\), \(G\), \(J\), \(K\) 
pcf 
pair correlation function 
Kinhom 
\(K\) for inhomogeneous point patterns 
Linhom 
\(L\) for inhomogeneous point patterns 
pcfinhom 
pair correlation for inhomogeneous patterns 
Finhom 
\(F\) for inhomogeneous point patterns 
Ginhom 
\(G\) for inhomogeneous point patterns 
Jinhom 
\(J\) for inhomogeneous point patterns 
localL 
GetisFranklin neighbourhood density function 
localK 
neighbourhood Kfunction 
localpcf 
local pair correlation function 
localKinhom 
local \(K\) for inhomogeneous point patterns 
localLinhom 
local \(L\) for inhomogeneous point patterns 
localpcfinhom 
local pair correlation for inhomogeneous patterns 
Ksector 
Directional \(K\)function 
Kscaled 
locally scaled \(K\)function 
Kest.fft 
fast \(K\)function using FFT for large datasets 
Kmeasure 
reduced second moment measure 
envelope 
simulation envelopes for a summary function 
varblock 
variances and confidence intervals 
for a summary function 
Related facilities:
plot.fv 
plot a summary function 
eval.fv 
evaluate any expression involving summary functions 
harmonise.fv 
make functions compatible 
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 
deriv.fv 
calculate derivative of a summary function 
pool.fv 
pool several estimates of 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 
nnmap 
nearest point image 
nnfun 
nearest point function 
density.ppp 
kernel smoothed density 
Smooth.ppp 
spatial interpolation of marks 
relrisk 
kernel estimate of relative risk 
sharpen.ppp 
data sharpening 
Summary statistics for a multitype point pattern:
A multitype point pattern is represented by an object X
of class "ppp"
such that marks(X)
is a factor.
relrisk 
kernel estimation of relative risk 
scan.test 
spatial scan test of elevated risk 
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 \(g_{i\bullet}\) 
pcfmulti 
general pair correlation function 
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 
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 
markcrosscorr 
mark crosscorrelation function 
markvario 
mark variogram 
Kmark 
markweighted \(K\) function 
Emark 
mark independence diagnostic \(E(r)\) 
Vmark 
mark independence diagnostic \(V(r)\) 
nnmean 
nearest neighbour mean index 
For marks of any type, there are the following:
Gmulti 
multitype nearest neighbour distribution 
Kmulti 
multitype \(K\)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 
Summary statistics for a point pattern on a linear network:
These are for point patterns on a linear network (class lpp
).
For unmarked patterns:
linearK 
\(K\) function on linear network 
linearKinhom 
inhomogeneous \(K\) function on linear network 
linearpcf 
pair correlation function on linear network 
For multitype patterns:
linearKcross 
\(K\) function between two types of points 
linearKdot 
\(K\) function from one type to any type 
linearKcross.inhom 
Inhomogeneous version of linearKcross 
linearKdot.inhom 
Inhomogeneous version of linearKdot 
linearmarkconnect 
Mark connection function on linear network 
linearmarkequal 
Mark equality function on linear network 
linearpcfcross 
Pair correlation between two types of points 
linearpcfdot 
Pair correlation from one type to any type 
linearpcfcross.inhom 
Inhomogeneous version of linearpcfcross 
Related facilities:
pairdist.lpp 
distances between pairs 
crossdist.lpp 
distances between pairs 
nndist.lpp 
nearest neighbour distances 
nncross.lpp 
nearest neighbour distances 
nnwhich.lpp 
find nearest neighbours 
nnfun.lpp 
find nearest data point 
density.lpp 
kernel smoothing estimator of intensity 
distfun.lpp 
distance transform 
envelope.lpp 
simulation envelopes 
rpoislpp 
simulate Poisson points on linear network 
It is also possible to fit point process models to lpp
objects.
See Section IV.
Summary statistics for a threedimensional point pattern:
These are for 3dimensional 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 multidimensional point pattern:
These are for multidimensional spacetime
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 
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 
III. MODEL FITTING (COX AND CLUSTER MODELS)
Cluster process models (with homogeneous or inhomogeneous intensity)
and Cox processes can be fitted by the function kppm
.
Its result is an object of class "kppm"
.
The fitted model can be printed, plotted, predicted, simulated
and updated.
kppm 
Fit model 
plot.kppm 
Plot the fitted model 
summary.kppm 
Summarise the fitted model 
fitted.kppm 
Compute fitted intensity 
predict.kppm 
Compute fitted intensity 
update.kppm 
Update the model 
improve.kppm 
Refine the estimate of trend 
simulate.kppm 
Generate simulated realisations 
vcov.kppm 
Variancecovariance matrix of coefficients 
coef.kppm

Extract trend coefficients 
formula.kppm

Extract trend formula 
parameters 
Extract all model parameters 
clusterfield 
Compute offspring density 
clusterradius 
Radius of support of offspring density 
Kmodel.kppm 
\(K\) function of fitted model 
For model selection, you can also use
the generic functions step
, drop1
and AIC
on fitted point process models.
The theoretical models can also be simulated,
for any choice of parameter values,
using rThomas
, rMatClust
,
rCauchy
, rVarGamma
,
and rLGCP
.
Lowerlevel fitting functions include:
lgcp.estK 
fit a logGaussian Cox process model 
lgcp.estpcf 
fit a logGaussian Cox process model 
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 
cauchy.estK 
fit a NeymanScott Cauchy cluster process 
cauchy.estpcf 
fit a NeymanScott Cauchy cluster process 
vargamma.estK 
fit a NeymanScott Variance Gamma process 
vargamma.estpcf 
fit a NeymanScott Variance Gamma process 
mincontrast 
lowlevel algorithm for fitting models 
IV. MODEL FITTING (POISSON AND GIBBS MODELS)
Types of models
Poisson point processes are the simplest models for point patterns. A Poisson model assumes that the points are stochastically independent. It may allow the points to have a nonuniform spatial density. The special case of a Poisson process with a uniform spatial density is often called Complete Spatial Randomness.
Poisson point processes are included in the more general class of Gibbs point process models. In a Gibbs model, there is interaction or dependence between points. Many different types of interaction can be specified.
For a detailed explanation of how to fit Poisson or Gibbs point process models to point pattern data using spatstat, see Baddeley and Turner (2005b) or Baddeley (2008).
To fit a Poisson or Gibbs point process model:
Model fitting in spatstat is performed mainly by the function
ppm
. Its result is an object of class "ppm"
.
Here are some examples, where X
is a point pattern (class
"ppp"
):
command  model 
ppm(X) 
Complete Spatial Randomness 
ppm(X ~ 1) 
Complete Spatial Randomness 
ppm(X ~ x) 
Poisson process with 
intensity loglinear in \(x\) coordinate  
ppm(X ~ 1, Strauss(0.1)) 
Stationary Strauss process 
ppm(X ~ x, Strauss(0.1)) 
Strauss process with 
It is also possible to fit models that depend on other covariates.
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 
parameters 
Extract all model parameters 
formula.ppm 
Extract the trend formula 
intensity.ppm 
Compute fitted intensity 
Kmodel.ppm 
\(K\) function of fitted model 
pcfmodel.ppm 
pair correlation of fitted model 
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 
Variancecovariance 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 
loglikelihood or logpseudolikelihood 
anova.ppm 
Analysis of deviance 
model.frame.ppm 
Extract data frame used to fit model 
model.images 
Extract spatial data used to fit model 
model.depends 
Identify variables in the model 
as.interact 
Interpoint interaction component of model 
fitin 
Extract fitted interpoint interaction 
is.hybrid 
Determine whether the model is a hybrid 
valid.ppm 
Check the model is a valid point process 
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.
X ~ 1 
No trend (stationary) 
X ~ x 
Loglinear trend \(\lambda(x,y) = \exp(\alpha + \beta x)\) 
where \(x,y\) are Cartesian coordinates  
X ~ polynom(x,y,3) 
Logcubic polynomial trend 
X ~ harmonic(x,y,2) 
Logharmonic polynomial trend 
X ~ Z 
Loglinear function of covariate Z 
The higher order (``interaction'') components are described by
an object of class "interact"
. Such objects are created by:
Poisson() 
the Poisson point process 
AreaInter() 
Areainteraction process 
BadGey() 
multiscale Geyer process 
Concom() 
connected component interaction 
DiggleGratton() 
DiggleGratton potential 
DiggleGatesStibbard() 
DiggleGatesStibbard potential 
Fiksel() 
Fiksel pairwise interaction process 
Geyer() 
Geyer's saturation process 
Hardcore() 
Hard core process 
HierHard() 
Hierarchical multiype hard core process 
HierStrauss() 
Hierarchical multiype Strauss process 
HierStraussHard() 
Hierarchical multiype Strausshard core process 
Hybrid() 
Hybrid of several interactions 
LennardJones() 
LennardJones potential 
MultiHard() 
multitype hard core process 
MultiStrauss() 
multitype Strauss process 
MultiStraussHard() 
multitype Strauss/hard core process 
OrdThresh() 
Ord process, threshold potential 
Ord() 
Ord model, usersupplied potential 
PairPiece() 
pairwise interaction, piecewise constant 
Pairwise() 
pairwise interaction, usersupplied potential 
Penttinen() 
Penttinen pairwise interaction 
SatPiece() 
Saturated pair model, piecewise constant potential 
Saturated() 
Saturated pair model, usersupplied potential 
Softcore() 
pairwise interaction, soft core potential 
Strauss() 
Strauss process 
StraussHard() 
Strauss/hard core point process 
Note that it is also possible to combine several such interactions
using Hybrid
.
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 
To inspect a quadrature scheme:
plot(Q) 
plot quadrature scheme Q 
print(Q) 
print basic information about 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 
To compute weights:
gridweights 
quadrature weights by the gridcounting rule 
Simulation and goodnessoffit for fitted models:
rmh.ppm 
simulate realisations of a fitted model 
simulate.ppm 
simulate realisations of a fitted model 
Point process models on a linear network:
An object of class "lpp"
represents a pattern of points on
a linear network. Point process models can also be fitted to these
objects. Currently only Poisson models can be fitted.
lppm 
point process model on linear network 
anova.lppm 
analysis of deviance for 
point process model on linear network  
envelope.lppm 
simulation envelopes for 
point process model on linear network  
fitted.lppm 
fitted intensity values 
predict.lppm 
model prediction on linear network 
linim 
pixel image on linear network 
plot.linim 
plot a pixel image on linear network 
eval.linim 
evaluate expression involving images 
linfun 
function defined on linear network 
V. MODEL FITTING (DETERMINANTAL POINT PROCESS MODELS)
Code for fitting determinantal point process models has recently been added to spatstat.
For information, see the help file for dppm
.
VI. MODEL FITTING (SPATIAL LOGISTIC REGRESSION)
Logistic regression
Pixelbased spatial logistic regression is an alternative technique for analysing spatial point patterns that is widely used in Geographical Information Systems. It is approximately equivalent to fitting a Poisson point process model.
In pixelbased logistic regression, the spatial domain is divided into small pixels, the presence or absence of a data point in each pixel is recorded, and logistic regression is used to model the presence/absence indicators as a function of any covariates.
Facilities for performing spatial logistic regression are provided in spatstat for comparison purposes.
Fitting a spatial logistic regression
Spatial logistic regression is performed by the function
slrm
. Its result is an object of class "slrm"
.
There are many methods for this class, including methods for
print
, fitted
, predict
, simulate
,
anova
, coef
, logLik
, terms
,
update
, formula
and vcov
.
For example, if X
is a point pattern (class
"ppp"
):
command  model 
slrm(X ~ 1) 
Complete Spatial Randomness 
slrm(X ~ x) 
Poisson process with 
intensity loglinear in \(x\) coordinate  
slrm(X ~ Z) 
Poisson process with 
Manipulating a fitted spatial logistic regression
anova.slrm 
Analysis of deviance 
coef.slrm 
Extract fitted coefficients 
vcov.slrm 
Variancecovariance matrix of fitted coefficients 
fitted.slrm 
Compute fitted probabilities or intensity 
logLik.slrm 
Evaluate loglikelihood of fitted model 
plot.slrm 
Plot fitted probabilities or intensity 
predict.slrm 
Compute predicted probabilities or intensity with new data 
There are many other undocumented methods for this class,
including methods for print
, update
, formula
and terms
. Stepwise model selection is
possible using step
or stepAIC
.
VII. SIMULATION
There are many ways to generate a random point pattern, line segment pattern, pixel image or tessellation in spatstat.
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 Matern Model I inhibition process 
rMaternII 
simulate the Matern Model II inhibition process 
rSSI 
simulate Simple Sequential Inhibition process 
rHardcore 
simulate hard core process (perfect simulation) 
rStrauss 
simulate Strauss process (perfect simulation) 
rStraussHard 
simulate Strausshard core process (perfect simulation) 
rDiggleGratton 
simulate DiggleGratton process (perfect simulation) 
rDGS 
simulate DiggleGatesStibbard process (perfect simulation) 
rPenttinen 
simulate Penttinen process (perfect simulation) 
rNeymanScott 
simulate a general NeymanScott process 
rMatClust 
simulate the Matern Cluster process 
rThomas 
simulate the Thomas process 
rLGCP 
simulate the logGaussian Cox process 
rGaussPoisson 
simulate the GaussPoisson cluster process 
rCauchy 
simulate NeymanScott process with Cauchy clusters 
rVarGamma 
simulate NeymanScott process with Variance Gamma clusters 
rcell 
simulate the BaddeleySilverman cell process 
runifpointOnLines 
generate \(n\) 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 
See also varblock
for estimating the variance
of a summary statistic by block resampling, and
lohboot
for another bootstrap technique.
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 
Simulationbased inference
envelope 
critical envelope for Monte Carlo test of goodnessoffit 
qqplot.ppm 
diagnostic plot for interpoint interaction 
scan.test 
spatial scan statistic/test 
studpermu.test 
studentised permutation test 
VIII. TESTS AND DIAGNOSTICS
Classical hypothesis tests:
quadrat.test 
\(\chi^2\) goodnessoffit test on quadrat counts 
clarkevans.test 
Clark and Evans test 
cdf.test 
Spatial distribution goodnessoffit test 
berman.test 
Berman's goodnessoffit tests 
envelope 
critical envelope for Monte Carlo test of goodnessoffit 
scan.test 
spatial scan statistic/test 
dclf.test 
DiggleCressieLoosmoreFord test 
mad.test 
Mean Absolute Deviation test 
dclf.progress 
Progress plot for DCLF test 
mad.progress 
Progress plot for MAD test 
Sensitivity diagnostics:
Classical measures of model sensitivity such as leverage and influence have been adapted to point process models.
leverage.ppm 
Leverage for point process model 
influence.ppm 
Influence for point process model 
dfbetas.ppm 
Parameter influence 
Diagnostics for covariate effect:
Classical diagnostics for covariate effects have been adapted to point process models.
parres 
Partial residual plot 
addvar 
Added variable plot 
rhohat 
Kernel estimate of covariate effect 
Residual diagnostics:
Residuals for a fitted point process model, and diagnostic plots based on the residuals, were introduced in Baddeley et al (2005) and Baddeley, Rubak and Moller (2011).
Type demo(diagnose)
for a demonstration of the diagnostics features.
diagnose.ppm 
diagnostic plots for spatial trend 
qqplot.ppm 
diagnostic QQ plot for interpoint interaction 
residualspaper 
examples from Baddeley et al (2005) 
Kcom 
model compensator of \(K\) function 
Gcom 
model compensator of \(G\) function 
Kres 
score residual of \(K\) function 
Gres 
score residual of \(G\) function 
psst 
pseudoscore residual of summary function 
psstA 
pseudoscore residual of empty space function 
psstG 
pseudoscore residual of \(G\) function 
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 
IX. DOCUMENTATION
The online manual entries are quite detailed and should be consulted first for information about a particular function.
The book Baddeley, Rubak and Turner (2015) is a complete course on analysing spatial point patterns, with full details about spatstat.
Older material (which is now outofdate but is freely available) includes Baddeley and Turner (2005a), a brief overview of the package in its early development; Baddeley and Turner (2005b), a more detailed explanation of how to fit point process models to data; and Baddeley (2010), a complete set of notes from a 2day workshop on the use of spatstat.
Type citation("spatstat")
to get a list of these references.
Licence
This library and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package.
Acknowledgements
Kasper Klitgaard Berthelsen, Ottmar Cronie, Yongtao Guan, Ute Hahn, Abdollah Jalilian, MarieColette van Lieshout, Tuomas Rajala, Suman Rakshit, Dominic Schuhmacher, Rasmus Waagepetersen and Hangsheng Wang made substantial contributions of code.
Additional contributions and suggestions from Monsuru Adepeju, Corey Anderson, Ang Qi Wei, Marcel Austenfeld, Sandro Azaele, Malissa Baddeley, Guy Bayegnak, Colin Beale, Melanie Bell, Thomas Bendtsen, Ricardo Bernhardt, Andrew Bevan, Brad Biggerstaff, Anders Bilgrau, Leanne Bischof, Christophe Biscio, Roger Bivand, Jose M. Blanco Moreno, Florent Bonneu, Julian Burgos, Simon Byers, YaMei Chang, Jianbao Chen, Igor Chernayavsky, Y.C. Chin, Bjarke Christensen, JeanFrancois Coeurjolly, Kim Colyvas, Robin Corria Ainslie, Marcelino de la Cruz, Peter Dalgaard, Mario D'Antuono, Sourav Das, Tilman Davies, Peter Diggle, Patrick Donnelly, Ian Dryden, Stephen Eglen, Ahmed ElGabbas, Belarmain Fandohan, Olivier Flores, David Ford, Peter Forbes, Shane Frank, Janet Franklin, FunwiGabga Neba, Oscar Garcia, Agnes Gault, Jonas Geldmann, Marc Genton, Shaaban Ghalandarayeshi, Julian Gilbey, Jason Goldstick, Pavel Grabarnik, C. Graf, Ute Hahn, Andrew Hardegen, Martin Bogsted Hansen, Martin Hazelton, Juha Heikkinen, Mandy Hering, Markus Herrmann, Paul Hewson, Kassel Hingee, Kurt Hornik, Philipp Hunziker, Jack Hywood, Ross Ihaka, Cenk Icos, Aruna Jammalamadaka, Robert JohnChandran, Devin Johnson, Mahdieh Khanmohammadi, Bob Klaver, Peter Kovesi, Mike Kuhn, Jeff Laake, Frederic Lavancier, Tom Lawrence, Robert Lamb, Jonathan Lee, George Leser, Li Haitao, George Limitsios, Andrew Lister, Ben Madin, Martin Maechler, Kiran Marchikanti, Jeff Marcus, Robert Mark, Peter McCullagh, Monia Mahling, Jorge Mateu Mahiques, Ulf Mehlig, Frederico Mestre, Sebastian Wastl Meyer, Mi Xiangcheng, Lore De Middeleer, Robin Milne, Enrique Miranda, Jesper Moller, Mehdi Moradi, Virginia Morera Pujol, Erika Mudrak, Gopalan Nair, Nader Najari, Nicoletta Nava, Linda Stougaard Nielsen, Felipe Nunes, Jens Randel Nyengaard, Jens Oehlschlaegel, Thierry Onkelinx, Sean O'Riordan, Evgeni Parilov, Jeff Picka, Nicolas Picard, Mike Porter, Sergiy Protsiv, Adrian Raftery, Suman Rakshit, Ben Ramage, Pablo Ramon, Xavier Raynaud, Matt Reiter, Ian Renner, Tom Richardson, Brian Ripley, Ted Rosenbaum, Barry Rowlingson, Jason Rudokas, John Rudge, Christopher Ryan, Farzaneh Safavimanesh, Aila Sarkka, Cody Schank, Katja Schladitz, Sebastian Schutte, Bryan Scott, Olivia Semboli, Francois Semecurbe, Vadim Shcherbakov, Shen Guochun, Shi Peijian, HaroldJeffrey Ship, IdaMaria Sintorn, Yong Song, Malte Spiess, Mark Stevenson, Kaspar Stucki, Michael Sumner, P. Surovy, Ben Taylor, Thordis Linda Thorarinsdottir, Berwin Turlach, Torben Tvedebrink, Kevin Ummer, Medha Uppala, Andrew van Burgel, Tobias Verbeke, Mikko Vihtakari, Alexendre Villers, Fabrice Vinatier, Sasha Voss, Sven Wagner, Hao Wang, H. Wendrock, Jan Wild, Carl G. Witthoft, Selene Wong, Maxime Woringer, Mike Zamboni and Achim Zeileis.
References
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